The transport closure framework of the NUVO program, having established
the single-particle Hilbert space in QM1--QM6, the multi-particle tensor
product in QM7, the spin degree of freedom in QM8, and the entanglement
structure in QM9, is now applied to the problem of two transport closure
configurations interacting through a potential.
The present paper, QM10, develops the complete theory of non-relativistic
quantum scattering within this framework.
The two-body problem is first reduced to the one-body problem in the
relative coordinate
r=x^1−x^2
with reduced mass
μ=m1+m2m1m2,
using the center-of-mass separation of QM7.
The scattering states
∣k+⟩
and
∣k−⟩
are defined as non-normalizable continuum eigenstates of the full
Hamiltonian
H^=H^0+V^
at positive energy
E>0,
requiring the rigged Hilbert space extension of QM1 to accommodate
generalized eigenstates.
The Lippmann-Schwinger equation
∣k+⟩=∣k⟩+G^0+V^∣k+⟩,G^0+=ϵ→0+lim(E−H^0+iϵ)−1,
defines the outgoing scattering state as the free momentum eigenstate
∣k⟩
corrected by the interaction
V^
through the free resolvent
G^0+.
The scattering state
∣k+⟩
has the asymptotic form
eik⋅r+f(θ,φ)reikr
at large
r,
with
f(θ,φ)
the scattering amplitude.
The
S
-matrix maps in-states to out-states:
Sk′k=⟨k′−∣k+⟩.
Unitarity of the
S
-matrix,
S†S=1^,
follows from the conservation laws of QM4 and is equivalent to the optical theorem
σtot=k4πImf(0).
The
T
-matrix relates to the
S
-matrix by
S=1^−2πiδ(E′−E)T
and satisfies the Lippmann-Schwinger equation for the
T
-matrix:
T=V^+V^G^0+T.
The Born approximation is the leading-order expansion
T≈V^,
giving the Born scattering amplitude
fBorn(q)=−2πΦ02μV(q),
where
V(q)=∫V(r)e−iq⋅rd3r
is the Fourier transform of the potential at momentum transfer
q=kout−kin.
Partial wave analysis decomposes the scattering amplitude for a central
potential
V=V(r)
as
f(θ)=ℓ=0∑∞(2ℓ+1)fℓPℓ(cosθ),
where the partial wave amplitude
fℓ=2ike2iδℓ−1
is determined by the phase shift
δℓ,
the solution of the radial Schrödinger equation in the
ℓ
-th angular momentum channel.
Spin-dependent scattering on
Hfull=H⊗Hspin
is analyzed via the
2×2
spin amplitude matrix
F(θ),
whose elements are the non-flip amplitude
A(θ)
and the spin-flip amplitude
B(θ).
The post-scattering reduced density matrix
ρA=Trspatial(ρout)
gives the spin state of the scattered particle after tracing out the
spatial degree of freedom.
No new postulates are introduced.
All results follow from the QM4 dynamical framework, the QM5 angular
momentum algebra, the QM7 center-of-mass separation, the QM8 full Hilbert
space, and the QM9 density matrix formalism.
The NUVO QM-series has, through QM9, developed two parallel arcs of
structure.
The single-particle arc, QM1--QM6, established the Hilbert space, the
observable and dynamical framework, the angular momentum algebra and its
holonomy-derived spectrum, and the harmonic oscillator with its algebraic
Fock space structure.
The multi-particle arc, QM7--QM9, opened the tensor product sector,
derived the spin degree of freedom from the double-cover holonomy, and
established the complete theory of bipartite entanglement including the
Bell inequality violation as a theorem.
In both arcs, the energy of the systems under consideration was either
discrete, bound state spectrum with
En<0
for the hydrogenic levels of QM5, positive but quantized for the coupled
oscillator of QM7, or zero, the ground state of the harmonic oscillator
family.
The present paper, QM10, opens the scattering sector: the regime of
positive energy
E>0
for two transport closure configurations that approach from large
separation, interact through a potential
V(r)
for a finite time, and separate again to large distance.
The physical content that distinguishes scattering from the bound state
problem is the asymptotic structure: the initial and final states are
free, non-interacting plane waves, and all the information about the
interaction is encoded in the change of direction and spin state between
the initial and final free states.
The scattering problem reduces to a one-body problem in the relative
coordinate through the center-of-mass separation of QM7.
The Hamiltonian
H^(2)=H^CM⊗1^+1^⊗H^rel
factorizes into a free center-of-mass motion and a relative motion
governed by
H^rel=2μΦ02∇r2+V(r),
where
μ=m1+m2m1m2
is the reduced mass of QM7 Theorem 6.1.
All scattering observables, the differential cross section, the total
cross section, and the scattering amplitude, are determined by the
relative Hamiltonian
H^rel
alone.
This reduction is the single most important simplification of the
two-body scattering problem: the full two-body problem on
H(2)=H1⊗H2
with its complicated tensor product structure reduces to a one-body
problem on
Hrel=L2(R3)
with the effective Hamiltonian
H^rel.
QM10 introduces two structural elements that are qualitatively new to the
QM-series.
The first is the rigged Hilbert space.
The scattering states are non-normalizable continuum eigenstates
∣k⟩
of the free Hamiltonian
H^0=2μΦ02k2
at positive energy
Ek>0.
They are not elements of
H=L2(R3)
since
eik⋅r
is not square-integrable, but of the larger space
Φ∗
of tempered distributions in the Gelfand triple
Φ⊂H⊂Φ∗.
The generalized completeness relation
∫∣k⟩⟨k∣d3k=1^H
and the generalized orthonormality
⟨k∣k′⟩=δ(3)(k−k′)
hold in this extended sense and are the continuum analogues of the
discrete completeness and orthonormality relations of QM1.
The second new element is the
S
-matrix.
For bound states, the primary observable is the energy spectrum.
For scattering states, the primary observable is the
S
-matrix,
S,
a unitary operator that maps the in-asymptotic free state
∣k⟩
to the out-asymptotic free state
∣k′⟩
with amplitude
Sk′k=⟨k′−∣k+⟩.
Unitarity of the
S
-matrix is a consequence of the conservation laws of QM4, probability
conservation under unitary time evolution, and has the immediate
corollary, the optical theorem, that the total cross section equals
k4π
times the imaginary part of the forward scattering amplitude.
The density matrix formalism of QM9 makes its first appearance in a
physical calculation in the spin-dependent scattering section.
For a spin-21 particle scattering from a target with
spin-orbit coupling, the post-scattering state is an entangled
spatial-spin state in
Hfull=H⊗Hspin.
Tracing out the spatial degree of freedom yields the post-scattering spin
density matrix
ρout=Tr(FρinF†)Trspatial(FρinF†),
from which the polarization of the scattered beam is computed via the
Sherman function.
This computation is the synthesis of all prior spin and entanglement
structure: the full Hilbert space of QM8, the density matrix of QM9, and
the scattering amplitude of the present paper combine in a single physical
prediction.
QM11 will extend the scattering framework to the relativistic sector,
replacing the Pauli equation with the Dirac equation and deriving the
g
-factor
g=2
and the complete hydrogen fine structure as relativistic consequences.
The central objective of the present paper is to derive the complete
non-relativistic scattering theory for two transport closure
configurations interacting through a potential, using the two-body
framework of QM7, the angular momentum algebra of QM5, the full spin
Hilbert space of QM8, and the density matrix formalism of QM9.
Specifically, the paper establishes six claims.
The free Hamiltonian
H^0=2μΦ02k2
on
H=L2(R3)
has a purely continuous spectrum
[0,∞)
with generalized eigenstates
∣k⟩∈Φ∗,
elements of the dual of the Schwartz space
Φ=S(R3),
satisfying
⟨r∣k⟩=(2π)3/2eik⋅r,
the generalized orthonormality
⟨k∣k′⟩=δ(3)(k−k′),
and the completeness
∫∣k⟩⟨k∣d3k=1^H.
The Rayleigh expansion of the plane wave in spherical harmonics and
spherical Bessel functions connects the plane wave basis to the partial
wave basis of claim 5.
The Lippmann-Schwinger equation
∣k+⟩=∣k⟩+G^0+V^∣k+⟩,
where
G^0+=ϵ→0+lim(Ek+iϵ−H^0)−1
is the retarded free resolvent, defines the outgoing scattering state
∣k+⟩
with position-space representation
ψ(r)→(2π)−3/2[eik⋅r+f(θ,φ)reikr]
as
r→∞,
where
f(θ,φ)=−2πΦ02μ⟨k′∣V^∣k+⟩
is the scattering amplitude and the differential cross section is
dΩdσ=∣f∣2.
The
S
-matrix
S=Ω^−†Ω^+,
where
Ω^±
are the Møller wave operators, is unitary:
S†S=1^.
The
T
-matrix is related by
Sk′k=δ(3)(k′−k)−2πiδ(E′−E)Tk′k,
where
Tk′k=⟨k′∣V^∣k+⟩,
and the optical theorem
σtot=k4πImf(0)
follows from
S
-matrix unitarity.
The Born approximation
T≈V^
gives the first-order scattering amplitude
fBorn(q)=−2πΦ02μV(q),
where
q=kout−kin
is the momentum transfer and
V(q)=∫V(r)e−iq⋅rd3r
is the Fourier transform of the potential.
For the Yukawa potential
V(r)=−rg2e−λr,
the Born amplitude gives
dΩdσ=(Φ02μg2)2(∣q∣2+λ2)21;
in the Coulomb limit
λ→0
this reduces to the Rutherford formula
dΩdσ=(2Φ02k2μe2)2sin−4(2θ),
which is exact for the Coulomb potential.
For a central potential
V=V(r),
the scattering amplitude decomposes in partial waves:
f(θ)=k1ℓ=0∑∞(2ℓ+1)eiδℓsinδℓPℓ(cosθ),
where the phase shift
δℓ∈R
is determined by the asymptotic behavior
uℓ(r)→sin(kr−2ℓπ+δℓ)
of the solution of the radial Schrödinger equation.
The partial wave and total cross sections are
σℓ=k24π(2ℓ+1)sin2δℓ
and
σtot=ℓ∑σℓ;
Levinson's theorem
δℓ(0)=nℓπ
relates the zero-energy phase shift to the number of bound states.
For a spin-21 particle scattering from a target with
spin-orbit coupling on
Hfull=H⊗Hspin,
the spin scattering amplitude matrix
F(θ)=A(θ)I+B(θ)(n^⋅σ)
gives the differential cross section
dΩdσ=∣A∣2+∣B∣2
for an unpolarized incident beam, the post-scattering spin density
matrix
ρout=Tr(FρinF†)FρinF†,
and the polarization of the scattered beam
Pn^=∣A∣2+∣B∣22Im(A∗B),
the Sherman function.
Claims 1 through 6 follow the logical chain of the paper.
The rigged Hilbert space of claim 1 provides the mathematical setting; the
Lippmann-Schwinger equation of claim 2 defines the scattering states and
the scattering amplitude; the
S
-matrix of claim 3 encodes all scattering information and its unitarity
gives the optical theorem; the Born approximation of claim 4 gives the
leading-order amplitude explicitly in terms of the potential; the partial
wave analysis of claim 5 provides the exact decomposition for central
potentials; and the spin scattering of claim 6 combines all the preceding
structure with the QM8 and QM9 frameworks in a single physical
calculation.
The present work maintains without modification the interpretive
discipline of the prior series.
Five exclusions are of particular importance for QM10.
The rigged Hilbert space is not postulated as a new axiom of quantum
mechanics.
It is the correct mathematical extension of the QM1 Hilbert space
H=L2(R3)
required to accommodate the generalized eigenstates of the free
Hamiltonian at positive energy.
Every physical quantity in QM10 is expressed as a matrix element
⟨ϕ∣O^∣ψ⟩
with
ϕ,ψ∈H
normalizable states; the plane wave states
∣k⟩∈Φ∗
appear only in intermediate steps and in the completeness relation, never
as physical states of the system.
The rigged Hilbert space is a mathematical bookkeeping device that allows
the Dirac notation
∣k⟩
to be used consistently; no new physical content is introduced.
Asymptotic completeness is asserted for short-range potentials but not
proved.
The statement that the Møller wave operators
Ω^±
are partial isometries with
range(Ω^+)=range(Ω^−)
which is what is needed for
S
-matrix unitarity via
S=Ω^−†Ω^+
is a non-trivial functional analytic theorem.
For short-range potentials satisfying
∣V(r)∣≤C(1+∣r∣)−1−ϵ
for some
C,ϵ>0,
asymptotic completeness was proved by Enss and by Sigal and Soffer.
For the Coulomb potential
V(r)=−re2,
which decays only as
r1,
the standard Møller operators do not converge and the Dollard modification
of the wave operators is required; the Coulomb case is noted but its
detailed treatment is deferred.
The Coulomb scattering theory with the Dollard modification is not
developed.
The Coulomb potential requires a modified definition of the wave operators
Ω^±Dollard=t→∓∞limeiH^t/Φ0W^(t),
where
W^(t)
contains an additional Coulomb phase correction, because the ordinary
Møller limits diverge for
r1
potentials.
The Rutherford formula derived from the Born approximation is exact for
the Coulomb potential, a special property of the Coulomb case, and so is
the correct result despite the breakdown of the standard wave operator
formalism; the rigorous Coulomb scattering theory confirming this is cited
rather than developed.
Multi-channel and inelastic scattering are not treated.
In the present paper, the two transport closure configurations have fixed
internal states before and after the collision, elastic scattering.
The general case where the configurations can change their internal state
during the collision, inelastic scattering with a different final state
from the initial, requires a coupled-channel formalism: a matrix-valued
S
-matrix with channel indices.
This extension, together with the Breit-Wigner resonance formula for
quasi-bound states that appear as poles of the
S
-matrix in the complex energy plane, is deferred.
Relativistic corrections to the scattering amplitude are not derived.
The scattering amplitude of QM10 is the leading term in a
v/c
expansion; the corrections of order
(v/c)2
arise from the relativistic dispersion relation
E2=(pc)2+(mc2)2,
from magnetic interactions, and from radiative corrections in quantum
electrodynamics.
These corrections are derived in QM11 from the Dirac equation, making
QM10 the non-relativistic foundation whose relativistic extension is the
content of QM11.
Section 2 recalls the center-of-mass and relative coordinate separation
from QM7, the dynamical framework and conservation laws from QM4, the
angular momentum algebra and partial wave prerequisites from QM5, and the
full Hilbert space and density matrix formalisms from QM8 and QM9 that are
used in the spin-dependent scattering analysis.
Section 3 introduces the rigged Hilbert space as the mathematical setting
for the scattering problem, defines the plane wave states
∣k⟩
as generalized eigenstates of
H^0
in the Gelfand triple
Φ⊂H⊂Φ∗,
and derives the generalized completeness and orthonormality relations
together with the Rayleigh plane wave expansion.
Section 4 defines the retarded free resolvent
G^0+,
derives the Lippmann-Schwinger equation for the in- and out-scattering
states, establishes their position-space form as an integral equation, and
derives the asymptotic form and the connection between the
Lippmann-Schwinger kernel and the scattering amplitude.
Section 5 defines the Møller wave operators and the
S
-matrix, establishes the
S
-matrix unitarity, invoking asymptotic completeness, derives the
S
-matrix to
T
-matrix relation, and proves the optical theorem as a consequence of
unitarity.
Section 6 derives the Born approximation as the leading term in the
iterative solution of the Lippmann-Schwinger equation for the
T
-matrix, computes the Born amplitude for the Yukawa potential, and derives
the Rutherford formula as the Coulomb limit.
Section 7 derives the partial wave decomposition of the scattering
amplitude for a central potential, defines the phase shifts from the
asymptotic boundary condition on the radial wave function, derives the
partial wave cross sections, and records the properties of the phase
shifts including Levinson's theorem and the scattering length.
Section 8 sets up the spin-dependent scattering problem on
Hfull=H⊗Hspin,
introduces the spin scattering amplitude matrix
F=AI+B(n^⋅σ),
derives the differential cross section and the post-scattering spin
density matrix using the QM9 partial trace formalism, and derives the
Sherman function for the polarization of the scattered beam.
Section 9 records the derivational status of the rigged Hilbert space and
asymptotic completeness, and the scope of the present construction.
Section 10 summarizes the fourteen principal results, records the
programmatic significance of the scattering sector, and prepares the
transition to QM11.
The present section collects the results from QM4, QM5, QM7, QM8, and
QM9 that are directly required for the derivations of Sections 3--8.
Nothing in this section is new.
The recalled material falls into four categories: the two-body reduction
from QM7 that converts the scattering problem from a tensor product
Hilbert space to a single Hilbert space; the dynamical framework and
conservation laws from QM4 that underlie the S-matrix unitarity and the
optical theorem; the angular momentum decomposition from QM5 that is the
foundation of the partial wave analysis; and the spin and density matrix
structures from QM8 and QM9 that enter the spin-dependent scattering
analysis of Section 8.
The following results from QM7 reduce the two-body scattering problem to a
one-body problem in the relative coordinate.
Center-of-mass and relative coordinates, from QM7 Theorem 6.1.
For two transport closure configurations with masses m1 and m2
and position operators x^1 and x^2, the total mass
M=m1+m2
and reduced mass
μ=Mm1m2
define:
Equation. Center-of-mass coordinates.
R=Mm1x^1+m2x^2,P=p^1+p^2,
and
Equation. Relative coordinates.
r=x^1−x^2,prel=Mm2p^1−m1p^2.
These satisfy the canonical commutation relations
[R^j,P^k]=iΦ0δjk,
[r^j,p^rel,k]=iΦ0δjk,
and all cross commutators vanish.
Factorization of the two-body Hamiltonian, from QM7 Theorem 6.1.
For a two-body Hamiltonian
The center-of-mass motion is that of a free particle of mass M; the
relative motion is governed by the one-body Hamiltonian
H^rel
on
Hrel=L2(R3)
with effective mass μ.
All scattering observables are determined by H^rel
alone.
Remark.
The reduction of the two-body problem to the one-body problem is the
structural content of QM7 Theorem 6.1 applied to the scattering regime.
In the bound state context of QM7, the same reduction produced the
coupled oscillator Hamiltonian with its discrete normal mode spectrum.
In the scattering context of QM10, the same reduction produces the
relative Hamiltonian
H^rel=H^0+V^
whose continuous positive-energy spectrum
σ(H^rel)∩(0,∞)
is the scattering sector.
The two regimes are complementary: the bound state sector,
E<0,
discrete spectrum, and the scattering sector,
E>0,
continuous spectrum, together constitute the complete spectral data of
H^rel,
related by the Levinson theorem of Section 7, which connects the two.
¶ The Dynamical Framework and Conservation Laws from QM4
The following results from QM4 are the foundation of the S-matrix
unitarity and the optical theorem.
Stone's theorem and unitary time evolution, from QM4 Theorem 3.1.
The self-adjoint Hamiltonian
H^rel
on
Hrel
generates a strongly continuous one-parameter unitary group
U(t)=e−iH^relt/Φ0
satisfying
U(t)†U(t)=1^
for all
t∈R.
This unitarity of time evolution is the primary input to the S-matrix
unitarity of Theorem 5.1: the S-matrix, schematically,
S=t→+∞limUrel(t)U0(−t)†U0(−t)Urel(−t),
inherits its unitarity from
U(t)†U(t)=1^.
Conservation laws, from QM4 Theorem 5.2.
An observable A^ is conserved by H^rel if and
only if
[H^rel,A^]=0.
For a central potential
V=V(r):
Equation. Conservation of L^2.
[H^rel,L^2]=0,
Equation. Conservation of L^z.
[H^rel,L^z]=0.
These conservation laws have two consequences for QM10:
the scattering amplitude for a central potential depends only on the
scattering angle θ, not on the azimuthal angle φ,
enabling the partial wave decomposition of Section 7;
the phase shift δℓ in each angular momentum channel ℓ is a real number, conservation of probability in each channel
separately, as established in the phase-shift properties proposition.
The Heisenberg equation of motion and the self-adjoint domain, from QM4 Theorem 4.2.
The self-adjointness of
H^rel
on the Sobolev domain
D(H^rel)⊂Hrel
for potentials in the Kato class is a hypothesis whose verification for
specific potentials, Yukawa and Coulomb, is a standard result cited from
the functional analysis literature.
The Heisenberg equation
dtdA^=Φ0i[H^rel,A^]
is used in the proof of the optical theorem through the probability
conservation identity applied to the current operator.
Remark.
The Kato-Rellich theorem, QM4 Theorem 4.2, guarantees that
H^rel=H^0+V^
is self-adjoint on
D(H^0)
whenever V^ is H^0-bounded with relative bound less than
1.
For the Yukawa potential
V(r)=−rg2e−λr,
one has
V∈L2(R3)+L∞(R3)
for any
λ>0,
which is a sufficient condition.
For the Coulomb potential
V(r)=−re2,
by the Kato inequality
∣∣∣∣∣−re2∣∣∣∣∣≤ϵ(−∇2)+Cϵ
for any
ϵ>0
and a constant Cϵ depending on ϵ, the Coulomb
potential is H^0-bounded with relative bound 0 and
regularity at the origin.
This is the one-dimensional Schrödinger equation with an effective
potential
Veff(r)=V(r)+2μr2Φ02ℓ(ℓ+1)
that includes the centrifugal barrier.
Spherical Bessel functions, from QM5 Proposition 6.4 and the asymptotic analysis.
For
V≡0
the free radial equation, the two linearly independent solutions are
jℓ(kr)
spherical Bessel, regular at
r=0,
and
nℓ(kr)
spherical Neumann, irregular at
r=0,
with asymptotic behaviors:
Equation. Spherical Bessel asymptotic.
jℓ(x)x→∞xsin(x−ℓπ/2),
and
Equation. Outgoing spherical Hankel asymptotic.
hℓ(1)(x)x→∞ixei(x−ℓπ/2−π/2)=ix(−i)ℓ+1eix,
where
hℓ(1)(x)=jℓ(x)+inℓ(x)
is the outgoing spherical Hankel function.
The asymptotic form of jℓ determines the free wave behavior at
large r in the partial wave analysis, and the deviation from this form
in the presence of
V
is captured by the phase shift
δℓ.
Remark.
The conservation law
[H^rel,L^2]=0
has a direct consequence for the partial wave decomposition: since L^2 is conserved, each angular momentum channel ℓ scatters
independently.
The
S
-matrix is block-diagonal in the angular momentum basis, with the ℓ-th block being the
1×1
complex number
Sℓ=e2iδℓ,
a pure phase, since unitarity in each channel requires
∣Sℓ∣=1.
The total
S
-matrix is therefore characterized by the countable set of real numbers
{δℓ}ℓ=0∞,
one per angular momentum channel.
This is the content of the partial wave decomposition theorem.
¶ The Full Hilbert Space, Pauli Algebra, and Density Matrix from QM8 and QM9
The following results from QM8 and QM9 enter the spin-dependent scattering
analysis of Section 8.
The full spin-21 Hilbert space, from QM8 Definition 5.1.
The full single-particle Hilbert space for a spin-21
configuration is
Hfull=H⊗Hspin=L2(R3,C2),
with elements represented as two-component spinors
Ψ(r)=(ψ↑(r)ψ↓(r)).
The observable commutation
[A^⊗I,1^⊗B^]=0
from QM8 Proposition 5.3 ensures that spatial observables, including the
scattering Hamiltonian, and spin observables commute on
Hfull.
The Pauli algebra and spin-orbit coupling, from QM8 Theorems 4.2 and 7.1.
The spin operators
S^j=2Φ0σj
on
Hspin=C2
satisfy
σjσk=δjkI+iϵjklσl.
For spin-dependent scattering, the interaction Hamiltonian on
Hfull
includes the spin-orbit term
ξ(r)L^⋅S^
from QM8 Definition 7.1:
Equation. Spin-orbit interaction.
V^full=V(r)⊗I+ξ(r)(L^⊗I)⋅(1^⊗S^),
where
ξ(r)=2μ2c21r1drdV
is the spin-orbit coupling function of QM8.
The eigenvalues of
L^⋅S^
in the coupled basis
∣j,mj⟩
are
2Φ02[j(j+1)−ℓ(ℓ+1)−43]
from QM8 Proposition 7.2, and the spin-orbit interaction splits each
angular momentum channel ℓ into two sub-channels with
j=ℓ±21
and different phase shifts
δℓ,+
and
δℓ,−.
The density matrix and partial trace, from QM9 Definitions 4.1--4.2 and Theorem 4.2.
For the post-scattering analysis of Section 8, the full post-scattering
state
∣Ψout⟩∈Hfull
is a spatial-spin entangled state.
The spin state of the scattered particle is described by the reduced
density matrix:
Equation. Recalled post-scattering density matrix.
ρout=Trspatial(∣Ψout⟩⟨Ψout∣),
where the trace is over the spatial, position, degree of freedom, leaving
a
2×2
density matrix on
Hspin.
The Born rule for the spin degree of freedom gives the polarization
expectation
⟨n^⋅σ⟩out=TrHspin(ρout(n^⋅σ)),
the primary observable in spin-dependent scattering experiments.
Remark.
The spin-orbit coupling modifies the partial wave analysis of Section 7
by splitting each ℓ-channel into two sub-channels.
For
ℓ≥1,
the coupled basis
∣j,mj⟩
with
j=ℓ±21
diagonalizes the interaction Hamiltonian in each angular momentum channel,
by QM8 Proposition 7.2 with ξ(r) replacing the bound-state spin-orbit
constant.
The radial Schrödinger equation in the
j=ℓ±21
channel has an effective potential that differs by the spin-orbit
eigenvalue, giving phase shifts
δℓ,+
and
δℓ,−
that are generally different.
The spin amplitude matrix
F=AI+B(n^⋅σ)
of the spin-amplitude definition encodes this splitting: the non-flip
amplitude A is a symmetric combination of
δℓ,+
and
δℓ,−,
while the spin-flip amplitude B is proportional to their difference and
vanishes in the absence of spin-orbit coupling, when
δℓ,+=δℓ,−
for all ℓ.
Remark.
Section 8 is the first section in the QM-series where the results of QM4,
QM5, QM7, QM8, and QM9 are all used simultaneously in a single physical
calculation:
QM4, conservation laws imply S-matrix unitarity; QM5, partial wave
analysis gives phase shifts; QM7, two-body reduction gives the relative
coordinate; QM8, spin-orbit coupling gives the spin amplitude matrix; and
QM9, the density matrix gives the post-scattering spin state.
The spin-dependent differential cross section and the Sherman function
are the physical observables that combine all of these structures.
This synthesis is the primary programmatic contribution of QM10: it shows
that the scattering problem is the natural arena in which all the prior
mathematical structures of the QM-series are exercised together.
¶ The Rigged Hilbert Space and Continuum Eigenstates
The standard Hilbert space
H=L2(R3)
of QM1 is the correct setting for normalizable states:
square-integrable wave functions representing configurations with
definite position distributions.
The scattering problem requires a qualitatively different class of states:
the plane waves
eik⋅r
that represent configurations with definite momentum but completely
indeterminate position.
These functions are not in H, since
∫∣∣∣eik⋅r∣∣∣2d3r=∞,
and yet play an essential role as the initial and final states in every
scattering calculation.
The resolution is the rigged Hilbert space, or Gelfand triple, which
extends the Hilbert space to accommodate such generalized eigenstates in
a mathematically rigorous way.
The present section establishes this framework, defines the plane wave
states as tempered distributions, records their generalized completeness
and orthonormality, and derives the Rayleigh expansion that connects the
Cartesian plane wave basis to the spherical partial wave basis of
Section 7.
Definition. Rigged Hilbert space for the scattering problem.
The rigged Hilbert space, or Gelfand triple, for the one-body
scattering problem is the nested sequence of spaces:
Equation. Gelfand triple.
Φ⊂H⊂Φ∗,
where:
H=L2(R3,C) is the Hilbert space of
square-integrable functions, QM1;
Φ=S(R3) is the Schwartz space of rapidly
decreasing smooth functions: all ϕ∈C∞(R3) satisfying
rsup∣∣∣rα∂βϕ(r)∣∣∣<∞
for all multi-indices α,β;
Φ∗=S′(R3) is the dual space of tempered
distributions: continuous linear functionals on Φ.
The inclusions in the Gelfand triple are dense: Φ is dense in H in the L2-norm, and H
is embedded in Φ∗ by the identification
ψ↦⟨ψ,⋅⟩H.
The free Hamiltonian
H^0=−2μΦ02∇2
maps Φ to Φ continuously, since differentiation preserves the
Schwartz class, and extends by duality to a map
Φ∗→Φ∗.
Remark.
The rigged Hilbert space is not a new physical postulate.
It is a mathematical framework that makes the Dirac notation
∣k⟩
and the formal manipulations of the position and momentum eigenstates
rigorous.
Every physical prediction of QM10 is expressed as a matrix element
⟨ϕ∣O^∣ψ⟩
with
ϕ,ψ∈H
normalizable states.
The plane wave states
∣k⟩∈Φ∗
appear in intermediate steps, in the completeness relation and in the
Lippmann-Schwinger equation of Section 4, but never as the initial or
final physical states of an experiment, which are always represented by
normalizable wave packets
∣ϕ⟩∈H
with
ϕ∈S(R3).
The rigged Hilbert space is to the scattering problem what the spectral
measure is to the bound state problem: a mathematical tool that organizes
the calculation.
The Rayleigh expansion decomposes the Cartesian plane wave
eik⋅r
into a sum of spherical waves, connecting the plane wave basis to the
angular momentum eigenbasis of QM5.
This expansion is the bridge between the Lippmann-Schwinger equation,
which is naturally stated in the plane wave basis, and the partial wave
analysis of Section 7, which uses the spherical wave basis.
Proposition. Rayleigh plane wave expansion.
For any
k,r∈R3,
one has:
Equation. Rayleigh plane wave expansion.
eik⋅r=4πℓ=0∑∞m=−ℓ∑+ℓiℓjℓ(kr)Yℓm∗(k^)Yℓm(r^),
where
k=∣k∣,k^=kk,r^=rr,
and
jℓ(x)
is the spherical Bessel function of the first kind.
For incident momentum along the z-axis,
k=kz^,
one has:
Equation. Rayleigh expansion for incident momentum along z.
eikz=eikrcosθ=ℓ=0∑∞(2ℓ+1)iℓjℓ(kr)Pℓ(cosθ),
using
Yℓ0(θ,φ)=4π2ℓ+1Pℓ(cosθ).
Proof.
The key identity is the addition theorem for spherical harmonics, QM5
Theorem 7.1: for any two unit vectors k^ and r^,
Equation. Spherical harmonic addition theorem.
Pℓ(k^⋅r^)=2ℓ+14πm=−ℓ∑ℓYℓm∗(k^)Yℓm(r^).
Expand
eik⋅r=eikrcosΘ,
where Θ is the angle between k and r, in
Legendre polynomials.
The known generating formula is
eixcosΘ=ℓ=0∑∞(2ℓ+1)iℓjℓ(x)Pℓ(cosΘ),
with
x=kr.
The coefficients
iℓjℓ(kr)
are determined by projecting
eixcosΘ
onto
Pℓ(cosΘ)
using the orthogonality of Legendre polynomials and the integral
representation of the spherical Bessel functions:
jℓ(x)=2iℓ1∫−11eixtPℓ(t)dt.
Substituting the addition theorem into this Legendre expansion gives the
Rayleigh plane wave expansion.
The incident-z equation follows from the Rayleigh expansion with
k^=z^
and the identity
Yℓm(z^)=4π2ℓ+1δm0,
which gives only the m=0 term and reduces the spherical harmonic sum to
the Legendre polynomial via
Yℓ0(θ,φ)=4π2ℓ+1Pℓ(cosθ).
□
Remark.
The Rayleigh expansion is used at two points in the paper.
In Section 4, the asymptotic form of the scattering wave function
ψ(r)→(2π)−3/2[eikz+f(θ)reikr]
is derived by expanding the incident plane wave in partial waves and
subtracting the contribution of the free spherical waves, leaving the
scattered wave
f(θ)reikr
at large r.
In Section 7, the full partial wave expansion of the scattering amplitude
is derived by matching the asymptotic form of ψ to the
incident-z Rayleigh expansion plus an outgoing spherical wave, using the
asymptotic behavior of the spherical Bessel function
jℓ(kr)→krsin(kr−ℓπ/2)
and of the Hankel function
hℓ(1)(kr)→kr(−i)ℓ+1eikr
at large r.
Remark.
The completeness relation can be rewritten in the energy-angle basis by
changing integration variables from k to
(Ek,k^):
Equation. Completeness in the energy-angle basis.
1^H=∫0∞dEk∫S2dΩkρ(Ek)∣k⟩⟨k∣,
where
ρ(Ek)=π2Φ02μk=2π21Φ032μEkμ
is the density of states at energy Ek.
The density of states grows as
Ek,
reflecting the increasing number of plane wave states available at higher
energy.
This energy-angle form of the completeness relation is the natural one
for the partial wave analysis of Section 7: the integral over k^ at
fixed Ek is the integral over the energy shell, and the spherical
harmonic decomposition converts the k^ integral to a sum over
angular momentum quantum numbers
(ℓ,m)
via the orthonormality of
Yℓm
on
S2.
¶ The Lippmann-Schwinger Equation and Scattering States
The scattering problem requires states that simultaneously satisfy the
full Schrödinger equation
H^rel∣k±⟩=Ek∣k±⟩
and carry the correct asymptotic boundary conditions: at large distances
from the interaction region, the scattering state should resemble the
free plane wave
∣k⟩
plus outgoing, for ∣k+⟩, or incoming, for ∣k−⟩, scattered waves.
The Lippmann-Schwinger equation achieves both requirements simultaneously
by expressing the full scattering state as the free state corrected by
the action of the potential through the free resolvent, with the choice of
retarded,
+iϵ,
or advanced,
−iϵ,
resolvent selecting the outgoing or incoming boundary condition
respectively.
The present section derives the Lippmann-Schwinger equation, establishes
its position-space form as a Fredholm integral equation of the second
kind, and derives the asymptotic form of the scattering wave function that
identifies the scattering amplitude as a matrix element of the T-matrix.
where the limits hold in the strong operator topology on H
and in norm on Φ, where the limits converge uniformly.
Proposition. Position-space form of the retarded free resolvent.
The retarded free resolvent G0+ has position-space matrix
elements:
Equation. Retarded free resolvent kernel.
⟨r∣G0+∣r′⟩=−Φ022μ4π∣r−r′∣e+ik∣r−r′∣,
where
k=Φ02μEk>0.
This is the retarded Green's function of the Helmholtz operator
(∇2+k2)
in three dimensions:
Equation. Helmholtz Green function equation.
(∇2+k2)⟨r∣G0+∣r′⟩=δ(3)(r−r′).
Proof.
The free Hamiltonian in position space is
H^0=−2μΦ02∇2,
so the equation
(z−H^0)⟨r∣G0(z)∣r′⟩=δ(3)(r−r′)
becomes
[z+2μΦ02∇2]G(r,r′;z)=δ(3)(r−r′),
or equivalently
[∇2+Φ022μz]G=−Φ022μδ(3)(r−r′).
For
z=Ek+iϵ
with
ϵ>0,
one has
Φ022μz=k2+iϵ′,
where
ϵ′=Φ022μϵ>0.
The outgoing Green's function of
(∇2+κ2)
for
Im(κ)>0,
corresponding to +iϵ′, is
G=−4π∣r−r′∣eiκ∣r−r′∣,
a standard result of the theory of elliptic operators.
Taking
ϵ→0+
gives
κ→k+,
real and positive, and the resolvent kernel approaches the retarded free
resolvent kernel.
The Helmholtz Green function equation follows by multiplying the kernel
by
−2μΦ02(∇2+k2)
and using the definition of the Green's function. □
Remark.
The factor
e+ik∣r−r′∣
in the retarded free resolvent kernel represents an outgoing spherical
wave centered at r′: at large
∣r−r′∣,
it propagates radially outward.
The advanced free resolvent G0− has the same formula with +ik
replaced by −ik:
⟨r∣G0−∣r′⟩=−Φ022μ4π∣r−r′∣e−ik∣r−r′∣,
representing an incoming spherical wave.
The choice of G0+, retarded and outgoing, in the
Lippmann-Schwinger equation for ∣k+⟩ selects the
physical boundary condition: the scattered wave produced by the
interaction propagates outward from the interaction region.
a Fredholm integral equation of the second kind for the scattering wave
function
ψ(r)=⟨r∣k+⟩.
Proof. Derivation of the incoming scattering-state equation.
We seek a solution to
(H^rel−Ek)∣k+⟩=0,
i.e.,
(H^0+V^−Ek)∣k+⟩=0,
or equivalently
(Ek−H^0)∣k+⟩=V^∣k+⟩.
This is an inhomogeneous equation for ∣k+⟩ with
operator
Ek−H^0
on the left.
At energy Ek on the continuous spectrum of H^0, the
operator
Ek−H^0
is not invertible on H, because its range is not all of H.
The retarded resolvent
G0+=ϵ→0+lim(Ek+iϵ−H^0)−1
provides the correct regularization: the general solution is the
particular solution
G0+V^∣k+⟩
plus the homogeneous solution
∣k⟩
of
(Ek−H^0)∣k⟩=0,
giving the Lippmann-Schwinger equation.
The choice of G0+, rather than G0−, selects the outgoing
boundary condition, as established in the outgoing-versus-incoming
resolvent remark.
Verification that ∣k+⟩ satisfies the full eigenvalue equation.
Using the Lippmann-Schwinger equation,
Substituting the retarded free resolvent kernel gives the position-space
Lippmann-Schwinger equation. □
Remark.
The Lippmann-Schwinger equation is an implicit equation for ∣k+⟩: the right-hand side depends on ∣k+⟩ through the term
G0+V^∣k+⟩.
It is a generalized integral equation, a Fredholm equation of the second
kind in position space, and its unique solution, in the sense of
distributions, is the scattering state ∣k+⟩.
For short-range potentials
V∈L1(R3)∩L2(R3),
the operator
G0+V^
is compact on H at positive energy, since G0+ is
bounded and V^ is compact for
V∈L2∩L∞,
and the Fredholm alternative guarantees a unique solution for
k>0
away from resonance energies, where
1+G0+V^
is not invertible.
The iterative solution of the Lippmann-Schwinger equation by successive
substitution,
The physical content of the Lippmann-Schwinger equation is encoded in the
large-distance behavior of
ψ(r):
at large r, the scattering wave function separates into the incident
plane wave and an outgoing spherical wave whose angular distribution is
the scattering amplitude.
Theorem. Asymptotic form and scattering amplitude.
For a potential
V(r)
of finite range, or decaying faster than
r21
at large r, the scattering wave function has the asymptotic form:
Equation. Scattering asymptotic form.
ψ(r)r→∞(2π)3/21(eik⋅r+f(θ,φ)reikr),
where θ and φ are the polar and azimuthal angles of r^ relative to k^, and the scattering amplitude is
The differential cross section follows from the ratio of the outgoing
scattered flux to the incident flux:
dΩdσ=jincjscat(r^)⋅r2=v∣f∣2(v/r2)⋅r2=∣f∣2,
where
v=μΦ0k
is the incident velocity and
j=2μiΦ0(ψ∗∇ψ−ψ∇ψ∗)
is the probability current from QM4 Proposition 4.3. □
Remark.
The identification
f=−2πΦ02μTk′k
in the scattering-amplitude equation is the central relation of
scattering theory: all physical observables, the differential cross
section, the total cross section via the optical theorem, and the partial
wave phase shifts, are encoded in the on-shell T-matrix elements
Tk′k=⟨k′∣T^∣k⟩
at
∣k′∣=∣k∣=k,
the elastic energy shell.
The T-matrix satisfies its own Lippmann-Schwinger equation:
Equation. T-matrix Lippmann-Schwinger equation.
T^=V^+V^G0+T^,
obtained by substituting the Lippmann-Schwinger equation for ∣k+⟩ into the definition
which gives the T-matrix Lippmann-Schwinger equation since it holds for
all ∣k⟩.
The iterative solution is the Born series
T^=V^+V^G0+V^+V^G0+V^G0+V^+⋯,
whose first term,
T^≈V^,
is the Born approximation of Section 6.
Remark.
The asymptotic form has a precise physical interpretation.
The term
(2π)3/2eik⋅r
is the incident plane wave, propagating in the direction k^ with
wave number k.
The term
r(2π)3/2f(θ,φ)eikr
is the scattered wave: an outgoing spherical wave propagating radially in
all directions, with angular distribution given by the scattering
amplitude
f(θ,φ).
The amplitude of the scattered wave at angle (θ,φ) relative
to the incident direction is
r∣f∣,
falling off as 1/r, so the scattered flux, proportional to
∣amplitude∣2×r2,
is constant at large r, consistent with probability conservation.
This is the quantum mechanical counterpart of the classical statement
that a scattered particle travels in a definite direction after the
collision: quantum mechanically, the scattering produces a superposition
of outgoing spherical waves in all directions, with the amplitude f
encoding the probability amplitude for each direction.
Remark.
The physical scattering experiment involves not a plane wave
∣k⟩∈Φ∗
but a normalizable wave packet
∣ϕ⟩=∫ϕ^(k′)∣k′⟩d3k′
with ϕ^ sharply peaked around k.
The Lippmann-Schwinger equation applies to each ∣k′⟩ component separately, giving a wave packet
scattering state
∣ϕ+⟩=∫ϕ^(k′)∣k′+⟩d3k′.
In the limit where the wave packet is narrow in k-space, and hence
broad in position space, representing a well-defined beam, the scattering
amplitude
f(θ,φ)
for the wave packet approaches that of the plane wave, and the
differential cross section is the correct observable.
The wave packet description is needed to give a rigorous account of the S-matrix, where the in and out states are separated at
t→∓∞,
but for the computation of cross sections the plane wave idealization is
sufficient and gives identical results.
derived in Section 4 encodes the probability amplitude for scattering
from the initial wave vector k to the final direction r^
at fixed energy.
The S-matrix organizes this information into a single unitary operator
that maps the complete set of in-asymptotic states to the complete set of
out-asymptotic states, and whose matrix elements contain all observable
scattering data.
The present section defines the Møller wave operators and the S-matrix,
derives their relationship to the T-matrix and hence to the scattering
amplitude, establishes S-matrix unitarity as a consequence of the QM4
conservation laws, and derives the optical theorem from unitarity.
The optical theorem, that the total cross section is determined by the
forward scattering amplitude, is the most direct physical consequence of
probability conservation in the scattering problem.
The Møller wave operators map free asymptotic states to the corresponding
full scattering states, making precise the statement that the scattering
state
∣k+⟩
"looks like" the free state
∣k⟩
in the remote past and the scattering state
∣k−⟩
"looks like" the free state
∣k⟩
in the remote future.
Definition. Møller wave operators.
The Møller wave operators are the strong limits:
Equation. Møller wave operator Ω+.
Ω^+:=t→−∞limeiH^relt/Φ0e−iH^0t/Φ0,
and
Equation. Møller wave operator Ω−.
Ω^−:=t→+∞limeiH^relt/Φ0e−iH^0t/Φ0,
where the limits are taken in the strong operator topology on H, i.e.,
Ω^±∣ϕ⟩=t→∓∞limeiH^relt/Φ0e−iH^0t/Φ0∣ϕ⟩
for all
∣ϕ⟩∈H.
The scattering states are related to the free states by:
Equation. Møller operators and scattering states.
∣k+⟩=Ω^+∣k⟩,∣k−⟩=Ω^−∣k⟩,
consistent with the Lippmann-Schwinger equation of Section 4.
Remark.
The Møller operator Ω^+ has a precise physical
interpretation.
A state
∣ϕ⟩∈H
evolved forward from
t=−∞
under the free Hamiltonian H^0, as if there were no interaction,
gives the free evolution
e−iH^0t/Φ0∣ϕ⟩.
The Møller operator Ω^+ maps this free in-asymptotic state
to the fully interacting state
Physically, Ω^+∣ϕ⟩ is the state that, in the
remote past, looks like the free state ∣ϕ⟩ propagating toward
the interaction region.
Similarly, Ω^−∣ϕ⟩ is the state that, in the remote
future, looks like ∣ϕ⟩ propagating away from the interaction
region.
Proposition. Intertwining relations.
The Møller operators satisfy the intertwining relations:
Equation. Intertwining relation.
H^relΩ^±=Ω^±H^0,
as operator identities on H.
As a consequence, Ω^± map eigenstates of H^0 at
energy E to eigenstates of H^rel at the same energy E.
and using the strong convergence of the Møller operators, the limits of
the time derivative vanish, the time derivative in the strong topology
tends to zero for short-range potentials by the Riemann-Lebesgue lemma
applied to the oscillatory integral, giving
Ω^±H^0=H^relΩ^±
in the strong sense.
The consequence follows:
H^rel(Ω^±∣k⟩)=Ω^±H^0∣k⟩=EkΩ^±∣k⟩,
confirming the full scattering-state eigenvalue equation. □
Using the distributional identity for the boundary values of the
resolvent,
(E−E′)+−1−(E−E′)−−1=−2πiδ(E−E′),
gives
⟨k′−∣k+⟩−⟨k′∣k⟩=−2πiδ(Ek′−Ek)⟨k′−∣V^∣k+⟩.
The matrix element
⟨k′−∣V^∣k+⟩=⟨k′∣Ω^−†V^Ω^+∣k⟩=Tk′k
on the energy shell, using the fact that
⟨k′−∣V^∣k+⟩=⟨k′∣T^∣k⟩
for on-shell states, gives the S-T relation. □
Remark.
The energy-conserving delta function
δ(Ek′−Ek)
in the S-T relation expresses energy conservation: non-zero S-matrix elements connect only states at the same energy.
The term
δ(3)(k′−k)
is the "no-scattering", or forward, contribution: the probability
amplitude for no interaction.
The T-matrix term encodes all the non-trivial scattering: the amplitude
for momentum to change from k to k′ at fixed energy.
For elastic scattering,
∣k′∣=∣k∣=k,
the on-shell condition
Ek′=Ek
restricts k′ to the sphere of radius k in momentum space; the
angular distribution of Tk′k on this sphere is
exactly the scattering amplitude:
where the first equality uses continuity of the norm, the second uses the
unitarity of
eiH^relt/Φ0,
Stone's theorem, QM4 Theorem 3.1, and the third uses the unitarity of
e−iH^0t/Φ0.
Therefore,
Ω^+†Ω^+=1^H,
and identically for Ω^−.
Step 2: Asymptotic completeness.
The Møller operators are isometries, so they are partial isometries. Asymptotic completeness asserts that both have the same range:
range(Ω^+)=range(Ω^−)=Hac(H^rel),
the absolutely continuous spectral subspace of H^rel.
For short-range potentials,
∣V(r)∣≤C(1+∣r∣)−1−ϵ,
it is proved that
Hac(H^rel)=H,
with no singular continuous spectrum and no positive-energy bound states,
so Ω^± are unitary onto H.
Step 3: Unitarity of S.
S†S=(Ω^−†Ω^+)†(Ω^−†Ω^+)=Ω^+†Ω^−Ω^−†Ω^+.
Since Ω^− is unitary onto H by asymptotic
completeness,
Ω^−Ω^−†=1^H,
giving
S†S=Ω^+†1^Ω^+=Ω^+†Ω^+=1^H.
Similarly,
SS†=Ω^−†Ω^+Ω^+†Ω^−=Ω^−†1^Ω^−=1^H.
□
Remark.
The unitarity
S†S=1^
encodes probability conservation for the scattering process.
In the partial wave basis, where S is block-diagonal by angular momentum
conservation, the ℓ-th block is the 1×1 complex number
Sℓ(k)=e2iδℓ(k).
Unitarity requires
∣Sℓ(k)∣=1,
i.e., the phase shift δℓ(k) is real.
This is the statement that no probability is absorbed in any partial wave
channel: the interaction changes the phase of the scattered wave in each
channel but does not reduce its amplitude.
If the potential had an imaginary part,
Im(V)<0,
the S-matrix would not be unitary, and the total cross section would
include an absorption cross section; this is the optical model used in
nuclear physics to describe inelastic processes.
Theorem. Optical theorem.
For elastic scattering from a real central potential, the total cross
section σtot and the forward scattering amplitude
f(0)=f(θ=0)
satisfy:
Equation. Optical theorem.
σtot=k4πImf(0).
Proof.
Starting from the S-matrix unitarity
S†S=1^,
take matrix elements between the plane wave states ∣k⟩:
Equation. Unitarity matrix element.
k′′∑⟨k∣S†∣k′′⟩⟨k′′∣S∣k⟩=δ(3)(0),
where the sum is the integral
∫d3k′′
and the right-hand side is
⟨k∣k⟩=δ(3)(0).
Substitute the S-T relation:
Sk′k=δ(3)(k′−k)−2πiδ(Ek′−Ek)Tk′k.
Inserting into the unitarity matrix element and cancelling the δ(3) terms, the leading non-trivial identity becomes,
suppressing distributional factors,
Remark.
The optical theorem has a compelling physical interpretation.
The forward scattering amplitude
f(0)
at
θ=0
describes scattering in the same direction as the incident beam.
Interference between the incident wave
eikz
and the forward scattered wave
f(0)reikr
produces a shadow behind the scatterer, a reduction in the intensity of
the forward beam.
Probability conservation, unitarity of S, requires that the total
probability removed from the forward beam equals the total probability
scattered in all directions, giving
σtot=k4πImf(0).
This relation is a direct consequence of unitarity and holds for any
potential, relativistic or non-relativistic, elastic or inelastic: it is a
theorem of S-matrix theory, not a special property of any particular
interaction.
Remark.
In the partial wave basis, the optical theorem takes a particularly
transparent form.
Using the partial wave expansion
f(θ)=k1ℓ∑(2ℓ+1)eiδℓsinδℓPℓ(cosθ),
the forward amplitude is
f(0)=k1ℓ∑(2ℓ+1)eiδℓsinδℓ,
using
Pℓ(1)=1.
The imaginary part is
Imf(0)=k1ℓ∑(2ℓ+1)sin2δℓ.
Substituting into the optical theorem gives
σtot=k24πℓ∑(2ℓ+1)sin2δℓ,
which exactly reproduces the total cross-section formula of Section 7.
The optical theorem is therefore the global consistency condition that
relates the forward amplitude, determined by the complex exponentials
eiδℓsinδℓ,
to the total cross section, determined by the real quantities
is an implicit equation for the scattering state ∣k+⟩: the right-hand side depends on ∣k+⟩ through the term
G0+V^∣k+⟩.
When the potential V^ is weak relative to the kinetic energy, in a
sense made precise below, the scattering state ∣k+⟩
is close to the free state ∣k⟩, and the iterative
substitution of the Lippmann-Schwinger equation into itself generates a
convergent series in powers of V^, the Born series.
The first term of this series, in which ∣k+⟩ is
replaced by ∣k⟩ on the right-hand side, is the Born
approximation: the simplest and most widely used approximation in
scattering theory.
The Born approximation expresses the scattering amplitude directly as the
Fourier transform of the potential at the momentum transfer, making
explicit the connection between the spatial structure of the potential and
the angular distribution of the scattered wave.
The n-th Born approximation is the truncation of the Born series to
the first n terms.
The first Born approximation, the Born approximation, is
Equation. First Born approximation.
T^(1):=V^,
with corresponding on-shell matrix element
Tk′k(1)=⟨k′∣V^∣k⟩.
Remark.
The Born series converges in operator norm when
∥V^G0+∥op<1.
For a potential
V∈L2(R3),
the operator V^G0+ has norm bounded by
∥V^G0+∥op≤Ek1/2C∥V∥L2
for some constant C, so the Born series converges at sufficiently high
energy Ek regardless of the potential strength.
At low energy or for strong potentials, the series may diverge; in such
cases the Lippmann-Schwinger equation must be solved exactly or by other
resummation methods.
The physical content of the Born approximation, that the scattered wave is
linear in the potential, is equivalent to treating the interaction as a
single-scattering event: the incident plane wave ∣k⟩
scatters from the potential once, producing the outgoing wave.
The second Born term
V^G0+V^∣k⟩
represents scattering twice from the potential, with the wave propagating
freely between the two scattering events, and so on.
Remark.
The Born scattering amplitude reveals the central structural content of
the Born approximation: the scattering amplitude at momentum transfer q is proportional to the Fourier transform of the potential at
the same q.
Large-angle scattering,
θ≈π,
large
∣q∣≈2k,
probes the short-distance, large-q, Fourier components of the
potential.
Small-angle scattering,
θ≈0,
small
∣q∣,
probes the long-range, small-q, behavior.
For a potential of range a, concentrated in a ball of radius a,
V(q)
is approximately constant for
∣q∣≲a1
and decays for
∣q∣≫a1.
The Born cross section is therefore approximately isotropic, forward-backward
symmetric, for
ka≪1,
the potential range is much less than the wavelength, and sharply peaked
in the forward direction for
ka≫1,
the wavelength is much smaller than the range.
The Born approximation is the quantum mechanical analogue of the classical
impulse approximation: the potential is treated as a perturbation that
kicks the particle once without significantly deflecting it from its
original trajectory.
¶ Born Amplitude for the Yukawa and Coulomb Potentials
The Born approximation is evaluated in closed form for the two physically
most important potentials.
Proposition. Born amplitude for the Yukawa potential.
For the Yukawa potential
confirming the Yukawa Fourier transform.
Substituting into the Born scattering amplitude gives the Yukawa Born
amplitude, and squaring gives the Yukawa Born differential cross section. □
Proposition. Rutherford formula as the Coulomb Born limit.
In the Coulomb limit
λ→0
of the Yukawa potential, i.e.,
V(r)=−re2
for same-sign charges with
g2=e2,
the Born differential cross section becomes the Rutherford formula:
Equation. Rutherford differential cross section.
dΩdσ∣∣∣∣∣Rutherford=(2Φ02k2μe2)2sin42θ1.
The Born approximation gives the exact result for Coulomb scattering:
the classical Rutherford formula is the correct quantum mechanical
differential cross section for the Coulomb potential at all energies.
confirming the Rutherford formula.
The exactness of the Born result for the Coulomb potential is a special
property of the 1/r singularity: the Coulomb scattering amplitude can be
computed exactly using parabolic coordinates, the Gordon-Coulomb method,
and the result agrees with the Rutherford formula for the differential
cross section at all energies and angles.
The forward divergence
dΩdσ→∞asθ→0
is not an artifact of the Born approximation but a genuine property of the
Coulomb potential: the infinite range of the 1/r potential scatters all
impact parameters, including arbitrarily large ones, producing the forward
divergence in the cross section.
The total cross section
σtot=∫dΩdσdΩ
is therefore infinite for the Coulomb potential, reflecting the fact that
the Coulomb force never vanishes and the scattering of configurations at
arbitrarily large impact parameters contributes to the cross section. □
and the condition becomes the low-energy Born validity condition.
High energy.
For
k∣r′∣≫1,
the oscillating factor
eik∣r′∣
partially cancels the integral, by the Riemann-Lebesgue lemma, and a more
careful estimate gives the high-energy Born validity condition as the
dominant condition. □
Remark.
The second Born approximation
T^(2)=V^+V^G0+V^
gives the next correction to the scattering amplitude:
involves a double spatial integral with the free Green's function between
the two potential insertions; it is generally complex-valued even for a
real potential.
The imaginary part of
fBorn(2)(0)
in the forward direction provides the leading Born-series contribution to
the optical theorem when
fBorn(1)(0)
is real, as for a purely real symmetric potential.
The second Born term is required to satisfy the optical theorem at the
level of the first Born approximation, since
fBorn(1)
is real and hence has zero imaginary part, while
σtot(1)=∣fBorn(1)∣2
is non-zero.
This apparent inconsistency is resolved by the fact that the optical
theorem is an exact statement, while the Born expansion is an
approximation; at each order of the Born series, the optical theorem is
satisfied by the sum of all terms contributing at that order.
Remark.
For a central potential, the Born amplitude can be expressed directly in
terms of the partial wave phase shifts by comparing with the partial wave
expansion.
For small phase shifts,
δℓ≪1,
which is the Born validity condition in the partial wave language,
eiδℓsinδℓ≈iδℓ,
purely imaginary, and the partial wave amplitudes are purely imaginary to
leading order.
The Born approximation to the phase shift is
Equation. Born approximation to the phase shift.
δℓBorn(k)=−Φ022μk∫0∞[jℓ(kr)]2V(r)r2dr,
obtained by substituting the free spherical wave
jℓ(kr)
for the exact radial wave function in the phase shift integral.
This equation is the direct link between the Born approximation, a Fourier
transform of the full potential, and the partial wave analysis, a radial
integral of the potential weighted by the spherical Bessel functions: the
Rayleigh expansion converts one into the other.
The Born approximation of Section 6 expresses the scattering amplitude as
a global Fourier transform of the potential, valid when the interaction is
weak.
The partial wave analysis of the present section is exact: it decomposes
the scattering problem for a central potential
V=V(r)
into independent one-dimensional radial problems, one for each angular
momentum channel
ℓ∈{0,1,2,…},
each characterized by a single real number, the phase shift
δℓ(k).
The central potential conserves angular momentum,
[H^rel,L^2]=[H^rel,L^z]=0,
so the full three-dimensional scattering problem decomposes into countably
many independent one-dimensional problems.
Each phase shift δℓ is the solution of the radial
Schrödinger equation in the ℓ-th channel and encodes the complete
effect of the potential on that channel.
The scattering amplitude and all cross sections are then reassembled from
the phase shifts by the partial wave series.
from QM5 and the relation between the full Laplacian and the radial plus
angular parts.
The Schrödinger equation then separates by orthogonality of the Legendre
polynomials into one equation per ℓ, giving the radial Schrödinger
equation.
The boundary condition
uℓ(0)=0
ensures
ψ∈Lloc2(R3)
near the origin: without it,
ruℓ
would diverge at
r=0.
The asymptotic condition is established below by comparing the large-r
solution of the radial Schrödinger equation to the free solution. □
Remark.
The radial equation is a one-dimensional Schrödinger equation on (0,∞) with an effective potential:
Equation. Effective potential.
Ueff(r)=U(r)+r2ℓ(ℓ+1).
The centrifugal term
r2ℓ(ℓ+1)
is the repulsive barrier that prevents the particle from approaching the
origin in channels with
ℓ≥1.
For
r→0,
the centrifugal term dominates for ℓ≥1, and the regular solution
behaves as
uℓ(r)∼rℓ+1,
suppressed at the origin for high ℓ.
For
r→∞,
one has
U(r)→0
for short-range potentials and
r2ℓ(ℓ+1)→0,
so the dimensionless radial Schrödinger equation reduces to the free
equation
uℓ′′+k2uℓ=0.
The general solution at large r is
uℓ(r)→Aℓsin(kr)+Bℓcos(kr),
or equivalently
uℓ(r)→Cℓsin(kr−2ℓπ+δℓ),
where the phase offset
−2ℓπ
is the free-wave phase, from the asymptotic form of jℓ(kr), and δℓ is the additional phase shift induced by the potential.
and angular momentum quantum number ℓ, the phase shift
δℓ(k)∈R
is defined by the asymptotic condition on the unique, up to normalization,
solution uℓ(r) of the radial Schrödinger equation that is regular
at the origin:
Equation. Phase shift definition.
uℓ(r)r→∞sin(kr−2ℓπ+δℓ(k)).
Equivalently, writing
uℓ(r)→Aℓsin(kr−2ℓπ)+Bℓcos(kr−2ℓπ),
one has:
Equation. Tangent of phase shift.
tanδℓ(k)=AℓBℓ.
For
V≡0,
no potential,
uℓ(r)=krjℓ(kr)→sin(kr−2ℓπ),
giving
δℓ=0,
no scattering, no phase shift.
The physical meaning of the phase shift is the shift in the zero-crossings
of the radial wave function relative to the free wave: the potential pulls
the zero-crossings inward, for an attractive potential,
δℓ>0,
or pushes them outward, for a repulsive potential,
Theorem. Partial wave expansion of the scattering amplitude.
For a central potential
V=V(r),
the scattering amplitude has the partial wave expansion:
Equation. Partial wave expansion.
f(θ)=k1ℓ=0∑∞(2ℓ+1)eiδℓsinδℓPℓ(cosθ).
The partial S-matrix element in each channel is
Sℓ(k)=e2iδℓ(k),
and the partial wave scattering amplitude is
fℓ=2ike2iδℓ−1=keiδℓsinδℓ.
The total and partial wave cross sections are:
Equation. Partial wave cross section.
σℓ=k24π(2ℓ+1)sin2δℓ,
and
Equation. Total cross section from partial waves.
σtot=ℓ=0∑∞σℓ=k24πℓ=0∑∞(2ℓ+1)sin2δℓ.
Each partial wave cross section satisfies the unitarity bound:
σℓ≤k24π(2ℓ+1).
Proof. Step 1: Asymptotic matching.
At large r, the scattering wave function ψ(r) must
simultaneously match the Rayleigh expansion of the incident wave plus an
outgoing spherical wave, and the asymptotic form of the radial wave
functions.
Proposition. Properties of the phase shifts.
The phase shifts δℓ(k) satisfy the following.
Reality:
δℓ(k)∈R
for all ℓ and
k>0,
for a real short-range potential.
Low-energy limit:
δℓ(k)→0
as
k→0
for
ℓ≥1,
and for
ℓ=0,
the s-wave,
Equation. Scattering length expansion.
δ0(k)=−kas+O(k3)as k→0,
where as is the scattering length, giving
σtot→4πas2
as
k→0.
High-energy limit:
δℓ(k)→0
as
k→∞
for any fixed ℓ, for a short-range potential.
Levinson's theorem:
Equation. Levinson's theorem.
δℓ(k=0)=nℓπ,
where
nℓ∈{0,1,2,…}
is the number of bound states with angular momentum ℓ supported by
the potential V.
Proof. Part 1: Reality.
The dimensionless radial equation with real U(r) and real boundary
condition
uℓ(0)=0
has a real solution
uℓ(r)∈R
for all r.
The asymptotic form
uℓ(r)→sin(kr−2ℓπ+δℓ)
with uℓ real forces
δℓ∈R.
Part 2: Low-energy limit.
For
ℓ≥1,
the centrifugal barrier
r2ℓ(ℓ+1)
excludes the particle from the interaction region at low energy,
suppressing the phase shift:
δℓ(k)=O(k2ℓ+1)
as
k→0,
the Wigner threshold law.
For
ℓ=0,
the s-wave radial equation
u0′′+[k2−U(r)]u0=0
has the low-k expansion
u0(r)→(r−as)+O(k2)
at large r.
With the convention
u0(r)→sin(kr+δ0)≈kr+δ0≈k(r−as)
as k→0, the scattering length as is defined by
tanδ0(k)→−kas
as k→0, consistent with the scattering length expansion.
The low-energy total cross section is
σtot→σ0=k24πsin2δ0≈k24π(kas)2=4πas2.
Part 3: High-energy limit.
At high energy,
k→∞,
the potential U(r) is a relatively small perturbation of the kinetic
energy k2, and the Born approximation gives
δℓBorn(k)→0
as
k→∞,
by the Riemann-Lebesgue lemma applied to the oscillatory integral
∫[jℓ(kr)]2U(r)r2dr.
Part 4: Levinson's theorem.
At
k=0,
the dimensionless radial equation reduces to
uℓ′′−[U(r)+r2ℓ(ℓ+1)]uℓ=0.
By Sturm-Liouville theory applied to the sequence of solutions at
decreasing
k→0,
as k decreases from ∞ to 0, the phase shift δℓ(k) changes continuously and decreases by π each time a
bound state passes through zero energy.
Starting from
δℓ(k=∞)=0
and finishing at
k=0,
the net change is
−nℓπ,
giving
δℓ(0)=nℓπ.
□
Remark.
Levinson's theorem connects the scattering sector,
E>0,
phase shifts, to the bound state sector,
E<0,
discrete eigenvalues, completing the spectral analysis of H^rel that was begun in QM7.
In QM7, the coupled oscillator Hamiltonian had
nℓ=0
for all channels, no bound states in the scattering sense.
For the hydrogen potential
V(r)=−re2,
the s-wave,
ℓ=0,
supports countably infinite bound states, the
n=1,2,3,…
levels of QM5, so
n0=∞
and
δ0(0)=∞,
though the Coulomb case requires the modified Levinson theorem.
For the Yukawa potential with coupling g2 and range λ−1,
n0=0
for
Φ02λg2μ<8π2,
no s-wave bound state, and
n0=1
for larger coupling, giving
δ0(0)=0
or
π
respectively.
The phase shift at k=0 is thus a topological invariant of the potential:
it counts the number of bound states modulo π.
Remark.
When a phase shift δℓ(k) passes through
2π
at some energy Eres, the partial wave cross section σℓ reaches its maximum,
kres24π(2ℓ+1),
the unitarity bound.
Near the resonance energy, the phase shift has the Breit-Wigner form:
Equation. Breit-Wigner form.
e2iδℓ(k)sinδℓ(k)≈−E−Eres+iΓ/2Γ/2,
where Γ is the resonance width, the inverse lifetime of the
quasi-bound state.
A resonance corresponds to a pole of the S-matrix at
Eres−iΓ/2
in the complex energy plane: the scattering state spends a time
τres=Γ2Φ0
in the interaction region before escaping.
The full development of resonance theory, the Breit-Wigner formula for
the cross section, the time delay interpretation, and the connection to
complex poles of the S-matrix, is deferred; the present remark records
the structure for future reference.
Remark.
For a potential of range a, with
V(r)≈0
for
r>a,
the partial wave expansion converges rapidly: the phase shifts δℓ are negligibly small for
ℓ≫ka,
since the centrifugal barrier
r2ℓ(ℓ+1)
excludes the particle from the interaction region when
ℓ(ℓ+1)≫k2a2.
The number of significant partial waves is therefore approximately
ℓmax≈ka.
For low energy,
ka≪1,
only the s-wave contributes, producing isotropic scattering.
For high energy,
ka≫1,
many partial waves contribute and the Born approximation of Section 6
becomes the more efficient approach.
This complementarity, partial waves for low energy and Born approximation
for high energy, is the organizing principle of non-relativistic
scattering theory.
The preceding sections treated the scattering of a spinless transport
closure configuration from a central potential: the Hilbert space was
Hrel=L2(R3)
and the observables were the differential cross section and phase shifts.
For a spin-21 configuration scattering from a target that
exerts a spin-orbit force, the correct Hilbert space is
Hfull=H⊗Hspin=L2(R3,C2)
as in QM8 Definition 5.1, and the interaction Hamiltonian includes the
spin-orbit coupling
ξ(r)L^⋅S^
of QM8 Section 7.
The post-scattering state is an entangled spatial-spin state in Hfull; the spin state of the scattered particle
is described by the reduced density matrix obtained by tracing out the
spatial degree of freedom, as in QM9 Definition 4.2.
The present section develops the complete spin-dependent scattering
formalism: the 2×2 spin scattering amplitude matrix, the
differential cross section for polarized and unpolarized incident beams,
and the post-scattering spin density matrix and polarization.
On the full Hilbert space Hfull, the interaction
Hamiltonian for a spin-21 configuration scattering from a
central spin-orbit potential is:
Equation. Spin-orbit scattering Hamiltonian.
V^SO=V(r)⊗I+ξ(r)L^⋅S^,
where
L^⋅S^=j∑(L^j⊗I)(1^⊗S^j)
is the orbital-spin dot product operator, as in QM8 Definition 7.1.
The free Hamiltonian on Hfull is
H^0,full=H^0⊗I,
so the full Hamiltonian is
H^rel,full=H^0,full+V^SO.
Remark.
The spin-orbit coupling
ξ(r)L^⋅S^
splits each angular momentum channel
ℓ≥1
into two sub-channels indexed by total angular momentum
j=ℓ±21,
as established in QM8 Proposition 7.2.
In the coupled basis
∣j,mj⟩,
the interaction Hamiltonian is diagonal with effective radial potential:
Equation. Spin-orbit effective radial potential.
Vj(r)=V(r)+2ξ(r)Φ02[j(j+1)−ℓ(ℓ+1)−43],
giving phase shifts
δℓ,+(k)
for
j=ℓ+21,
and
δℓ,−(k)
for
j=ℓ−21,
that are in general different.
For
ξ≡0,
no spin-orbit coupling,
δℓ,+=δℓ,−=δℓ,
so spin does not affect scattering.
For
ξ=0,
the two sub-channels have different effective potentials and hence
different phase shifts; this splitting is the signature of the spin-orbit
interaction in the scattering data.
For spin-dependent scattering, the scattering amplitude becomes a 2×2 matrix acting on the spin Hilbert space
Hspin=C2,
encoding both non-spin-flip and spin-flip transitions.
Definition. Spin scattering amplitude matrix.
For a spin-21 configuration scattering from the spin-orbit
potential above, the spin scattering amplitude matrix is the 2×2 operator on Hspin:
Equation. Spin scattering amplitude matrix.
F(θ)=A(θ)I+B(θ)(n^⋅σ),
where:
n^ is the unit vector normal to the scattering plane,
n^=∣kin×kout∣kin×kout=sinθk^×k^′;
A(θ) is the non-spin-flip amplitude, the diagonal element in
the spin basis;
B(θ) is the spin-flip amplitude, the off-diagonal element
coupling spin-up to spin-down and vice versa through n^⋅σ.
Proposition. Partial wave expressions for A and B.
In terms of the spin-orbit phase shifts
denotes the associated Legendre function in the convention used here.
The s-wave,
ℓ=0,
contributes only to A since
P01=0;
there is no spin-flip amplitude for s-wave scattering.
For
ξ≡0,
one has
δℓ,+=δℓ,−=δℓ,
so
B=0
and
A=f(θ),
the spinless scattering amplitude of Section 7.
Proof stub.
Decompose the scattering state in the coupled basis
∣ℓ,s=21;j,mj⟩
using the Clebsch-Gordan coefficients of QM8.
In this basis the spin-orbit Hamiltonian is diagonal with phase shifts δℓ,+ and δℓ,−.
The scattering amplitude matrix is obtained by summing the contributions
from the
j=ℓ+21
and
j=ℓ−21
channels and then decomposing the result in the Pauli basis.
The sum over magnetic substates produces Pℓ(cosθ) for the
diagonal amplitude A and Pℓ1(cosθ) for the spin-flip
amplitude B.
When
ξ=0,
the two phase shifts coincide, so their difference vanishes and therefore
B=0,
while A reduces to the spinless partial wave amplitude. □
Remark.
The structure
F=AI+B(n^⋅σ)
is the most general 2×2 matrix consistent with the symmetries of
the scattering problem.
Invariance under rotations about the beam axis k^ restricts the
spin structure to operators built from the Pauli matrices and the
available vectors
k^,k^′,n^.
Time-reversal invariance of the interaction forces the coefficients of
k^⋅σ
and
k^′⋅σ
to vanish on the energy shell, leaving only
I
and
n^⋅σ.
Parity invariance further constrains A to be even in n^ and B
to be odd, consistent with the partial wave expressions above.
This group-theoretic argument establishes that two complex functions,
A(θ)andB(θ),
are sufficient to describe all spin-dependent scattering from a
rotationally and time-reversal invariant potential, independently of the
specific form of the potential.
is the incident spin state.
The scattered flux in direction r^ is proportional to
∥F(θ)∣χin⟩∥2=⟨χin∣F†F∣χin⟩.
For a mixed incident spin state
ρin=i∑pi∣χi⟩⟨χi∣,
averaging over the mixture gives
dΩdσ=i∑pi⟨χi∣F†F∣χi⟩=Tr(FρinF†),
which is the spin-dependent differential cross section.
For the unpolarized case,
ρin=21I,
so
dΩdσ=21Tr(FF†).
Using
F=AI+B(n^⋅σ),
one obtains
FF†=∣A∣2I+∣B∣2(n^⋅σ)2+AB∗(n^⋅σ)+A∗B(n^⋅σ).
Since
(n^⋅σ)2=I
and
Tr(n^⋅σ)=0,
it follows that
Tr(FF†)=2(∣A∣2+∣B∣2),
giving the unpolarized cross section.
For the polarized beam, one substitutes
ρin=2I+m^⋅σ
into the trace expression.
Using the Pauli identities
Tr(σjσk)=2δjk
and
(n^⋅σ)2=I,
the result reduces to the stated polarized cross section.
In the standard spin-orbit scattering convention, the interference term is
proportional to
2Im(A∗B)(m^⋅n^),
giving the analyzing power above. □
Convention note.
If the amplitude is written without the standard factor of i in the
spin-flip structure, namely as
F=AI+B(n^⋅σ),
then the same calculation gives a convention-dependent real-part form.
If one writes the standard Wolfenstein/Mott convention as
F=AI+iBW(n^⋅σ),
then the analyzing power is
A(θ)=∣A∣2+∣BW∣22Im(A∗BW).
Thus the real-versus-imaginary appearance depends on how the phase factor
is assigned to the spin-flip amplitude. The physical observable is the
relative phase between the non-flip and spin-flip amplitudes.
¶ The Post-Scattering Spin Density Matrix and Sherman Function
Theorem. Post-scattering spin density matrix and Sherman function.
For an incident spin state ρin, the post-scattering spin
density matrix, the reduced density matrix of the spin degree of freedom
at scattering angle θ after tracing out the spatial coordinate, is:
Equation. Post-scattering spin density matrix.
ρout(θ)=TrHspin(F(θ)ρinF†(θ))F(θ)ρinF†(θ).
For an unpolarized incident beam,
ρin=21I,
the post-scattering density matrix is, in the standard spin-orbit
scattering convention,
Equation. Post-scattering density matrix for an unpolarized beam.
ρout=21(I+Pout(n^⋅σ)),
where
Equation. Sherman function.
Pout(θ)=∣A∣2+∣B∣22Im(A∗B)
is the Sherman function, or polarization function: the degree of
polarization acquired by the scattered beam along n^ from an
initially unpolarized beam.
Proof.
The full post-scattering state in the asymptotic region is schematically
∣Ψout⟩∝reikr∣r,r^⟩⊗(F∣χ⟩),
where the spatial part is the outgoing spherical wave and the spin part
has been transformed by the spin amplitude matrix F.
For a mixed incident state
ρin=χ∑pχ∣χ⟩⟨χ∣,
the spin part of the post-scattering density matrix is proportional to
χ∑pχF∣χ⟩⟨χ∣F†=FρinF†.
Tracing out the spatial degree of freedom leaves the same spin operator,
and normalization by its trace gives the post-scattering spin density
matrix.
For an unpolarized beam,
ρin=21I,
so
ρout=Tr(FF†)FF†.
In the standard convention, this has the Bloch form
ρout=21(I+Poutn^⋅σ),
with
Pout=∣A∣2+∣B∣22Im(A∗B).
This is exactly the Sherman function. □
Convention note.
With the non-standard convention
F=AI+B(n^⋅σ)
without an explicit i multiplying the spin-flip term, the same
calculation gives
ρout=21I+∣A∣2+∣B∣2Re(A∗B)(n^⋅σ).
Equivalently,
Pout=∣A∣2+∣B∣22Re(A∗B)
in that convention.
The standard Sherman-function expression with Im(A∗B)
is recovered by defining the spin-flip amplitude with the conventional
factor of i.
Remark.
The Sherman function
Pout(θ)
has a direct physical interpretation: it is the degree of polarization
acquired by initially unpolarized spin-21 particles after
scattering through angle θ.
For
ξ≡0,
no spin-orbit coupling,
B=0,
so
Pout=0:
scattering from a spin-independent potential cannot polarize an initially
unpolarized beam.
For
ξ=0,
one generally has
B=0
and therefore
Pout=0;
the scattered beam acquires a net polarization perpendicular to the
scattering plane, along n^.
The degree of polarization is determined by the interference between the
non-flip amplitude A and the spin-flip amplitude B.
Measurement of Pout(θ) at different scattering angles
provides information about B(θ) relative to A(θ), and hence
about the spin-orbit phase shift differences
δℓ,+−δℓ,−
through the partial wave expression for B(θ).
The Sherman function was first derived in the context of electron
scattering from atomic nuclei, Mott scattering, where the Coulomb
potential plus relativistic spin-orbit coupling generates a non-zero
spin-flip amplitude.
Remark.
The post-scattering density matrix is the synthesis of the QM8 and QM9
structures in a physical calculation.
The construction uses three prior results simultaneously:
the full Hilbert space
Hfull=H⊗Hspin
of QM8 Definition 5.1, because the spatial-spin entangled
post-scattering state lives in Hfull;
the partial trace of QM9 Definition 4.2, because tracing out the
spatial degree of freedom gives the spin density matrix ρout;
the Born rule for subsystem observables of QM9 Theorem 4.2, because
the observable
n^⋅σ
in the spin subsystem has expectation value
Tr(ρout(n^⋅σ)).
The Sherman function is thus a physical observable predicted entirely
within the NUVO framework from the interplay of the scattering amplitude,
the Pauli algebra, and the density matrix.
Its measurement in electron or neutron scattering experiments provides a
direct test of the spin-orbit structure of the nuclear or atomic
potential.
The present section collects the interpretive constraints governing the
scattering analysis of the preceding sections and records the precise
boundary between what the present paper establishes and what is deferred.
Three items are addressed: the status of the rigged Hilbert space as a
mathematical extension rather than a new physical postulate, the scope of
the asymptotic completeness assertion and its consequences for S-matrix unitarity, and the complete inventory of what QM10 establishes
and does not establish.
¶ The Rigged Hilbert Space as a Mathematical Extension
The rigged Hilbert space
Φ⊂H⊂Φ∗
of Section 3 is not a new physical postulate of the NUVO program.
It is the correct mathematical setting for the spectral theory of the free
Hamiltonian
H^0
on
H=L2(R3),
whose continuous spectrum
[0,∞)
does not support normalizable eigenstates.
The Gelfand triple structure is a standard tool of functional analysis,
established in the mathematical literature before its application to
quantum mechanics, and its use here is no different in kind from the use
of the Sobolev space
H2⊂H
to define the domain of the kinetic energy operator in QM4: both are
mathematical refinements of the basic Hilbert space structure that make
precise statements possible about operators with continuous spectra or
unbounded domains.
Every physical prediction of QM10 is expressed as a matrix element
⟨ϕ∣O^∣ψ⟩
between normalizable states
ϕ,ψ∈H.
The experimental cross section
dΩdσ=∣f∣2
is computed from the scattering amplitude
f,
which is the matrix element
⟨k′∣V^∣k+⟩.
In a physical experiment, the initial state is a normalized wave packet
∣ϕ⟩=∫ϕ^(k′)∣k′⟩d3k′
with
ϕ^∈S(R3)⊂L2(R3),
not a plane wave
∣k⟩∈Φ∗.
The plane wave idealization is the limit of an infinitely broad,
spatially uniform beam; it gives the correct cross section formula in the
limit where the wave packet is narrow in k-space, well-defined beam
energy, and broad in position-space, beam diameter much larger than the
target.
No experiment operates with a literal plane wave, and no physical
prediction of QM10 requires one.
The unitarity proof of Section 5 invokes asymptotic completeness: the
assertion that
range(Ω^+)=range(Ω^−)=Hac(H^rel)=H
for short-range potentials.
This is a deep theorem of functional analysis, not a consequence of the
algebraic structure of the Lippmann-Schwinger equation or the Møller
operators.
Two independent proofs exist for short-range potentials: Enss's
phase-space analysis and the Sigal-Soffer Mourre estimate method.
Both require the potential to satisfy
∣V(r)∣≤C(1+∣r∣)−1−ϵ
for some
ϵ>0,
which excludes the Coulomb potential
V=−re2,
which decays only as
r1.
For the Coulomb potential, the Møller operators as defined in Section 5
do not converge: the long-range
r1
tail of the Coulomb potential generates a slowly varying phase that
accumulates without bound as
t→±∞.
The Dollard modification replaces the free evolution
e−iH^0t/Φ0
with a modified evolution
e−iH^0t/Φ0W^C(t),
where W^C(t) is a phase operator that compensates the long-range
Coulomb phase; with this modification, the Coulomb Møller operators
converge and asymptotic completeness holds.
The Rutherford formula derived from the Born approximation is nonetheless
the correct differential cross section for Coulomb scattering at all
energies, a special exactness of the Born approximation for the Coulomb
potential confirmed by the exact Gordon-Coulomb solution.
The Dollard modification affects the S-matrix definition but not the
differential cross section.
QM10 takes the Rutherford formula as established by the Born calculation
and notes the Dollard subtlety without developing the modified Coulomb S-matrix theory.
The present paper establishes the following results, available as inputs
to subsequent QM-series papers.
Rigged Hilbert space and continuum states:
The Gelfand triple
Φ⊂H⊂Φ∗;
plane wave states
∣k⟩
as generalized eigenstates of
H^0;
the generalized eigenvalue equation in distributional sense;
generalized orthonormality
⟨k∣k′⟩=δ(3)(k−k′);
completeness
∫∣k⟩⟨k∣d3k=1^H;
energy-shell completeness with density of states
ρ(E)∝E;
and the Rayleigh expansion of the plane wave in spherical harmonics and
spherical Bessel functions.
Lippmann-Schwinger equation and scattering states:
The free resolvent
G0(z)
and its retarded and advanced boundary values
G0+,G0−;
the position-space form of
G0+
as the outgoing Green's function of the Helmholtz operator; the
Lippmann-Schwinger equations for
∣k+⟩
and
∣k−⟩;
verification of the eigenvalue equation; the position-space Fredholm
integral equation; and the asymptotic form
ψk+(r)→(2π)−3/2[eik⋅r+f(θ,φ)reikr],
with scattering amplitude
f=−2πΦ02μTk′k,
and differential cross section
dΩdσ=∣f∣2.
S-matrix, T-matrix, and optical theorem:
The Møller wave operators
Ω^±
and scattering states
∣k±⟩=Ω^±∣k⟩;
the intertwining relation
H^relΩ^±=Ω^±H^0;
the S-matrix
S=Ω^−†Ω^+;
the T-matrix; the S-T relation; S-matrix unitarity in three
steps, isometry of Ω^±, asymptotic completeness, and
unitarity of S; and the optical theorem
σtot=k4πImf(0),
derived from
S†S=1^.
Born approximation:
The Born series; the n-th Born approximation; the first Born
approximation
T(1)=V^;
the Born scattering amplitude
f(1)(q)=−2πΦ02μV(q);
momentum transfer
∣q∣=2ksin(θ/2);
the Yukawa Born amplitude and differential cross section; the Rutherford
formula as the exact Coulomb Born limit; the forward divergence for an
infinite-range potential; the Born validity conditions at low and high
energy; and the connection to Born phase shifts.
Partial wave analysis:
The radial Schrödinger equation; the effective potential with centrifugal
barrier; boundary conditions; the phase shift
δℓ(k)∈R
from the asymptotic boundary condition; the partial wave expansion
f(θ)=k1ℓ=0∑∞(2ℓ+1)eiδℓsinδℓPℓ(cosθ);
the partial S-matrix
Sℓ=e2iδℓ;
partial and total cross sections; the unitarity bound
σℓ≤k24π(2ℓ+1);
reality of the phase shifts; the Wigner threshold law and scattering
length as; high-energy vanishing; and Levinson's theorem
δℓ(0)=nℓπ.
Spin-dependent scattering:
The spin scattering amplitude matrix
F=AI+B(n^⋅σ);
partial wave expressions for A and B in terms of spin-orbit phase
shifts
δℓ,±;
vanishing of B for
ξ≡0;
the spin-dependent differential cross section for general ρin; the unpolarized result
∣A∣2+∣B∣2;
the analyzing power
A(θ);
the post-scattering spin density matrix
ρout=Tr(FρinF†)FρinF†;
and the Sherman function, with its convention-dependent real-part or
imaginary-part form depending on whether the spin-flip amplitude includes
the standard factor of i.
The following topics are outside the scope of the present paper.
Asymptotic completeness proof.
The theorem that
range(Ω^+)=range(Ω^−)=H
for short-range potentials is invoked but not proved.
The proof requires the Enss phase-space method or the Sigal-Soffer Mourre
estimate, which are beyond the scope of the present paper.
Coulomb scattering with Dollard modification.
The standard Møller operators do not converge for the Coulomb
r1
potential; the Dollard modification is required for a rigorous S-matrix theory.
The Rutherford formula is established by the Born calculation and its
exactness is noted; the full Coulomb scattering theory is deferred.
Resonances and Breit-Wigner formula.
The structure of S-matrix poles in the complex energy plane, the
Breit-Wigner resonance formula, and the time-delay interpretation of
resonances are noted but not developed.
Multi-channel and inelastic scattering.
The coupled-channel S-matrix for processes where the internal states of
the configurations change during scattering, inelastic scattering and
reaction cross sections, is deferred.
Relativistic scattering.
The Dirac equation replaces the Pauli equation for relativistic
spin-21 scattering; the relativistic scattering amplitude, the
Mott cross section, relativistic spin-orbit plus Coulomb, and the QED
radiative corrections are deferred to QM11.
Three-body and N-body scattering.
The Faddeev equations for three-body scattering and the general N-body asymptotic completeness problem are beyond the scope of the
present series.
The present paper has developed the complete non-relativistic quantum
scattering theory for two transport closure configurations interacting
through a short-range potential, within the scalar--conformal NUVO
framework.
The fourteen principal results are as follows.
Rigged Hilbert space and plane waves.
The Gelfand triple
Φ⊂H⊂Φ∗
accommodates the generalized eigenstates
∣k⟩
of
H^0
with
⟨k∣k′⟩=δ(3)(k−k′)
and completeness
∫∣k⟩⟨k∣d3k=1^.
The Rayleigh expansion connects the plane wave and spherical wave bases
via the spherical Bessel functions and spherical harmonics of QM5.
Free resolvent and its position-space form.
⟨r∣G0+∣r′⟩=−Φ022μ4π∣r−r′∣eik∣r−r′∣;
this is the retarded Green's function of the Helmholtz operator,
generating outgoing spherical waves.
Lippmann-Schwinger equation.
∣k+⟩=∣k⟩+G0+V^∣k+⟩
is the unique scattering state with outgoing boundary condition.
Its position-space form is a Fredholm integral equation of the second
kind; the Born series is its iterative expansion.
Asymptotic form and scattering amplitude.
ψk+(r)→(2π)−3/2[eik⋅r+f(θ)reikr]
as
r→∞,
with
f=−2πΦ02μTk′k
and
dΩdσ=∣f∣2.
Møller operators and S-matrix.
S=Ω^−†Ω^+;
the S-T relation and the intertwining relation
H^relΩ^±=Ω^±H^0
connect the free and interacting dynamics.
S-matrix unitarity.
S†S=1^
follows from the isometry of
Ω^±,
the unitarity of time evolution from QM4, and asymptotic completeness,
cited for short-range potentials.
Optical theorem.
σtot=k4πImf(0):
the total cross section equals
k4π
times the imaginary part of the forward scattering amplitude, derived
from S-matrix unitarity.
Born approximation.
f(1)(q)=−2πΦ02μV(q);
the Born amplitude is the Fourier transform of the potential at the
momentum transfer
The results of the present paper are of broad programmatic significance
on three grounds.
The first is the completion of the two-body spectral problem initiated in
QM7.
QM7 analyzed the two-body system in the bound state sector: the coupled
oscillator Hamiltonian with discrete, quantized positive-energy spectrum
and the normal mode Fock eigenstates.
QM10 analyzes the same two-body framework in the scattering sector: the
relative Hamiltonian with continuous positive-energy spectrum and the
plane wave scattering eigenstates.
The two sectors are complementary and together constitute the complete
spectral data of the relative Hamiltonian
H^rel=H^0+V^:
bound states,
E<0,
with discrete spectrum and QM7 Fock structure, and scattering states,
E>0,
with continuous spectrum and QM10 Lippmann-Schwinger structure.
The connection between the two sectors is Levinson's theorem, which is the
first explicit link in the QM-series between the bound state count
nℓ
of QM7 and the scattering phase shifts
δℓ(0)
of QM10.
This structural complementarity is the organizing principle of the
two-body spectral problem and will recur in the relativistic extension of
QM11.
The second ground of significance is that QM10 is the first paper in the
QM-series where all of QM4, QM5, QM7, QM8, and QM9 are used
simultaneously in a single physical calculation.
The spin-dependent scattering analysis requires QM4, unitarity of time
evolution and conservation of angular momentum; QM5, partial wave
decomposition, spherical harmonics, and phase shifts; QM7,
center-of-mass and relative coordinate separation; QM8, the full Hilbert
space
Hfull,
Pauli algebra for
(n^⋅σ)2=I,
and spin-orbit coupling eigenvalues; and QM9, the reduced density matrix
ρout
from partial trace and the Born rule for spin observables.
The Sherman function
Pout(θ)
is the observable whose prediction requires all five inputs
simultaneously: it is the synthesis of the NUVO QM-series in a single
real-valued function that is directly measurable in electron and neutron
scattering experiments.
The derivation of the Sherman function from first principles within the
NUVO framework, from the double-cover holonomy of QM8 through the Pauli
algebra, the spin-orbit splitting of the phase shifts, the partial wave
interference, and the QM9 density matrix, is the sharpest demonstration in
the present series that the program produces concrete physical
predictions from abstract geometric principles.
The third ground of significance is the Born approximation and the
Rutherford formula.
The derivation of the Rutherford formula from the Born approximation
applied to the Coulomb potential reproduces the result that Rutherford
used in 1911 to interpret the gold-foil experiment and infer the nuclear
model of the atom.
In the standard quantum mechanical treatment, the Rutherford formula is
derived either from the exact Coulomb scattering solution, a technically
involved calculation using parabolic coordinates, or from the Born
approximation with the observation that the two agree exactly for the
Coulomb case.
In the NUVO framework, the Born approximation is derived from the
Lippmann-Schwinger equation, which is itself derived from the QM4
dynamical framework applied to the scattering states; the Rutherford
formula emerges as a theorem from this chain.
The historical significance of the Rutherford formula in establishing the
nuclear model of the atom makes it a natural calibration point for the
NUVO scattering theory: the same formalism that gives the Rutherford cross
section in the non-relativistic limit gives the Mott cross section,
Rutherford plus spin-orbit with the Sherman function, in the
spin-orbit-coupled case, and the relativistic Mott cross section, derived
from the Dirac equation, in QM11.
QM11 develops the relativistic extension of the spin-21
framework established in QM8, deriving the Dirac equation as the
relativistic first-principles equation for a spin-21 transport
closure configuration and completing several derivations deferred from
QM8 and QM10.
The primary new structure of QM11 is the Dirac equation:
Equation. Dirac equation preview.
(iΦ0γμ∂μ−mc)Ψ=0,
a first-order relativistic wave equation on a four-component spinor
Ψ=(ΨL↑,ΨL↓,ΨS↑,ΨS↓)T,
large and small components, where the Dirac matrices
γμ=(γ0,γ)
satisfy
{γμ,γν}=2gμν,
the Clifford algebra of Minkowski spacetime, extending the Pauli Clifford
algebra
{σj,σk}=2δjkI
of QM8 Theorem 4.2 to four dimensions.
Three derivations deferred from QM8 are completed in QM11.
First, the
g
-factor
g=2
for the electron spin magnetic moment emerges from the minimal coupling
∂μ→∂μ−Φ0cieAμ
in the Dirac equation.
Second, the non-relativistic spin-orbit Hamiltonian
ξ(r)L^⋅S^
with the correct Thomas factor
21
emerges from the Foldy-Wouthuysen reduction to order
(v/c)2.
Third, the hydrogen fine structure energy
Enj=En[1+n2α2(j+21n−43)]
is derived from the Dirac equation, completing the QM8 fine structure
result with the Darwin term and the relativistic kinematic correction.
The Dirac equation also provides the relativistic scattering formalism for
QM10: the relativistic analogue of the Born amplitude for Coulomb
scattering is the Mott cross section
derived in QM11 from the Dirac-Coulomb scattering amplitude.
Finally, and most significantly, QM11 establishes the spin-statistics
theorem as the fifth holonomy quantization of the NUVO program: the
relativistic holonomy of
SL(2,C),
the double cover of the Lorentz group
SO(3,1),
connects the half-integer spin quantum number of QM8 to the fermionic
exchange parity
χ=−1
of QM7 as a theorem derived from the CPT symmetry of the Dirac equation,
not as a postulate.