The harmonic oscillator occupies a central structural position
in quantum mechanics: it is the simplest non-trivial dynamical
system, the foundation of quantum field theory, and the generating
model for coherent states, squeezed states, and the semiclassical
limit.
In the scalar--conformal NUVO framework, the harmonic oscillator
Hamiltonian
H^osc=2mp^2+21mω2x^2
is a special case of the self-adjoint Hamiltonians established in QM4 with
a Kato-class potential
V(x)=21mω2x2.
The present paper derives its complete structure as a sequence
of theorems from the canonical commutation relations of QM1
and the dynamical framework of QM4.
The central algebraic tool is the pair of ladder operators
a^=2mωΦ0p^+imωx^
and
a^†=2mωΦ0p^−imωx^,
which satisfy
[a^,a^†]=1^
and decompose the Hamiltonian as
H^osc=Φ0ω(N^+211^),
where
N^=a^†a^
is the number operator.
The complete eigenvalue spectrum
En=(n+21)Φ0ω
for
n∈{0,1,2,…}
and the energy eigenstates, Hermite-Gaussian functions, are derived
from the ladder operator algebra by the same technique as the angular
momentum spectrum in QM5.
The energy-time uncertainty relation of QM3 is applied to the
harmonic oscillator to establish the zero-point energy
E0=21Φ0ω
as a structural consequence: a state of strictly zero energy would
violate the uncertainty bound, so the ground state energy is necessarily
positive.
Coherent states ∣α⟩ are defined as eigenstates of
the annihilation operator
a^∣α⟩=α∣α⟩
for
α∈C,
and are shown to be the Gaussian minimum-uncertainty states of QM3
with the oscillator zero-point width
ℓ0=mωΦ0,
propagating under the harmonic oscillator dynamics without change of
shape.
The displacement operator representation
∣α⟩=D^(α)∣0⟩
is established, and the overcompleteness of the coherent state family
is derived.
The paper closes with the three-dimensional isotropic harmonic
oscillator, whose eigenstates are expressed in spherical coordinates
as products of radial Laguerre functions and spherical harmonics
Yℓm(θ,φ)
from QM5.
The energy shell structure
EN=(N+23)Φ0ω
with degeneracy
dN=2(N+1)(N+2)
is derived from the combined radial and angular quantum number constraints.
No new postulates are introduced.
All results follow from the QM1 canonical commutation relations,
the QM4 dynamical framework, the QM3 uncertainty relations, and
the QM5 angular momentum structure.
The scalar--conformal NUVO program has now established its complete
algebraic foundation, QM1 through QM3, its dynamical framework
(QM4), and its first major physical sector, QM5: angular momentum
and the hydrogen spectrum.
The present paper, QM6, develops the second major physical sector:
the quantum harmonic oscillator.
The oscillator occupies a distinctive position within the QM-series
that is different from the hydrogen atom.
The hydrogen atom is the primary physical validation of the NUVO
framework, whose energy spectrum was derived in the Q-series and
whose full wave function structure was completed in QM5.
The harmonic oscillator is a structural template: the simplest
dynamical system with a discrete non-degenerate spectrum, the
generating model for the coherent state theory of QM6 through
QM11, and the foundational object from which quantum field theory
builds its multi-excitation structure.
In the scalar--conformal NUVO program, the oscillator Hamiltonian
H^osc=2mp^2+21mω2x^2
is a special case of the self-adjoint Hamiltonians established in
QM4 Theorem 4.2, with the harmonic potential
V(x)=21mω2x2
satisfying the Kato-class regularity conditions of QM4 Definition 3.2.
Its complete structure is derived from prior results without new
physical assumptions.
QM6 closes three program arcs opened in earlier papers and depends
on their results in structurally specific ways.
The first arc runs from QM3 to QM6.
QM3 Theorem 6.1 identified the Gaussian minimum-uncertainty states
as those saturating the Cauchy--Schwarz bound in the Robertson
inequality, and showed that every Gaussian width σ>0
gives a minimum-uncertainty state.
The oscillator dynamics of QM4 and QM6 together select a preferred
width: only the Gaussian with
σ=2ℓ0=2mωΦ0
retains its shape under the harmonic oscillator time evolution.
This dynamical selection of a preferred minimum-uncertainty state
is the coherent state of QM6, and the program arc from QM3 to QM6
is precisely the arc from algebraic characterization, minimum
uncertainty, to dynamical characterization, shape preservation.
The second arc runs from QM4 to QM6.
QM4 established the Ehrenfest theorem: the centroid
(⟨x⟩(t),⟨p⟩(t))
of any closure state follows the classical equations of motion.
For the harmonic oscillator, the classical equation is
x¨+ω2x=0,
so the centroid traces a classical oscillation.
The distinctive feature of coherent states, established in QM6, is
that not only the centroid but the entire Gaussian profile propagates
classically: the widths Δx and Δp are constant
in time, and the state at each time is a displaced Gaussian identical
in shape to the initial state.
The third arc runs from QM5 to QM6.
The ladder operator technique introduced in QM5 for angular momentum---
raising and lowering the L^3 eigenvalue by Φ0
while preserving the L^2 eigenvalue---recurs here for
energy, raising and lowering the energy eigenvalue by
Φ0ω.
The algebraic structure is the same; the spectrum is different,
semi-infinite rather than finite, reflecting the absence of an
upper bound on the energy eigenvalue analogous to the upper bound
∣m∣≤ℓ
for angular momentum.
QM6 is thus the energy-sector analogue of QM5, and the two papers
together establish the two canonical ladder algebras of the
QM-series.
The harmonic oscillator is not merely a physical system but a
structural template whose algebraic skeleton propagates forward
throughout the series.
QM7 treats the coupled two-oscillator system and the normal mode
transformation, which is the simplest instance of a linear
canonical transformation and the precursor of the multi-mode
structure of quantum field theory.
QM9 constructs entangled coherent states as non-factorizable
two-mode coherent state superpositions, using the single-mode
coherent state theory of the present paper.
QM10 uses the oscillator algebra to model radiation field modes
in the derivation of scattering cross-sections with photon emission
and absorption.
QM11 extends the oscillator to the relativistic Klein--Gordon
oscillator and establishes the connection to the free field modes
of the RQM-series.
In each of these, the specific results of the present paper---the
Fock state basis, the coherent state family, the displacement
operator, and the overcompleteness relation---are the structural
inputs rather than background material.
The central objective of the present paper is to derive the complete
structure of the harmonic oscillator in the scalar--conformal NUVO
transport closure framework from the canonical commutation relations
of QM1, the dynamical framework of QM4, the uncertainty relations
of QM3, and the angular momentum structure of QM5.
Specifically, the paper establishes six claims.
The harmonic oscillator Hamiltonian decomposes as
H^osc=Φ0ω(N^+211^),
where the number operator
N^=a^†a^
is self-adjoint and non-negative, and the annihilation and creation
operators
a^=2mωΦ0mωx^+ip^
and
a^†=(a^)†
satisfy
[a^,a^†]=1^,
derived from the canonical commutation relation
[x^,p^]=iΦ0
of QM1.
The complete eigenvalue spectrum of H^osc is
σ(H^osc)={(n+21)Φ0ω:n∈{0,1,2,…}},
derived by the same ladder termination argument as QM5:
non-negativity of N^ forces the eigenvalue sequence to
terminate at n=0, the vacuum
a^∣0⟩=0,
and integer steps above give
n∈Z≥0.
Each eigenvalue is non-degenerate, and the Fock states ∣n⟩
form a complete orthonormal basis for H.
The zero-point energy
E0=21Φ0ω
is derived as a structural consequence of the Heisenberg uncertainty
relation of QM3: any state of the oscillator satisfies
⟨H^osc⟩≥ωΔxΔp≥21Φ0ω,
with equality achieved only by the Gaussian ground state.
The non-zero ground state energy is not postulated but derived.
The position-space energy eigenstates are the Hermite-Gaussian
functions:
Ψn(x)∝Hn(ℓ0x)exp(−2ℓ02x2),
derived by applying
n!(a^†)n
to the ground state
Ψ0(x)∝exp(−2ℓ02x2)
and recognizing the resulting polynomial factor as the Hermite
polynomial Hn(ξ).
Orthonormality follows from self-adjointness of H^osc;
completeness from the spectral theorem of QM1.
Coherent states ∣α⟩ are defined as eigenstates
of a^:
a^∣α⟩=α∣α⟩
for
α∈C.
They are shown to be equivalent to three characterizations: the
Gaussian minimum-uncertainty states of QM3 with oscillator width
σ=2ℓ0;
the states generated from the vacuum by the displacement operator
∣α⟩=D^(α)∣0⟩;
and the states that evolve under H^osc dynamics
without change of shape, with Δx(t) and Δp(t)
constant.
The coherent state family is overcomplete in H, satisfying
the resolution of the identity
π1∫C∣α⟩⟨α∣d2α=1^.
The three-dimensional isotropic harmonic oscillator
H^osc(3)=j=1∑3[2mp^j2+21mω2(x^j)2]
has energy levels
EN=(N+23)Φ0ω
for
N∈{0,1,2,…}
with degeneracy
dN=2(N+1)(N+2),
derived from the Cartesian product of three 1D spectra.
In spherical coordinates, the eigenstates factorize as
unrℓ(r)Yℓm(θ,φ)
with the angular factor given by the QM5 spherical harmonics and
the radial factor expressed in terms of associated Laguerre polynomials.
Claims 1 through 6 are logically ordered.
The ladder decomposition of claim 1 is the algebraic input to
the spectrum derivation of claim 2.
The zero-point energy of claim 3 connects claim 2 to the QM3
uncertainty structure and motivates the identification of the ground
state in claim 4.
The Hermite-Gaussian structure of claim 4 is the position-space
realization of the abstract Fock states of claim 2.
The coherent states of claim 5 are identified as the distinguished
subfamily of the QM3 minimum-uncertainty family that is dynamically
stable under the oscillator of claims 1--4.
The three-dimensional extension of claim 6 combines the 1D structure
of claims 1--4 with the QM5 angular momentum structure to give the full
3D oscillator spectrum and eigenstates.
The present work maintains without modification the interpretive
discipline established throughout the prior series.
Four exclusions are of particular importance for QM6.
The energy spectrum is not postulated.
In many presentations of the harmonic oscillator, the energy
eigenvalues
En=(n+21)ℏω
are stated as a result to be verified by solving the Schrödinger equation,
with the zero-point energy
21ℏω
sometimes described as an empirical feature or a consequence of the
uncertainty principle invoked without derivation.
In the NUVO framework, the spectrum is derived from the ladder
operator algebra via the termination argument of the oscillator-spectrum
theorem: non-negativity of the number operator N^ forces the
spectrum to terminate at n=0, and the integer-step structure of the
ladder above the ground state gives the complete spectrum.
The zero-point energy is then derived in the zero-point-energy theorem
as a structural lower bound from the Heisenberg relation of QM3, not
introduced as a separate assumption.
The Hermite polynomials are not introduced as known special functions.
The polynomial factor
Hn(ℓ0x)
in the energy eigenfunction emerges from applying (a^†)n
to the ground state in position space; the resulting function is recognized
as the n-th Hermite polynomial after the fact, not before.
The Rodrigues formula for Hermite polynomials, which is the classical
definition, emerges from this application of the creation operator
as a consequence rather than as an input.
Coherent states are not defined by their Gaussian form.
The Gaussian form of the coherent state position-space representation
is derived as a consequence of the eigenvalue condition
a^∣α⟩=α∣α⟩,
not assumed.
The equivalence of the three characterizations of coherent states---
eigenstate of a^, Gaussian minimum-uncertainty state, and
displaced vacuum---is established as a theorem rather than taken as
a definition.
The three-dimensional oscillator spectrum is not assumed.
The degeneracy
dN=2(N+1)(N+2)
and the parity selection rule, ℓ and N must have the same
parity, are derived from the explicit quantum number structure of the
Cartesian and spherical decompositions, not stated without derivation.
Section 2 recalls the harmonic oscillator Hamiltonian
and its self-adjointness from QM4, the canonical commutation
relation from QM1, the Gaussian minimum-uncertainty states and
the uncertainty bound from QM3, the Ehrenfest theorem from QM4,
and the spherical harmonics from QM5.
Section 3 introduces the annihilation and creation
operators a^ and a^†, derives the fundamental
commutation relation
[a^,a^†]=1^
from the CCR, decomposes the Hamiltonian as
Φ0ω(N^+211^),
and derives the commutation relations of the number operator with a^ and a^†.
Section 4 derives the complete spectrum
{(n+21)Φ0ω:n≥0}
from the ladder algebra, establishes the zero-point energy
from the uncertainty principle, and records the Fock state matrix
elements.
Section 5 derives the ground state as the Gaussian solving
a^Ψ0=0
in position space, the excited states as Hermite-Gaussian functions,
and establishes their orthonormality and completeness.
Section 6 defines coherent states as eigenstates of a^, derives their Fock-state expansion and Gaussian position
representation, constructs them via the displacement operator,
derives their shape-preserving time evolution, and establishes the
overcompleteness of the coherent state family.
Section 7 develops the three-dimensional isotropic
harmonic oscillator in both Cartesian and spherical representations,
derives the energy shell structure and degeneracy, identifies the
spherical harmonic angular factors from QM5, and records the shell
quantum number constraints and parity selection rule.
Section 8 records the role of the oscillator as
a structural template, the explicit comparison of the QM5 and QM6
ladder algebras, and the scope of the present construction.
Section 9 summarizes the sixteen principal results,
records the programmatic significance of the coherent state
construction and the zero-point energy derivation, and prepares
the transition to QM7.
The present section collects the results from QM1, QM3, QM4, and
QM5 that are directly required for the derivations of Sections 3--7.
Nothing in this section is new.
The section serves two purposes: making the logical dependencies
explicit before the main derivations begin, and recording the
specific forms in which prior results will be applied so that the
proofs of Sections 3--7 can refer back to numbered equations rather
than repeating the recalled content.
The harmonic oscillator Hamiltonian in one spatial dimension is
Equation. Harmonic oscillator Hamiltonian.
H^osc:=2mp^2+21mω2x^2,
where x^ is multiplication by x,
p^=−iΦ0∂x,
m>0 is the mass parameter, and ω>0 is the angular frequency
of the oscillator.
Self-adjointness, from QM4 Theorem 4.2.
The potential
V(x)=21mω2x2
is a Kato-class potential, in the sense of QM4 Definition 3.2: it is
locally square-integrable and satisfies the Kato--Rellich bound with
respect to the kinetic operator
T^=−2mΦ02∂x2.
By QM4 Theorem 4.2, the operator H^osc is
self-adjoint on the Sobolev domain
H2(R)∩D(x^2)
and bounded below.
Unitary time evolution, from QM4 Theorem 3.1.
Stone's theorem, applied to the self-adjoint generator H^osc, gives the strongly continuous unitary group
U^(t)=e−iH^osct/Φ0
on H, well-defined for all t∈R.
Every initial state
Ψ0∈H
evolves to
Ψ(t)=U^(t)Ψ0,
with
∥Ψ(t)∥H=1
for all t if
∥Ψ0∥H=1.
The Ehrenfest equations for the oscillator, from QM4 Proposition 5.1.
For any normalized closure state
Ψ(t)=U^(t)Ψ0∈D(x^)∩D(p^),
the Ehrenfest theorem of QM4 applied to H^osc gives
the coupled equations:
Equation. Ehrenfest equation for position.
dtd⟨x⟩(t)=m⟨p⟩(t),
Equation. Ehrenfest equation for momentum.
dtd⟨p⟩(t)=−mω2⟨x⟩(t).
These are exactly the classical harmonic oscillator equations of
motion with no quantum correction.
The exactness---the absence of higher-moment corrections that appear
for non-quadratic potentials---is a special property of the quadratic
potential: since
∂x2∂2V=mω2
is constant, all higher moments decouple from the centroid equations.
The general solution is:
Equation. Ehrenfest solution for position.
⟨x⟩(t)=⟨x⟩(0)cos(ωt)+mω⟨p⟩(0)sin(ωt),
Equation. Ehrenfest solution for momentum.
⟨p⟩(t)=⟨p⟩(0)cos(ωt)−mω⟨x⟩(0)sin(ωt),
valid for any initial state.
Remark.
The Ehrenfest theorem establishes that the centroid
(⟨x⟩(t),⟨p⟩(t))
follows the classical trajectory for any initial state Ψ0.
It does not, however, determine the time evolution of the widths Δx(t) and Δp(t).
For a general initial state, the widths evolve non-trivially and
the Gaussian shape is not preserved.
The special property of coherent states, established later in the
coherent-state evolution theorem, is that their widths are constant:
Δx(t)=2ℓ0
and
Δp(t)=2p0
for all t.
This shape-preservation property, combined with the Ehrenfest centroid
motion, means that a coherent state at time t is a displaced copy of
the coherent state at t=0, with the displacement following the
classical orbit.
The Ehrenfest theorem is a necessary but not sufficient condition for
this behavior; the full coherent-state analysis of Section 6 is required.
both trivially satisfied since x^ commutes with itself and p^ commutes with itself.
The three relations above are the complete algebraic input to the
derivation of
[a^,a^†]=1^
in the ladder-operator commutator lemma: the oscillator ladder algebra
is a direct consequence of the position-momentum CCR, just as the angular
momentum commutation algebra of QM5 was a direct consequence of the
three-dimensional CCR.
Remark.
The connection between the CCR
[x^,p^]=iΦ0
and the ladder commutation relation
[a^,a^†]=1^
is a canonical transformation in the algebraic sense.
The annihilation operator
a^=2mωΦ0mωx^+ip^
is a complex linear combination of x^ and p^ chosen so
that the commutator [a^,a^†] comes out as a scalar
multiple of the identity---specifically as 1^ rather
than iΦ01^---by absorbing the factor Φ0
into the normalization of the ladder operators.
This normalization choice makes the algebra of a^ and a^† universal: it does not depend on m or ω,
unlike the CCR which contains Φ0.
The energy spectrum then follows from this universal algebra combined
with the specific Hamiltonian decomposition
The following results from QM3 are used directly in Sections 4--6.
The uncertainty bound and its saturation, from QM3 Theorem 3.2 and Proposition 3.4.
For any normalized state
Ψ∈H,
one has
Equation. Heisenberg uncertainty bound.
Δx⋅Δp≥2Φ0,
with equality if and only if
(p^−⟨p⟩)Ψ=iμ(x^−⟨x⟩)Ψ
for some μ∈R with μ>0 for a normalizable state,
the saturation condition of QM3 Proposition 3.4.
Minimum-uncertainty states are Gaussian, from QM3 Theorem 6.1.
For μ>0, the unique normalized solution of the saturation condition
is the Gaussian closure state
The right-hand side of the energy lower bound setup is then bounded
below by
2Φ0ω
using the AM--GM inequality and the Heisenberg uncertainty bound, as
established later in the zero-point-energy theorem.
Remark.
The use of QM3 in QM6 is specifically structured.
QM3 characterizes the minimum-uncertainty states algebraically:
for any σ>0, the Gaussian
ΨG,(σ,⟨x⟩,⟨p⟩)
saturates the Heisenberg bound.
QM6 adds a dynamical filter: among all Gaussians, the harmonic
oscillator dynamics selects
σ=2ℓ0
as the unique width preserved by the time evolution.
The transition from QM3 to QM6 is therefore from a one-parameter
family of minimum-uncertainty states, parametrized by σ,
to a zero-parameter family of preferred states, the single width
2ℓ0,
with only the center
(⟨x⟩,⟨p⟩)
remaining as a free parameter.
This transition is the content of the coherent-state evolution theorem.
¶ Angular Momentum and Spherical Harmonics from QM5
The following results from QM5 are used in Section 7 for the
three-dimensional harmonic oscillator.
Rotational symmetry and angular momentum conservation, from QM4 Proposition 7.3 and QM5 Theorems 3.1--3.2.
For the isotropic three-dimensional harmonic oscillator
H^osc(3)=j∑[2mp^j2+21mω2(x^j)2],
the potential
V(x)=21mω2∣x∣2
is rotationally symmetric.
By QM4 Proposition 7.3:
[H^osc(3),L^j]=0
for all j, so
[H^osc(3),L^2]=0
and the eigenstates can be chosen to be simultaneous eigenstates of
H^osc(3),L^2,L^3.
L^2 in spherical coordinates, from QM5 Proposition 6.1.
The decomposition of the three-dimensional Laplacian in spherical
coordinates gives:
separating the kinetic operator into a purely radial part and a
centrifugal part
2mr2L^2.
This decomposition, combined with the spherical potential
V(x)=21mω2r2,
enables the separation of variables in Section 7.
Spherical harmonics as angular closure eigenstates, from QM5 Theorem 6.2 and Proposition 6.3.
The joint eigenstates of L^2 and L^3 on the unit
sphere S2 are the spherical harmonics
orthonormal on S2, and complete in L2(S2).
Their parity under
x→−x
is
(−1)ℓ,
from QM5 Proposition 6.3.
Remark.
The spherical harmonics from QM5 appear in QM6 not as background
material but as active structural components.
In the spherical representation of the 3D oscillator, Section 7, the
angular factor of each eigenstate is exactly
Yℓm(θ,φ)
from QM5 Theorem 6.2.
The parity selection rule of Section 7, the requirement that ℓ and
the total excitation number N have the same parity, is a direct
consequence of the parity
(−1)ℓ
of the spherical harmonics and the parity of the radial oscillator
functions.
This is the first instance in the QM-series of a result from a physical
sector paper, QM5, serving as a direct structural input to a subsequent
physical sector paper, QM6, rather than merely as a formal prerequisite.
The pattern will continue: QM6 results feed into QM7, QM5 and QM6 results
together feed into QM8, and so forth.
¶ Ladder Operators and the Algebra of the Oscillator
The canonical commutation relation
[x^,p^]=iΦ01^
of QM1 encodes the complete algebraic structure of the harmonic
oscillator when rewritten in terms of two non-self-adjoint operators
adapted to the oscillator's natural scales.
These are the ladder operators a^ and a^†,
which decompose the Hamiltonian into a form
Φ0ω(N^+211^)
from which the entire spectrum can be read off algebraically.
The derivations in the present section are the one-dimensional energy
analogues of the angular momentum ladder derivations of QM5 Section 4:
the same technique of converting a commutation relation among
self-adjoint operators into a raising-lowering algebra among
non-self-adjoint operators, and then reading the spectrum from
the termination of the ladder.
This non-self-adjoint character is essential to the raising and
lowering action: a self-adjoint operator commuting with H^osc would preserve each energy eigenspace, whereas a^ and a^† map between adjacent eigenspaces,
as established in the number-operator commutator lemma and used in
the oscillator-spectrum theorem.
The inverse relations expressing x^ and p^ in terms
of a^ and a^† will be used in Sections 5--6:
Equation. Inverse ladder relations.
x^=2ℓ0(a^+a^†),p^=2ip0(a^†−a^).
These follow by adding and subtracting the annihilation and creation
operator definitions.
¶ The Fundamental Commutation Relation [a^,a^†]=1^
Lemma. Fundamental commutation relation of the oscillator.
On S(R):
Equation. Oscillator ladder commutator.
[a^,a^†]=1^.
Proof.
Expand the commutator using the definitions of a^ and a^†:
differs from the angular momentum commutation relations of QM5 in one
fundamental way: the right-hand side is a scalar multiple of the
identity operator, not another element of the algebra.
This means the oscillator ladder algebra
{1^,a^,a^†}
is the Weyl--Heisenberg algebra, while the angular momentum algebra
{L^1,L^2,L^3}
is the Lie algebra of SO(3).
In the Weyl--Heisenberg algebra,
[a^,a^†]=1^
implies that repeated application of a^† never returns
to the starting state, so the spectrum is semi-infinite, bounded below
and unbounded above.
In the SO(3) algebra,
[L^j,L^k]=iΦ0ϵjklL^l
implies that repeated application of L^+ eventually reaches a
maximum, because
∣m∣≤ℓ,
so the spectrum is finite within each ℓ-multiplet.
The two algebras produce qualitatively different spectral structures
from the same ladder technique.
is the central algebraic result of QM6.
It expresses H^osc entirely in terms of the number
operator
N^=a^†a^,
making the spectrum of H^osc directly readable
from the spectrum of N^: if
N^∣n⟩=n∣n⟩,
then
H^osc∣n⟩=(n+21)Φ0ω∣n⟩.
The task of finding the spectrum of H^osc reduces
to finding the spectrum of N^, which is accomplished by the
ladder operator argument of Section 4.
The
21Φ0ω
offset from the zero-point energy is an immediate consequence of the
211^
term in the Hamiltonian decomposition, which in turn arises from the
imω[x^,p^]
term in the computation of a^†a^: it is the
canonical commutation relation that produces the zero-point energy.
This shows that the annihilation operator in the Heisenberg picture
oscillates at frequency ω, which is the key identity used in
the proof of the coherent-state evolution theorem to show that coherent
states evolve as coherent states.
Remark.
The four commutation relations of the number-operator commutator lemma
are the energy-sector analogues of the angular momentum ladder
commutation relations
[L^3,L^+]=Φ0L^+
and
[L^3,L^−]=−Φ0L^−
of QM5 Lemma 4.1.
In both cases, the commutation relation of the “diagonal” operator,
here N^, in QM5 the component L^3, with the raising
operator has the raising operator itself on the right-hand side with
a positive coefficient; and with the lowering operator with a negative
coefficient.
This is the algebraic signature of a ladder structure.
The difference is in the coefficient: QM5 gives
[L^3,L^+]=Φ0L^+
with coefficient Φ0, while QM6 gives
[N^,a^†]=a^†
with coefficient 1, dimensionless.
The dimensionless coefficient in QM6 reflects the fact that N^
measures eigenvalues in units of Φ0ω, via
H^osc=Φ0ω(N^+21),
while L^3 measures eigenvalues in units of Φ0 directly.
The algebra established in Section 3---the decomposition
H^osc=Φ0ω(N^+211^)
and the commutation relations
[N^,a^]=−a^
and
[N^,a^†]=a^†
---contains the complete spectral information of the harmonic oscillator.
The present section extracts this information by the ladder termination
argument, derives the zero-point energy as a consequence of the
uncertainty principle, and records the matrix elements of the ladder
operators in the Fock basis.
The logical structure is identical to the angular momentum spectrum
derivation of QM5 Section 4: non-negativity of the diagonal operator
forces termination of the lowering sequence, the termination condition
identifies the ground state, and the ladder above the ground state
generates the complete spectrum.
The difference from QM5 is that the spectrum here is semi-infinite,
with no upper termination, as anticipated in the earlier ladder-algebra
comparison.
Step 2: Raising and lowering action.
From the number-operator commutation lemma,
[N^,a^]=−a^
and
[N^,a^†]=a^†.
By the same argument as QM5 Proposition 4.2, if
N^∣n⟩=n∣n⟩,
then, when non-zero,
N^(a^∣n⟩)=(n−1)(a^∣n⟩)
and
N^(a^†∣n⟩)=(n+1)(a^†∣n⟩).
Thus a^ lowers the eigenvalue by 1 and a^† raises it by 1.
Step 3: Termination and identification of the ground state.
Starting from any eigenvalue n≥0 and applying a^ repeatedly,
the sequence of eigenvalues
n,n−1,n−2,…
must terminate at a non-negative value, by Step 1.
Let nmin be the termination point, so
a^∣nmin⟩=0.
Then
N^∣nmin⟩=a^†a^∣nmin⟩=a^†⋅0=0,
giving
nmin=0.
Since the eigenvalue decreases by 1 at each step and terminates
at
nmin=0,
the starting eigenvalue n must be a non-negative integer:
n∈{0,1,2,…}.
Step 4: Non-degeneracy.
Each eigenvalue
n∈{0,1,2,…}
is non-degenerate.
Suppose
N^Ψ=nΨ
for some normalized Ψ.
Then
a^nΨ
is an eigenstate of N^ with eigenvalue 0, by applying Step 2 n times, so
a^nΨ∝∣0⟩.
But the vacuum ∣0⟩ satisfying
a^∣0⟩=0
is unique up to phase: in position space this condition is a first-order
ordinary differential equation with a one-dimensional solution space,
as established in the ground-state theorem.
Therefore Ψ is uniquely determined, up to phase, by n,
confirming non-degeneracy.
The spectrum of H^osc follows from the Hamiltonian
decomposition:
En=Φ0ω(n+21).
□
Remark.
The harmonic oscillator spectrum
σ(H^osc)={(n+21)Φ0ω:n≥0}
and the angular momentum spectrum
σ(L^2)={ℓ(ℓ+1)Φ02:ℓ≥0}
of QM5 both arise from ladder termination arguments, but their structures
reflect the different algebras.
The oscillator spectrum is equally spaced: consecutive eigenvalues
are separated by a fixed gap
Φ0ω
at every level.
This equal spacing is a consequence of
[N^,a^†]=a^†
with unit coefficient: each application of a^† raises
by exactly 1 unit regardless of the current level.
The angular momentum spectrum
ℓ(ℓ+1)Φ02
is not equally spaced: the gap between consecutive levels grows as ℓ increases.
This is because the SO(3) commutation algebra
[L^j,L^k]=iΦ0ϵjklL^l
involves the generators themselves on the right, so the step size
effectively increases with ℓ.
The two spectral structures represent the two canonical types of
quantum spectra: the equally-spaced oscillator spectrum, from the
Weyl--Heisenberg algebra, and the quadratically-growing angular
momentum spectrum, from the SO(3) algebra.
Equality conditions.
Equality in the zero-point lower bound requires equality in both
inequalities.
Equality in the Heisenberg bound requires
Δx⋅Δp=2Φ0,
i.e. Ψ is a Gaussian minimum-uncertainty state, by QM3 Theorem 6.1.
Equality in the AM--GM step requires a=b, i.e.
2m(Δp)2=2mω2(Δx)2,
or equivalently,
Δp=mωΔx.
Combining this with the saturation condition
Δx⋅Δp=2Φ0
gives
Δx⋅mωΔx=2Φ0,
so
(Δx)2=2mωΦ0=2ℓ02,
and therefore
Δx=2ℓ0.
The minimum-uncertainty state with
σ=Δx=2ℓ0
is the Gaussian
ΨG,(ℓ0/2,⟨x⟩,⟨p⟩)
of QM3 Theorem 6.1.
For this to also achieve
⟨H^osc⟩=2Φ0ω,
one further needs
⟨x⟩2=0
and
⟨p⟩2=0,
so that
⟨A2⟩=(ΔA)2
in the energy expectation.
Thus
⟨x⟩=0,⟨p⟩=0.
The ground state is therefore the specific Gaussian with
σ=2ℓ0,⟨x⟩=0,⟨p⟩=0,
which is the state derived in the ground-state theorem.
Its uniqueness, up to phase, confirms the non-degeneracy of the ground
state established in the oscillator-spectrum theorem. □
Remark.
The zero-point energy theorem establishes the zero-point energy as a
structural consequence of the Heisenberg uncertainty relation of QM3
rather than as a computational result read off from the spectrum.
The derivation makes the physical content precise: a state of zero
energy in the harmonic oscillator would require
Δx=0
and
Δp=0,
that is, perfect spatial localization at the equilibrium point and
perfect rest, which would violate
Δx⋅Δp≥2Φ0
from QM3.
The minimum non-zero energy compatible with this bound is
E0=2Φ0ω,
achieved by the Gaussian ground state.
In the NUVO framework, the zero-point energy is therefore not a mysterious
quantum feature but an algebraic consequence of the canonical commutation
relation: the same CCR that gives
[a^,a^†]=1^
and hence the
211^
in
H^osc=Φ0ω(N^+211^)
also gives
Δx⋅Δp≥2Φ0,
and both are manifestations of the same underlying commutation structure.
¶ Matrix Elements of the Ladder Operators in the Fock Basis
The matrix elements of a^ and a^† in the Fock
basis are determined by normalization, in exact parallel with the matrix
elements of L^− and L^+ in the ∣ℓ,m⟩
basis established in QM5 Proposition 5.3.
Proposition. Matrix elements of the oscillator ladder operators.
In the Fock basis {∣n⟩}n≥0:
Equation. Creation operator matrix element.
a^†∣n⟩=n+1∣n+1⟩,
Equation. Annihilation operator matrix element.
a^∣n⟩=n∣n−1⟩(n≥1),a^∣0⟩=0.
The Fock state ∣n⟩ is generated from the vacuum ∣0⟩
by repeated application of a^†:
Equation. Fock state from vacuum.
∣n⟩=n!(a^†)n∣0⟩.
Proof. Creation operator.
By the oscillator-spectrum theorem, a^†∣n⟩ is an
eigenstate of N^ with eigenvalue n+1, from the number-operator
commutator lemma, so
a^†∣n⟩=cn+∣n+1⟩
for some constant cn+.
Compute
∣cn+∣2=∥∥∥a^†∣n⟩∥∥∥H2.
Then
∥∥∥a^†∣n⟩∥∥∥H2=⟨n∣a^a^†∣n⟩.
From
[a^,a^†]=1^,
one has
a^a^†=a^†a^+1^=N^+1^.
Therefore
⟨n∣a^a^†∣n⟩=⟨n∣(N^+1^)∣n⟩=n+1.
Choosing the positive real square root gives
cn+=n+1,
so
a^†∣n⟩=n+1∣n+1⟩.
Annihilation operator.
Similarly,
a^∣n⟩=cn−∣n−1⟩
for n≥1.
Compute
∣cn−∣2=∥a^∣n⟩∥H2=⟨n∣N^∣n⟩=n.
Choosing the positive real square root gives
cn−=n,
so
a^∣n⟩=n∣n−1⟩.
For n=0,
a^∣0⟩=0
by the ground state condition established in the proof of the
oscillator-spectrum theorem.
Generation formula.
Apply a^† repeatedly to ∣0⟩, using the creation
operator matrix element at each step:
a^†∣0⟩=1∣1⟩,
a^†∣1⟩=2∣2⟩,
and continuing to
a^†∣n−1⟩=n∣n⟩.
Composing gives
(a^†)n∣0⟩=n!∣n⟩,
which rearranges to
∣n⟩=n!(a^†)n∣0⟩.
□
Remark.
The matrix elements
a^†∣n⟩=n+1∣n+1⟩
and
a^∣n⟩=n∣n−1⟩
are structurally simpler than their angular momentum analogues from
QM5 Proposition 5.3.
The QM5 matrix elements involve the square root
ℓ(ℓ+1)−m(m±1),
which depends on both quantum numbers ℓ and m and vanishes at
the termination points
m=±ℓ.
The oscillator matrix elements involve simply n and n+1, which depend only on the single quantum number n and
never vanish, except
a^∣0⟩=0,
the ground state.
The simplicity reflects the simpler algebra: the oscillator ladder raises n by 1 unconditionally, with no upper bound, while the angular
momentum ladder must vanish at the multiplet boundary.
The generation formula
∣n⟩=n!(a^†)n∣0⟩
has a particularly clean form: the n! denominator accumulates
exactly the product
1⋅2⋯n
from the n applications of a^†.
Remark.
The inverse relations
x^=2ℓ0(a^+a^†)
and
p^=2ip0(a^†−a^),
combined with the oscillator ladder matrix elements, give the matrix
elements of x^ and p^ in the Fock basis:
⟨n′∣x^∣n⟩=2ℓ0(nδn′,n−1+n+1δn′,n+1),
⟨n′∣p^∣n⟩=2ip0(n+1δn′,n+1−nδn′,n−1).
These matrix elements show that x^ and p^ connect only
adjacent Fock states,
Δn=±1,
which is the origin of the dipole selection rule for the harmonic
oscillator: only transitions between adjacent energy levels are allowed
under a linear coupling
V=−qEx^.
This selection rule will be used in QM10 in the analysis of radiation
emission and absorption by an oscillating charge.
The abstract Fock states ∣n⟩ established in Section 4 are elements
of the abstract Hilbert space H.
Their position-space representations---the wave functions
ψn(x)=⟨x∣n⟩
---are derived in the present section by translating the abstract
ladder operator conditions into differential equations on L2(R).
The ground state is determined by solving
a^ψ0=0
as a first-order ordinary differential equation; the excited states
are obtained by applying the position-space form of a^†
to the ground state n times, and the polynomial factor generated by
these applications is recognized as the n-th Hermite polynomial.
The connection to the Gaussian minimum-uncertainty states of QM3 is
made explicit:
ψ0
is the specific Gaussian with width
σ=2ℓ0
identified in the proof of the zero-point-energy theorem.
Theorem. Ground state wave function.
The harmonic oscillator ground state ψ0 is the unique, up to
overall phase, normalized solution of
a^ψ0=0
in L2(R), with position-space representation
Equation. Ground state wave function.
ψ0(x)=(πℓ02)1/41exp(−2ℓ02x2).
This is the Gaussian minimum-uncertainty state of QM3 Theorem 6.1 with
width
σ=2ℓ0,
mean position
⟨x⟩=0,
and mean momentum
⟨p⟩=0,
satisfying
Δx=2ℓ0,Δp=2p0,Δx⋅Δp=2Φ0.
Proof.
The condition
a^ψ0=0
in position space, using the position-space annihilation operator, becomes
21(ℓ0x+ℓ0∂x)ψ0(x)=0.
This rearranges to the first-order ordinary differential equation:
Equation. Ground state ODE.
dxdψ0=−ℓ02xψ0.
This is separable:
ψ0dψ0=−ℓ02xdx.
Integrating both sides gives
lnψ0(x)=−2ℓ02x2+C,
so
ψ0(x)=Aexp(−2ℓ02x2)
for a constant A∈C.
The normalization condition
∫R∣ψ0(x)∣2dx=1
fixes
∣A∣2∫Rexp(−ℓ02x2)dx=∣A∣2πℓ0=1,
giving
∣A∣=(πℓ02)−1/4.
Choosing A real and positive yields the ground state wave function.
Uniqueness up to phase follows from the one-dimensional solution space
of the ground-state ODE on L2(R): the only square-integrable
solutions are constant multiples of the Gaussian, since
exp(−2ℓ02x2)
decays at ±∞.
Verification of minimum uncertainty.
The standard deviations follow from the Gaussian form:
Δx=σ=2ℓ0,
the root-mean-square width of the closure density ∣ψ0∣2,
which is a Gaussian of variance ℓ02/2, and
Δp=2σΦ0=2⋅ℓ0/2Φ0=2ℓ0Φ0=2p0,
using
ℓ0p0=Φ0.
Therefore
Δx⋅Δp=(2ℓ0)(2p0)=2ℓ0p0=2Φ0,
confirming saturation of the Heisenberg bound. □
Remark.
The ground state wave function is consistent with all prior results
in the following senses.
It achieves the zero-point energy bound of the zero-point-energy theorem,
with equality conditions
⟨x⟩=⟨p⟩=0
and
σ=2ℓ0.
It is the Gaussian minimum-uncertainty state of QM3 Theorem 6.1,
confirmed by
Δx⋅Δp=2Φ0.
It is the vacuum of the ladder algebra, satisfying
a^ψ0=0
by construction.
These three characterizations---energy minimizer, uncertainty minimizer,
and ladder vacuum---all select the same unique state.
Their coincidence is not accidental: all three are manifestations of
the same canonical commutation relation.
¶ Excited State Wave Functions and Hermite Polynomials
The excited states ψn for n≥1 are obtained by applying
n!(a^†)n
to the ground state, using the generation formula from the previous
section.
In position space, the operator
a^†=2ξ−∂ξ
applied repeatedly to the Gaussian generates a family of polynomials
times the Gaussian, which are identified as the Hermite polynomials.
Theorem. Excited state wave functions and Hermite polynomials.
The n-th energy eigenstate ψn, with
En=(n+21)Φ0ω,
has position-space representation
Equation. Excited state wave function.
ψn(x)=2nn!πℓ01Hn(ℓ0x)exp(−2ℓ02x2),
where Hn(ξ) is the Hermite polynomial of degree n in ξ,
defined by the Rodrigues formula
is confirmed by the standard orthonormality integral for Hermite
polynomials:
∫−∞∞Hn(ξ)2e−ξ2dξ=π2nn!.
□
Remark.
The Hermite polynomials Hn(ξ) satisfy the recurrence relation
Hn+1(ξ)=2ξHn(ξ)−2nHn−1(ξ)
and the differential equation
Hn′′(ξ)−2ξHn′(ξ)+2nHn(ξ)=0,
the Hermite ordinary differential equation, which is the position-space
form of the oscillator eigenvalue equation
H^oscψn=Enψn
after substituting the excited-state wave function.
The first three cases are
H0(ξ)=1,
H1(ξ)=2ξ,
and
H2(ξ)=4ξ2−2.
Thus the n=0, n=1, and n=2 wave functions are:
ψ0(x)=(πℓ02)1/41exp(−2ℓ02x2),
ψ1(x)=(πℓ02)1/42x/ℓ0exp(−2ℓ02x2),
ψ2(x)=8(πℓ02)1/44(x/ℓ0)2−2exp(−2ℓ02x2).
The parity of ψn under x→−x is
(−1)n.
Even n states are symmetric and odd n states are antisymmetric.
This follows from
Hn(−ξ)=(−1)nHn(ξ),
a consequence of the Rodrigues formula.
The parity selection rule for the harmonic oscillator---dipole
transitions are allowed only between states of opposite parity, i.e.
between adjacent n values---follows from this property combined with
the
Δn=±1
selection rule noted in the position-momentum matrix-element remark.
Proposition. Orthonormality and completeness of the Fock states.
The Fock states
{ψn}n≥0
form a complete orthonormal basis for
H=L2(R,C):
Equation. Oscillator completeness.
⟨ψn′,ψn⟩H=δn′n,n=0∑∞∣ψn⟩⟨ψn∣=1^H.
Proof. Orthonormality.
For n=n′, the states ψn and ψn′ are eigenstates
of the self-adjoint operator H^osc with distinct
eigenvalues
En=En′.
Eigenstates of a self-adjoint operator corresponding to distinct
eigenvalues are orthogonal.
For n=n′, unit normalization was verified in the proof of the
excited-state theorem via the standard Hermite orthogonality integral.
Completeness.
The operator H^osc is a self-adjoint operator on H, bounded below, with purely discrete spectrum
σ(H^osc)={(n+21)Φ0ω:n≥0}.
By the spectral theorem for self-adjoint operators, QM1 Theorem 6.1,
the family of eigenstates corresponding to the complete discrete spectrum
forms a complete orthonormal basis for H.
Since each eigenvalue En is non-degenerate and the spectrum is
exactly
{(n+21)Φ0ω:n≥0},
the family {ψn}n≥0 is precisely this complete basis,
giving the resolution of the identity. □
Remark.
The completeness of the Fock states in
H=L2(R)
means that any closure state
Ψ∈H
can be expanded as
Ψ=n=0∑∞cnψn
with coefficients
cn=⟨ψn,Ψ⟩H
and
n∑∣cn∣2=1.
This Fock basis expansion is the oscillator analogue of the spherical
harmonic expansion of angular closure states on S2 established
in QM5 Proposition 6.3.
In both cases, completeness is grounded in the spectral theorem of QM1
applied to a self-adjoint operator with discrete spectrum rather than
in the theory of special functions.
The Fock basis expansion will be used directly in Section 6 to derive
the coherent state expansion and in Section 7 to construct the
three-dimensional oscillator eigenstates.
The Fock states ∣n⟩ established in Sections 4 and 5 are the
eigenstates of the energy---they have definite energy but maximally
uncertain phase---and they form the natural basis for perturbation theory
and matrix mechanics.
However, for the analysis of the quantum-classical correspondence and
for the physical sectors of QM9 through QM11, a different family of
states is more natural: the coherent states, which are the states
closest to classical oscillation.
The present section defines coherent states as eigenstates of the
annihilation operator, derives their Fock-state expansion and
position-space Gaussian form, constructs them via the displacement
operator, establishes their shape-preserving time evolution, and proves
the overcompleteness of the coherent state family.
Five results are established in sequence, each building on the previous:
the Fock expansion leads to the Gaussian form, which connects to the
displacement operator construction, which enables the time evolution
analysis, which in turn motivates the overcompleteness.
Definition. Coherent states as eigenstates of a^.
A coherent state∣α⟩ with parameter α∈C is a normalized eigenstate of the annihilation
operator:
Equation. Coherent-state eigenvalue equation.
a^∣α⟩=α∣α⟩,⟨α∣α⟩=1.
Remark.
Since a^ is not self-adjoint, its eigenvalues need not be real.
The eigenvalue α in the coherent-state eigenvalue equation is in
general a complex number,
α∈C,
and every α∈C is an eigenvalue of a^.
This contrasts with the self-adjoint operators H^osc
and N^, whose eigenvalues are the real numbers
(n+21)Φ0ω
and n, respectively.
The parametrization of coherent states by
α∈C
a two-real-parameter family, versus the parametrization of Fock states by
n∈Z≥0,
a one-integer-parameter family, reflects the fact that the coherent state
family is overcomplete: there are continuously many more coherent states
than Fock states, as established in the overcompleteness proposition.
The Fock-basis expansion of the coherent state is derived from the
eigenvalue condition combined with the matrix elements of the oscillator
ladder operators.
Lemma. Fock-basis expansion of coherent states.
The unique normalized eigenstate of a^ with eigenvalue α∈C has the Fock-basis expansion
Equation. Coherent-state Fock expansion.
∣α⟩=e−∣α∣2/2n=0∑∞n!αn∣n⟩.
Proof.
Write
∣α⟩=n=0∑∞cn∣n⟩
in the Fock basis, using completeness.
Apply the annihilation operator using the annihilation-operator matrix
element:
a^∣α⟩=n=1∑∞cnn∣n−1⟩=n=0∑∞cn+1n+1∣n⟩.
The eigenvalue equation
a^∣α⟩=α∣α⟩
requires
n=0∑∞cn+1n+1∣n⟩=αn=0∑∞cn∣n⟩.
Comparing coefficients of ∣n⟩ gives:
Equation. Coherent-state coefficient recurrence.
cn+1n+1=αcn,⇒cn+1=n+1αcn.
Solving the recurrence by induction:
cn=n!αnc0.
The normalization condition
n=0∑∞∣cn∣2=1
gives
∣c0∣2n=0∑∞n!∣α∣2n=∣c0∣2e∣α∣2=1,
so
∣c0∣=e−∣α∣2/2.
Choosing c0 real and positive, fixing the overall phase, gives
c0=e−∣α∣2/2
and hence the coherent-state Fock expansion. □
¶ Equivalence to Gaussian Minimum-Uncertainty States
Theorem. Coherent states are Gaussian minimum-uncertainty states.
The position-space representation of the coherent state ∣α⟩
is the Gaussian
Thus D^(α)∣0⟩ is a normalized eigenstate of a^
with eigenvalue α.
By uniqueness of the coherent state for each α,
D^(α)∣0⟩=eiϑ∣α⟩
for some phase ϑ.
The phase is fixed to ϑ=0 by comparing with the Fock expansion,
or by the Baker--Campbell--Hausdorff factorization
D^(α)=e−∣α∣2/2eαa^†e−αa^,
which gives the same expansion directly. □
Remark.
In terms of the position and momentum operators, using the inverse
relations, one has
αa^†−αa^=Φ0i(⟨p⟩αx^−⟨x⟩αp^),
so the displacement operator takes the form
D^(α)=exp[Φ0i(⟨p⟩αx^−⟨x⟩αp^)].
This is the unitary operator that displaces the vacuum ∣0⟩,
a Gaussian centered at the origin, to a Gaussian centered at
(⟨x⟩α,⟨p⟩α)
in phase space.
The action of D^(α) is therefore a phase-space translation:
it moves the center of the Gaussian without changing its shape, which is
consistent with the fact that all coherent states have the same width
is a coherent state up to an overall phase, the coherent-state Gaussian
theorem applies and gives
Δx(t)=2ℓ0,Δp(t)=2p0
for all t.
Centroid trajectory.
From the coherent-state means applied to
α(t)=∣α∣ei(ϕα−ωt),
one obtains
⟨x⟩(t)=ℓ02Re(α(t))=ℓ02∣α∣cos(ϕα−ωt),
and
⟨p⟩(t)=p02Im(α(t))=−p02∣α∣sin(ϕα−ωt).
These are the coherent-state position and momentum evolution equations,
and they are the classical harmonic oscillator trajectories, confirming
consistency with the Ehrenfest theorem. □
Remark.
The coherent-state evolution theorem establishes the dynamical
characterization of coherent states that completes the program arc from
QM3.
For a general Gaussian state with width
σ=2ℓ0,
the harmonic oscillator dynamics cause the width to oscillate:
Δx(t)
alternates between
σ
and
2mωσΦ0=2σℓ02
at twice the oscillator frequency.
Only for
σ=2ℓ0
is
2σℓ02=σ,
so that the two extremes coincide and the width is constant.
The QM3 minimum-uncertainty family, all Gaussians, is dynamically filtered
by the oscillator to the coherent state family, the single width
2ℓ0,
with α as the remaining free parameter encoding the classical
initial conditions.
This is the precise content of the program arc from QM3 to QM6.
Proposition. Overcompleteness of the coherent state family.
The coherent states are not mutually orthogonal:
Equation. Coherent-state overlap.
⟨α′∣α⟩=exp(α′α−2∣α′∣2−2∣α∣2),
and the family
{∣α⟩:α∈C}
satisfies the resolution of the identity:
Equation. Coherent-state resolution of the identity.
π1∫C∣α⟩⟨α∣d2α=1^H,
where
d2α=d(Reα)d(Imα).
The coherent state family is therefore overcomplete: it spans H via the resolution of the identity but is not linearly
independent, since each coherent state has non-zero overlap with all
others.
Proof. Overlap.
Using the Fock expansions of ∣α′⟩ and ∣α⟩:
which is the matrix element of 1^H in
the Fock basis, confirming the coherent-state resolution of the identity. □
Remark.
The resolution of the identity for coherent states differs structurally
from the orthonormal resolutions established in earlier papers.
The Fock state resolution
n=0∑∞∣n⟩⟨n∣=1^
is a discrete sum over orthonormal states.
The spherical harmonic resolution
ℓ,m∑∣Yℓm⟩⟨Yℓm∣=1^L2(S2)
from QM5 is a discrete sum over orthonormal angular eigenstates.
The coherent state resolution
π1∫C∣α⟩⟨α∣d2α=1^H
is a continuous integral with the factor 1/π, rather than a sum, over
a non-orthogonal family with the phase-space area measure d2α.
The 1/π factor arises because the coherent states are overcomplete
by exactly a factor of π per unit phase-space area, reflecting the
Heisenberg phase-space uncertainty cell area
Δx⋅Δp=2Φ0.
The overcompleteness relation is the foundation of the coherent-state path
integral and the Bargmann--Fock representation, both of which will be used
in QM10 and the field-theoretic extensions.
The one-dimensional harmonic oscillator of Sections 3--6 is now extended
to three spatial dimensions.
The three-dimensional isotropic harmonic oscillator
H^osc(3)=j=1∑3[2mp^j2+21mω2(x^j)2]
admits two equivalent treatments: the Cartesian product of three
independent one-dimensional oscillators, which makes the spectrum and
degeneracy immediately transparent; and the spherical coordinate
separation, which exploits the rotational symmetry of the isotropic
potential and expresses the eigenstates as products of radial functions
and the QM5 spherical harmonics
Yℓm(θ,φ).
The two representations are complementary: the Cartesian representation
counts states by quantum number triples
(nx,ny,nz),
while the spherical representation labels them by the physically
meaningful angular momentum quantum numbers
(nr,ℓ,m).
Both yield the same spectrum
EN=(N+23)Φ0ω
and the same degeneracy
dN=2(N+1)(N+2),
providing a consistency check on both derivations.
¶ The Three-Dimensional Hamiltonian and Its Symmetry
Proposition. Rotational symmetry of the isotropic oscillator.
The three-dimensional isotropic harmonic oscillator Hamiltonian
which gives the Cartesian oscillator energies.
The product states form a complete orthonormal basis for H(3) by the completeness of each one-dimensional Fock
basis. □
Proposition. Degeneracy of the 3D oscillator energy shells.
The number of linearly independent eigenstates with total excitation
number N is
using the spherical-coordinate form of the three-dimensional oscillator
Hamiltonian.
Since L^2 acts only on the angular variables and Yℓm is its eigenfunction, the operator L^2 acting
on
Rnrℓ(r)Yℓm(θ,φ)
yields
ℓ(ℓ+1)Φ02
times the same product.
The radial derivative operator acts only on Rnrℓ(r).
Dividing through by
Yℓm(θ,φ)=0
and substituting
Rnrℓ(r)=rR(r)
to convert the spherical Laplacian to the one-dimensional form gives
the three-dimensional oscillator radial equation, which is independent
of m, confirming the (2ℓ+1)-fold degeneracy in m for each ℓ. □
Theorem. Complete eigenstructure of the 3D isotropic oscillator.
The radial equation has normalizable solutions if and only if
EN=(N+23)Φ0ω
with
Equation. Three-dimensional oscillator quantum number constraint.
The quantum number structure of the complete 3D oscillator theorem is
recorded shell by shell for the lowest four energy levels.
Remark. Shell structure of the 3D isotropic oscillator.
N
EN/(Φ0ω)
d_
(nr,ℓ) values
Angular content
0
3/2
1
(0,0)
ℓ=0: one s-state
1
5/2
3
(0,1)
ℓ=1: three p-states
2
7/2
6
(1,0),(0,2)
ℓ=0: one s; ℓ=2: five d
3
9/2
10
(1,1),(0,3)
ℓ=1: three p; ℓ=3: seven f
The parity of the N-th shell is
(−1)N:
states in even shells,
N=0,2,4,…,
are parity-even and states in odd shells,
N=1,3,5,…,
are parity-odd.
Remark.
The parity of the eigenstate
ΨNℓm(x)
under
x→−x
is
(−1)ℓ,
arising from the spherical harmonic parity of QM5 Proposition 6.3:
Yℓm(π−θ,φ+π)=(−1)ℓYℓm(θ,φ).
The radial function
Rnrℓ(r)
depends only on
r=∣x∣
and is parity-even.
Since
ℓ≡N(mod2),
the parity of the full eigenstate is
(−1)ℓ=(−1)N,
so all states in a given shell have the same parity.
This uniform shell parity is a direct consequence of the constraint
ℓ≡N(mod2)
and the QM5 spherical harmonic parity---two results from different
papers combining to give a unified physical statement.
Remark.
The degeneracy
dN=2(N+1)(N+2)
of the 3D isotropic oscillator is larger than the angular momentum
degeneracy
2ℓ+1
of a single ℓ-multiplet, because multiple ℓ values contribute
to each shell.
For example, the
N=2
shell has
d2=6
states: one s-state,
ℓ=0,m=0,
and five d-states,
ℓ=2,m=−2,−1,0,1,2.
This multi-ℓ degeneracy within each shell is analogous to the
accidental n2 degeneracy of the hydrogen atom established in QM5
Theorem 7.2, and has a similar origin: the isotropic harmonic oscillator
potential
V∝r2,
like the Coulomb potential
V∝r1,
has a higher symmetry than SO(3).
For the 3D oscillator, this higher symmetry is SU(3), the
special unitary group in three dimensions, which generates transitions
between different ℓ values within the same shell N.
The full derivation of the SU(3) symmetry and its consequences
for the oscillator spectrum are beyond the scope of the present paper
and are deferred as a structural extension of the series.
Remark.
The three-dimensional oscillator analysis is the first instance in the
QM-series in which the results of QM5, spherical harmonics, angular
momentum eigenvalues, and parity, are used as direct structural inputs
to the derivation of a physical spectrum, rather than merely as formal
prerequisites.
The angular factor
Yℓm(θ,φ)
in the eigenstate
ΨNℓm(r,θ,φ)=Rnrℓ(r)Yℓm(θ,φ),
the centrifugal term
2mr2ℓ(ℓ+1)Φ02
in the radial equation, and the parity
(−1)ℓ
are all QM5 results applied here without re-derivation.
This cross-paper dependence is the normal pattern from QM6 onward:
each new physical sector paper builds directly on the structural results
established in prior sector papers, and the program accumulates physical
results by layering new derivations on top of established ones rather
than re-deriving from first principles.
The present section collects the interpretive constraints governing
the harmonic oscillator analysis of the preceding sections and records
the scope of the present construction relative to the remainder of the
QM-series.
Three items are addressed: the role of the oscillator as a structural
template throughout the series and beyond, the explicit comparison
between the QM5 angular momentum ladder algebra and the QM6 energy
ladder algebra, and the precise boundary between what the present paper
establishes and what is deferred.
The harmonic oscillator occupies a position in the QM-series that is
qualitatively different from the hydrogen atom of QM5.
The hydrogen atom is the primary physical validation target of the
scalar--conformal NUVO framework: its energy spectrum was derived in
the Q-series and its full eigenstate structure was completed in QM5,
constituting the first complete physical system derived within the
program.
The harmonic oscillator is instead a structural template: its algebraic
skeleton---the ladder algebra
[a^,a^†]=1^,
the number operator decomposition
H^osc=Φ0ω(N^+21),
and the coherent state family---recurs throughout every subsequent paper
in the QM-series and in the field-theoretic extensions beyond it.
Four specific structural roles are worth recording explicitly.
The oscillator ladder as the energy-sector template.
The ladder technique introduced in QM5 for angular momentum---raising
and lowering the L^3 eigenvalue by Φ0 while preserving
the L^2 eigenvalue---reappears here as raising and lowering
the energy eigenvalue by
Φ0ω.
In both cases the technique extracts the complete spectrum from a
commutation relation without solving a differential equation.
In QM8, the same technique will be applied to the spin algebra
[S^j,S^k]=iΦ0ϵjklS^l
to derive the spin spectrum.
In QM10, a modified ladder technique, the Lippmann--Schwinger equation,
will be applied to the scattering sector.
The present paper establishes the oscillator version as the simplest
non-trivial instance of the ladder technique with a semi-infinite
spectrum; subsequent papers adapt and extend it.
Coherent states and the quantum-classical correspondence.
The Ehrenfest theorem of QM4 established that the centroid of any closure
state follows the classical equations of motion.
For the harmonic oscillator, the coherent-state evolution theorem goes
further: the coherent states are the unique family for which not only the
centroid but the entire Gaussian profile propagates classically, with
the width
Δx(t)=2ℓ0
constant in time.
This is the sharpest instance in the QM-series of the quantum-classical
correspondence: a coherent state is, in a precise sense, as classical as
a quantum state can be.
The coherent states will appear in QM9 as the building blocks of entangled coherent states, non-factorizable superpositions of coherent
states in two-mode systems; in QM10 as the reference states for
scattering cross-sections involving coherent radiation fields; and in
QM11 as the classical limit of relativistic field configurations.
The oscillator as the foundation of quantum field theory.
The scalar--conformal NUVO program develops quantum mechanics in the
QM-series; the RQM-series and the field-theoretic extensions beyond it
treat the relativistic and multi-mode cases.
In that broader context, the harmonic oscillator is the elementary object
from which all of quantum field theory is built: each mode of a free
quantum field is an independent harmonic oscillator, the vacuum of the
field is the tensor product
k⨂∣0k⟩
of the ground states of all modes, and a field excitation is a Fock state
a^k†∣0k⟩
for a specific mode k.
The Fock basis established in the present paper, the creation and
annihilation operators, and the coherent state overcompleteness relation
π1∫C∣α⟩⟨α∣d2α=1^
are the tools used throughout quantum field theory.
Recording this role explicitly here allows the connections to be made
precisely when the field-theoretic extension is developed, without
retroactively re-deriving the oscillator structure.
The zero-point energy and the uncertainty principle.
The zero-point-energy theorem established the zero-point energy
E0=21Φ0ω
as a structural consequence of the Heisenberg uncertainty relation: a
state of zero energy would require exact position and exact momentum
simultaneously, violating
Δx⋅Δp≥2Φ0.
In quantum field theory, the vacuum energy of each field mode is its
zero-point energy
21Φ0ωk,
and the total vacuum energy is
k∑21Φ0ωk.
The structure of the zero-point energy and its connection to the
uncertainty principle, established here for the single-mode oscillator,
therefore underlies one of the most fundamental---and most
computationally significant---features of quantum field theory.
¶ Comparison of the Oscillator and Angular Momentum Ladders
The angular momentum ladder of QM5 and the oscillator ladder of the
present paper are the two canonical ladder algebras of the QM-series.
Their structural similarities and differences are recorded here in a
form that makes both papers easier to use as references.
Property
QM5 angular momentum
QM6 oscillator
Diagonal operator
L^3 self-adjoint
N^=a^†a^ self-adjoint
Ladder operators
L^+=L^1+iL^2, \hat{L}_
a^†, \hat
Fundamental relation
[\hat{L}_{+},\hat{L}_{-}]=2\Phi_{0}\hat{L}_
[a^,a^†]=1^
Ladder commutators
[\hat{L}_{3},\hat{L}_{+}]=\Phi_{0}\hat{L}_
[\hat{N},\hat{a}^{\dagger}]=+\hat{a}^
[\hat{L}_{3},\hat{L}_{-}]=-\Phi_{0}\hat{L}_
[\hat{N},\hat{a}]=-\hat
Spectrum type
Finite: m\in\
Semi-infinite: n\in\
Upper termination
Yes: $\hat{L}_
\ell,\ell\rangle=0$
Lower termination
Yes: $\hat{L}_
\ell,-\ell\rangle=0$
Quantization source
Integer holonomy, QM5 Theorem 5.1
$\hat
Eigenvalue step size
Φ0 per step
1 per step, dimensionless
Degeneracy per eigenvalue
2ℓ+1 for each L^2 eigenvalue
1 non-degenerate
Underlying algebra
SO(3) Lie algebra
Weyl--Heisenberg algebra
Non-self-adjoint operators
\hat{L}_{+}^{\dagger}=\hat{L}_
\hat{a}^{\dagger}=(\hat{a})^
The most significant structural difference between the two algebras is
the presence or absence of upper termination.
In the angular momentum case, the operator L^2 provides an
upper bound on the L^3 eigenvalue: since
L^2=L^12+L^22+L^32
and all terms are non-negative, the L^3 eigenvalue mΦ0 is bounded by
∣m∣Φ0≤ℓ(ℓ+1)Φ0.
This forces both upper and lower termination, giving the finite multiplet
structure.
In the oscillator case, there is no operator analogous to L^2
that bounds N^ from above: the number operator has no upper bound,
so the ladder extends infinitely upward.
Only the non-negativity of N^, which bounds it from below, forces
the existence of the vacuum state.
The absence of holonomy quantization in QM6 is also structurally
significant.
In QM5, the integer character of the angular momentum quantum numbers
was derived from the holonomy condition: the azimuthal transport closure
state must be single-valued under
φ→φ+2π,
selecting
m∈Z.
In QM6, no such geometric condition is required: the eigenvalues of N^ are forced to be non-negative integers by the non-negativity
of N^ and the unit step size of the ladder, the canonical
commutation relation
[a^,a^†]=1^,
without any topological input.
This is consistent with the physical picture: the oscillator quantum
number n is not a holonomy winding number but a count of excitation
quanta.
The present paper establishes the complete algebraic and spectral
structure of the one-dimensional harmonic oscillator, the coherent
state theory, and the three-dimensional isotropic oscillator
eigenstructure.
The following results are established and available as inputs to
subsequent QM-series papers.
Algebraic structure:
The ladder operator commutation relation
[a^,a^†]=1^,
the Hamiltonian decomposition
H^osc=Φ0ω(N^+211^),
and the number operator commutation relations.
Spectral structure:
The complete spectrum
σ(H^osc)={(n+21)Φ0ω:n≥0}
with non-degenerate eigenvalues, the zero-point energy
E0=21Φ0ω
from the uncertainty principle, and the Fock state matrix elements
a^†∣n⟩=n+1∣n+1⟩
and
a^∣n⟩=n∣n−1⟩.
Position-space eigenstates:
The ground state Gaussian
ψ0(x)∝e−x2/(2ℓ02),
the excited state Hermite-Gaussians
ψn(x)∝Hn(x/ℓ0)e−x2/(2ℓ02),
and the completeness of the Fock basis in
L2(R).
Coherent state theory:
The eigenstate definition
a^∣α⟩=α∣α⟩,
the Fock expansion, the Gaussian minimum-uncertainty equivalence with
width
σ=2ℓ0,
the displacement operator construction
∣α⟩=D^(α)∣0⟩,
the shape-preserving time evolution
U^(t)∣α⟩=e−iωt/2∣αe−iωt⟩,
and the overcompleteness resolution of the identity
π1∫∣α⟩⟨α∣d2α=1^.
Three-dimensional oscillator:
The rotational symmetry
[H^osc(3),L^j]=0,
the Cartesian product eigenstructure, the spherical separation with
Rnrℓ(r)Yℓm(θ,φ)
eigenstates, and the complete quantum number structure
N=2nr+ℓ,
EN=(N+23)Φ0ω,
dN=2(N+1)(N+2),
together with the parity
(−1)N
of each energy shell.
The following topics are outside the scope of the present paper and are
deferred.
Squeezed states.
A squeezed state is a state with
Δx<2ℓ0
and correspondingly
Δp>2p0,
achieved by a Bogoliubov transformation that mixes a^ and a^†.
Such states are minimum-uncertainty states in the QM3 sense but lie
outside the coherent state family because their width differs from
2ℓ0.
Their derivation requires the squeeze operator
S^(ξ)=exp[21(ξ∗a^2−ξ(a^†)2)]
and the associated Bogoliubov transformation, which is developed in the
context of quantum optics beyond the current scope.
Anharmonic corrections.
The harmonic oscillator is the leading-order approximation to any
potential near a stable equilibrium.
Anharmonic corrections---cubic and quartic perturbations to the
potential---shift the energy levels away from the equally spaced
structure of the oscillator spectrum theorem.
Their treatment requires the perturbation theory developed in a
subsequent paper and is outside the scope of the present derivation.
Field-mode quantization.
The application of the oscillator algebra to quantize each mode of a
free classical field---giving the Fock space of quantum field
theory---is the central step in the transition from the QM-series to the
RQM-series and the field-theoretic extensions.
The present paper establishes the single-mode oscillator structure that
each field mode will inherit; the multi-mode tensor product construction
is the content of the field-theoretic extension.
The Jaynes--Cummings model.
The coupling of a two-level atom to a single field mode---the
Jaynes--Cummings model of quantum optics---involves the interaction
between the oscillator ladder algebra of QM6 and the spin algebra of QM8.
Its analysis requires both the coherent state theory of the present paper
and the spin structure of QM8, and is deferred to the appropriate point
in the series.
Remark.
The present paper completes the transition from the algebraic foundation
papers, QM1--QM3, and the first physical sector, QM4--QM5, to the
dynamical and multi-particle sector, QM6--QM11.
QM1--QM3 established the state space, superposition structure, and
uncertainty relations without dynamics.
QM4--QM5 established the dynamics, Schrödinger equation, and the
rotational sector, angular momentum and hydrogen spectrum.
QM6 establishes the oscillator sector and, through the coherent state
theory, the most precise expression of the quantum-classical
correspondence available within the non-relativistic framework.
The results of QM5, spherical harmonics, and QM6, Fock states and
coherent states, together provide the structural inputs for all
subsequent physical sector papers.
The present paper has derived the complete structure of the
scalar--conformal NUVO harmonic oscillator from the canonical
commutation relations of QM1 and the dynamical framework of QM4,
without postulating the spectrum, the Hermite polynomial eigenfunctions,
or the coherent state properties.
The sixteen principal results are as follows.
Ladder operators from the CCR.
The annihilation and creation operators
a^=2mωΦ0mωx^+ip^
and
a^†=(a^)†
are non-self-adjoint operators on
S(R)⊂H
satisfying
[a^,a^†]=1^,
derived by direct computation from the canonical commutation relation
[x^,p^]=iΦ0
of QM1.
Hamiltonian decomposition and the number operator.
The harmonic oscillator Hamiltonian decomposes as
H^osc=Φ0ω(N^+211^),
where
N^=a^†a^
is the self-adjoint non-negative number operator.
The commutation relations
[N^,a^]=−a^,
[N^,a^†]=+a^†,
and their Hamiltonian equivalents
[H^osc,a^]=−Φ0ωa^,
[H^osc,a^†]=+Φ0ωa^†
establish the raising and lowering action.
The
21Φ0ω
offset in the decomposition arises directly from the CCR.
Complete spectrum of the harmonic oscillator.
The spectrum is
σ(H^osc)={(n+21)Φ0ω:n∈{0,1,2,…}},
derived by the four-step ladder termination argument: non-negativity
of N^ bounds the spectrum from below, the lower termination
condition
a^∣0⟩=0
identifies the ground state, the unit step of the ladder gives integer
eigenvalues, and non-degeneracy follows from the uniqueness of the
vacuum.
Each eigenvalue is non-degenerate.
Zero-point energy from the uncertainty principle.
For any normalized state,
⟨H^osc⟩≥21Φ0ω,
with equality if and only if Ψ is the ground state.
The bound is derived by combining
⟨A2⟩≥(ΔA)2
with the AM--GM inequality and the Heisenberg bound
Δx⋅Δp≥2Φ0
of QM3.
The equality condition
Δp=mωΔx
combined with Robertson saturation uniquely identifies the ground
state as the Gaussian with
σ=2ℓ0,
⟨x⟩=⟨p⟩=0.
Fock state matrix elements and the generation formula.
a^†∣n⟩=n+1∣n+1⟩
and
a^∣n⟩=n∣n−1⟩
for n≥1, with
a^∣0⟩=0,
derived by normalization computations using the identity
a^a^†=N^+1^.
The generation formula
∣n⟩=n!(a^†)n∣0⟩
follows by induction.
Ladder operators in position space.
a^=2ξ+∂ξ
and
a^†=2ξ−∂ξ
in terms of the dimensionless variable
ξ=ℓ0x,
derived by substituting the position-space momentum
p^=−iΦ0∂x
into the ladder-operator definitions.
Ground state wave function.
The condition
a^ψ0=0
in position space is a first-order ordinary differential equation
∂xψ0=−ℓ02xψ0,
whose unique normalized solution is the Gaussian
ψ0(x)=(πℓ02)−1/4exp(−2ℓ02x2).
This is the QM3 minimum-uncertainty Gaussian with
σ=2ℓ0,
⟨x⟩=⟨p⟩=0,
and saturation
Δx⋅Δp=2Φ0.
Excited state wave functions and Hermite polynomials.
The n-th eigenstate is
ψn(x)=2nn!πℓ01Hn(ℓ0x)exp(−2ℓ02x2),
where the Hermite polynomial Hn(ξ) is generated by the identity
(ξ−∂ξ)ne−ξ2/2=Hn(ξ)e−ξ2/2,
derived from the factorization
ξ−∂ξ=−eξ2/2∂ξe−ξ2/2.
The Rodrigues formula
Hn(ξ)=(−1)neξ2∂ξne−ξ2
emerges from this factorization.
Completeness and orthonormality of the Fock basis.
⟨ψn′,ψn⟩H=δn′n
from self-adjointness of H^osc and distinct
eigenvalues, and
n=0∑∞∣n⟩⟨n∣=1^H
from the spectral theorem of QM1 applied to the purely discrete
spectrum of H^osc.
Coherent states as eigenstates of a^.
The coherent state ∣α⟩ with
α∈C
satisfies
a^∣α⟩=α∣α⟩
and has the unique normalized Fock expansion
∣α⟩=e−∣α∣2/2n=0∑∞n!αn∣n⟩,
derived from the recurrence
cn+1n+1=αcn
and normalization.
Coherent states are Gaussian minimum-uncertainty states.
The position-space representation of ∣α⟩ is the Gaussian
of QM3 with width
σ=2ℓ0,
mean position
⟨x⟩α=ℓ02Re(α),
and mean momentum
⟨p⟩α=p02Im(α),
derived by solving the coherent-state ODE and normalizing.
The widths
Δx=2ℓ0
and
Δp=2p0
are independent of α.
Displacement operator construction.
The unitary displacement operator
D^(α)=exp(αa^†−αa^)
satisfies
D^(α)a^D^(α)†=a^−α1^
by the Baker--Campbell--Hausdorff lemma applied to the commutator
[αa^†−αa^,a^]=−α1^,
and generates coherent states from the vacuum:
∣α⟩=D^(α)∣0⟩.
Shape-preserving time evolution of coherent states.
U^(t)∣α⟩=e−iωt/2∣αe−iωt⟩,
derived via the Heisenberg-picture identity
U^†(t)a^U^(t)=a^e−iωt
and confirmed by the Fock expansion.
The widths
Δx(t)=2ℓ0
and
Δp(t)=2p0
are constant; the centroid follows the classical trajectory.
The overall phase
e−iωt/2
arises from the zero-point energy.
Overcompleteness of the coherent state family.
The overlap
⟨α′∣α⟩=exp(α′α−2∣α′∣2−2∣α∣2)
shows coherent states are not orthogonal.
The resolution of the identity
π1∫C∣α⟩⟨α∣d2α=1^H
is established by computing the Fock matrix elements in polar
coordinates and using the Gaussian integral
∫0∞r2n+1e−r2dr=2n!.
Three-dimensional isotropic oscillator.
The Hamiltonian
H^osc(3)=j∑[2mp^j2+21mω2(x^j)2]
commutes with all L^j by rotational symmetry.
The Cartesian product structure gives
EN=(N+23)Φ0ω
with
dN=2(N+1)(N+2)
from stars-and-bars counting.
The spherical separation gives eigenstates
The results of the present paper are of broad programmatic significance
for the scalar--conformal NUVO series on three distinct grounds.
The first is the completion of the QM3-to-QM6 program arc.
QM3 identified the Gaussian minimum-uncertainty states algebraically:
for any width
σ>0,
the Gaussian closure state
ΨG,(σ,⟨x⟩,⟨p⟩)
saturates the Heisenberg bound
Δx⋅Δp=2Φ0.
The algebraic characterization established a one-parameter family of
optimal states without selecting among them.
The harmonic oscillator dynamics of the present paper acts as a
dynamical filter on this family: of all the Gaussians, only those with
the specific width
σ=2ℓ0
retain their Gaussian form under the time evolution U^(t).
All other Gaussians undergo “breathing” oscillation of their width at
frequency 2ω.
The dynamical selection of
σ=2ℓ0
is therefore the dynamical characterization that completes the algebraic
characterization of QM3: the coherent states are the
minimum-uncertainty states that are also dynamically stable under the
harmonic oscillator.
The two characterizations, algebraic saturation of Cauchy--Schwarz and
dynamical shape preservation, together determine the coherent states
uniquely, and the program arc from QM3 to QM6 is the arc from the
algebraic to the dynamical characterization.
The second ground of programmatic significance is the derivation of the
zero-point energy from the uncertainty principle.
The zero-point-energy theorem establishes that no state of the harmonic
oscillator can have energy less than
2Φ0ω.
The proof uses no spectral information about H^osc;
it derives the bound from the Heisenberg relation of QM3 alone, via the
AM--GM inequality.
The zero-point energy is therefore not a computational artifact of the
energy eigenvalues but a structural consequence of the canonical
commutation relation: the same CCR that gives
[a^,a^†]=1^
and the
211^
in
H^osc=Φ0ω(N^+211^)
also gives
Δx⋅Δp≥2Φ0,
which forces
⟨H^osc⟩≥2Φ0ω.
This is one of the NUVO program's cleanest demonstrations that quantum
phenomena arise from transport closure geometry---specifically from the
holonomy structure encoded in the CCR---rather than from postulates about
state spaces.
The third ground is the first active use of QM5 results in a subsequent
physical sector paper.
The three-dimensional oscillator analysis of Section 7 does not merely
require QM5 as a logical prerequisite; it uses the spherical harmonics
Yℓm(θ,φ)
as the explicit angular eigenstates in the eigenstate factorization
ΨNℓm=Rnrℓ(r)Yℓm(θ,φ),
and uses the parity
(−1)ℓ
of the spherical harmonics from QM5 to derive the shell parity
(−1)N
of the 3D oscillator.
The quantum number constraint
N=2nr+ℓ
arises from the interaction between the oscillator's energy structure,
the radial quantum number nr, and the angular momentum structure,
the orbital quantum number ℓ.
This interaction between dynamical and rotational structure is the
pattern that recurs throughout the remainder of the QM-series: QM7
couples oscillators, QM8 couples orbital and spin angular momentum,
QM9 entangles multi-particle states, and in each case the QM5 angular
momentum structure and the QM6 oscillator structure are both active
ingredients.
The present paper completes the treatment of single-particle quantum
mechanics for the two canonical physical systems of the non-relativistic
NUVO framework: the hydrogen atom, QM4--QM5, and the harmonic oscillator,
QM4--QM6.
The next paper, QM7, opens the multi-particle sector.
The fundamental new object in QM7 is the tensor product Hilbert space
H(2)=H⊗H
for two-particle systems, or more generally
H(N)=j=1⨂NH
for N-particle systems.
The tensor product construction is not merely a formal extension of the
single-particle framework: it introduces genuinely new physical content
through the entanglement structure of states that do not factorize as
products
Ψ1⊗Ψ2,
which is the subject of QM9.
The harmonic oscillator of the present paper enters QM7 in a specific
and concrete way: through the coupled harmonic oscillator system with
Hamiltonian
H^coupled=H^osc(1)+H^osc(2)+κx^1x^2.
Here H^osc(1) and H^osc(2)
are the individual oscillator Hamiltonians and
κx^1x^2
is a bilinear coupling.
The normal mode transformation---a linear canonical transformation that
diagonalizes H^coupled into two independent
oscillators with shifted frequencies
ω±=ω1±mω2κ
---is the simplest instance of a linear canonical transformation.
After the normal mode transformation, each normal mode is an independent
harmonic oscillator whose Fock states and coherent states are given by
the present paper.
The coupled oscillator provides the first example in the QM-series where
a many-body problem is solved by a transformation that decouples the
degrees of freedom, and it serves as the conceptual prototype for the
more complex decouplings---the Bogoliubov transformation in many-body
physics, the mode decomposition in quantum field theory---that appear in
the later papers.