harmonic-oscillator-quantization

For Schrodinger eq, harmonic oscillator potential

This commutation relation is partially similar to complexification eigenvalue technique (ref-13, p.20–30) (will be used in the treatment of angular momentum operator)

Complexify to get the characteristic operator of , obtaining [ladder-operator]

This commutation relation indicates that has eigenvalues with uniform spacing

Question Try to generalize the technique here to (if possible) classical harmonic oscillator

The lowest energy state of the eigenstates given by the ladder operator of harmonic oscillator quantization satisfies [harmonic-oscillator-ground-state]

Calculate the action of the operator, obtaining the lowest energy of

high order energy states (normalized)

Energy is

Using the eigenstate can be written as

is called Hermite polynomial

For quantum harmonic oscillator, even for static wave function, there are possible characteristic energies

Warning Don’t assume that since the lowest energy is non-zero , there is energy out of nowhere, because the energy of a static hydrogen atom can still be negative

It can be proven that this eigenstate series orthogonally expands

For example, prove that expands using Fourier transform method

Assume that the orthogonal , define

Fourier transform

All order derivatives at are equal to zero

The quadratic form interpretation of the expansion coefficients is the probability in . The expected energy is

In addition to causing the eigenvalues of to be uniformly spaced, the ladder operators also satisfy , which allows them to correspond to

metric symmetric tensor space
symmetric polynomial space

They also satisfy

Since it is not , the situation for different states to complexified eigenvalue technique

[Gaussian-integral]

Holds for . contains dense points, consistent with the uniqueness of analytic continuation. Analytically continued to . But note that has a double branch.

Diagonalize the quadratic form into on Euclidean type using an orthonormal basis.

[why-pi-in-Gaussian-integral]

This might provide a clue as to why appears in the Stirling approximation of the factorial .

The characteristic polynomial of the harmonic oscillator ODE is , the prototype is , so and complex numbers are introduced, which leads to circles, and thus . is related to the ground state of the quantum harmonic oscillator. For simplicity, is omitted. The general momentum operator actually corresponds to a phase change . If we add a scaling factor to the momentum operator, then the momentum operator can correspond to a phase change . At this time, the ground state may also become where contains a factor, and its integral is directly normalized, without needing to add a scaling factor. Similarly, for Feynman path integrals, using this method may no longer require additional normalization factors or Zeta function regularization.

The appearance of in Stirling’s approximation might also be similar. One should ask where the factorial (or its reciprocal) with the scaling factor comes from, for example, from the volume calculation of spheres and spherical surfaces.

Another revelation is that the appearing in the kernel of the Feynman path integral quantization of the harmonic oscillator corresponds to the property of the factorial function , where an additional scaling factor also appears. Therefore, should the modified factorial function satisfy ?

If the solution of the harmonic oscillator uses fixed starting positions , then

where

Action

where

For time only depending on the difference

[path-integral-quantization]

cf. (ref-28, ch.path-integral-formalism)

Propagator represents constructing unitary using Feynman path integrals and Lagrangian.

For free field

Decomposed into classical path and gap ,

Boundary is zero

==>

Now

Where, due to the classical path of a free particle being a straight line

Boundary is zero

==>

Now

As a generalization of Gaussian integral

Quadratic form
. To simplify notation, use

==> Eigenvalues . orthogonal eigenfunctions . Expansion of orthogonal eigenfunctions or Fourier expansion

Diagonalize with orthogonal basis. Now

Using the normalization from why-pi-in-Gaussian-integral, part of the infinite product becomes . The final result is

The total result is

For the harmonic oscillator, similar to

Using integration by parts

Quadratic form
. For simpler notation, use

==> Eigenvalues . orthogonal eigenfunctions . orthogonal eigenfunction expansion or Fourier expansion

Diagonalize using an orthogonal basis and a generalization of the Gaussian integral. The difference from the free field is the emergence of a new infinite product (cf. Euler-reflection-formula)

The result is

The total result is

Is this method not generalizable to the hydrogen atom problem? It is said that there is a method to transform the hydrogen atom problem into the path integral of the harmonic oscillator under the symmetry.

Question Inspired by the classification of regularity in the discussion of the proof of Stokes’ theorem, it seems like the concepts of propagator and Sobolev systems are quite compatible

[eigen-decomposition]

Characteristic equation given by

Decomposition of given by characteristic orthonormal basis

According to wiki:Path_integral_formulation, then let perform Taylor expansion, where corresponds to energy level

There should be a kind of “spectral theory”, “spectral measure”, that can define through normalization techniques

In addition to finite-dimensional and discrete countably infinite-dimensional spaces , one can also consider integral infinite-dimensional spaces . In this case, the “basis” is not required to be in , for example . However, the decomposition coefficients of the function with respect to the basis should be in ,

Regarding field quantization

One perspective is path integral quantization of fields.

[field-path-integral-quantization] Question Since the harmonic oscillator can perform path integration by eigenvalue diagonalization & generalized Gaussian integration, why don’t KG eq (or Dirac eq) which are similar to harmonic oscillator eq also perform eigenvalue diagonalization & generalized Gaussian integration for path integration? Moreover, the Lagrangian of the harmonic oscillator is very similar to the Lagrangian of KG eq. Then time corresponds to spacetime , and position corresponds to field value . But I don’t support using a rectangle in for path integration; a setting more compatible with should be used. For example, perhaps the time interval should correspond to the spacetime quadratic interval ? Not using because it is not translation invariant?

Another (?) viewpoint is field quantization using field operators

recall Klein--Gordon-equation Consider plane wave solutions

But this depends on the decomposition of time and space, which is not conducive to generalization to general spacetime manifolds

For the Dirac field, other structures need to be used. Use two eigenvalues of Pauli-matrix (ref-18, p.305–308)

After adding parity, it is

note-math