For Schrodinger eq, harmonic oscillator potential
Calculate all ฮด action Lie bracket
This commutation relation is partially similar to complexification eigenvalue technique (ref-13, p.20โ30) (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
[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
Regarding field quantization
One perspective is path integral quantization of fields.
[field-path-integral-quantization] Question Since the harmonic oscillator can be path-integrated by eigenvalue diagonalization & generalized Gaussian integral, why don't KG eq (or Dirac eq) similar to the harmonic oscillator eq also perform eigenvalue diagonalization & generalized Gaussian integral path integral?
Another (?) perspective is field operator quantization of fields.
recall Klein--Gordon-equation Consider the plane wave solution
If we ignore plane waves, even if we might lose accuracy, we get an ODE
This is the -valued point particle harmonic oscillator equation, with frequency
Then the point particle harmonic oscillator can be quantized
recall linear-superposition-of-KG-eq superposition of KG plane waves on the hyperboloid
Question Homomorphize the quantum harmonic oscillator of point particles to the quantum harmonic oscillator of the KG field, with coefficient constraint (Sobolev), the coefficient constraint of multiple raising and lowering corresponds to symmetric tensors, the entire space is (where is the space of harmonic oscillator quantization)
Question Is this tensor based on the module?
Vacuum is something lower than zero order
The energy operator of the KG field quantum harmonic oscillator can also be expressed in the form of KG field operator + energy of the KG field, even if this is not calculating the energy of the KG field (field operator after differential plane wave expansion)
For the Dirac field, it seems that there is no need to do what the KG field does, because "once" quantization already exists, but it is finite-dimensional, a representation comes from the two eigenvalues of Pauli-matrix (ref-18, p.305โ308)
After adding parity, it is
recall linear-superposition-of-Dirac-eq superposition of Dirac plane waves on
Question Homomorphically map it to the quantum field of the Dirac field, plus coefficient restriction (Sobolev), the coefficient restriction of multiple raising and lowering corresponds to alternating tensor, the entire space is
Question Is this tensor based on the module?
Vacuum is something lower than zero order
The energy operator of the Dirac field quantum field can also be expressed in the form of Dirac field operator + energy of the Dirac field, even if this is not calculating the energy of the Dirac field