Quantum harmonic oscillator in phase space

Continuing the physics posts, today I want to show how to solve a standard quantum mechanics problem: the harmonic oscillator, but do it in the unusual phase space formulation of quantum mechanics. It is not until we can solve concrete problems that we truly understand a new formalism. For today's result I will follow a pedagogical paper by Emile Grgin and Guido Sandri from 1996.

The Hamiltonian of the harmonic oscillator is obviously:

\(H=\frac{1}{2m} p^2 + \frac{m\omega^2}{2} x^2\)

and it is advantageous to work with dimensionless variables as follows:

\(p=\sqrt{\hbar \omega m} \eta\)
\(x = \sqrt{\hbar / \omega m} \xi\)
\(H = \hbar \omega \chi\)

If we recall the abstract products \(\alpha\) and \(\sigma\) in phase space:

\(\alpha = \frac{2}{\hbar} sin(\frac{\hbar}{2} \overleftrightarrow{\nabla})\)
\(\sigma = cos(\frac{\hbar}{2} \overleftrightarrow{\nabla}))\)

where

\(\overleftrightarrow{\nabla} = \frac{\overleftarrow{\partial}}{\partial \eta}\frac{\overrightarrow{\partial}}{\partial \xi} - \frac{\overleftarrow{\partial}}{\partial \xi}\frac{\overrightarrow{\partial}}{\partial \eta}\)
\(f\overleftrightarrow{\nabla}g = \{f,g\} \): the Poisson bracket

with \(\eta\) and \(\xi\) cannonical variables (dimensionless momenta and position) and the funny arrows are used as a reminder of how they act: on the left or on the right arguments.

In phase space if \(u\) is a state, the condition for a pure state is:

\(u=u\sigma u\)

the expectation value for an observable \(f\) and a state \(u\) is:

\(\langle f {\rangle}_{u} = \int f \sigma u ~dx dp\)

and the characteristic equation corresponding to the eigenvalue-eigenvector problem in standard Hilbert space formalism is:

\(f \sigma u_{\lambda} = \lambda u_{\lambda}\)

In the Hilbert space formulation, the time-independent Schrodinger equation in the dimensionless parameters reads:

\(\frac{1}{2}(\xi^2 - {\partial_{\xi}}^2) \psi = \lambda \psi\)

and the solution is well known:

\({|\psi|}^2 = \frac{1}{\sqrt{\pi}2^n n!} H_n^2 (\xi) e^{-\xi^2}\)
\(\lambda = n+1/2\)
with \(H\) the Hermite polynomials.

Now what do we get in the phase space formulation? The Hamiltonian is:

\(\chi = \frac{1}{2}(\xi^2 + \eta^2)\)

and we can compute \(\chi \alpha\) and \(\chi \sigma\):

\(\chi \alpha = \eta \partial_{\xi} - \xi \partial_{\eta}\)
\(\chi \sigma = \frac{1}{2}(\xi^2 - \hbar^2 / 4 \partial^2_{\xi}) + \frac{1}{2}(\eta^2 - \hbar^2 / 4 \partial^2_{\eta})\)

because higher powers of \(\overleftrightarrow{\nabla}\) vanish as they apply on the left side as well.

\(\chi \alpha \) is the same as in classical mechanics and \(\chi \sigma \) is a double copy of the Hamiltonian in the Hilbert space formalism!!! And this is the key to prove the equivalence between the two approaches. What you get from the characteristic equation is a sum of the products of the Hermite polynomials which by some Hermite polynomials identity magic reduces to the desired result. But those are mathematical details relevant only to test one's knowledge of functional analysis. The full mathematical details can be found here and here. We are after the conceptual ideas.

For the problem above the link between the phase space state \(u\) and the Hilbert state \(\psi\) is given by:

\({|\psi(\xi)|}^2 = \frac{1}{2\pi}\int_{-\infty}^{+\infty} u(\xi, \eta) ~d\eta\)

What I presented so far is the quantum mechanics formulation in phase space using the observable product \(\sigma\). But the products \(\sigma\) and \(\alpha\) can be combined to create an associative product \(\star = \sigma + J \frac{\hbar}{2}\alpha\) known as the Moyal, or the star product.

Jose Enrique Moyal

Recalling that \(u\sigma u = u\), similarly in the star product formulation the pure states \(F\) respect:

\((2 \pi \hbar) F\star F = F\)

and the eigenvalue-eigenvector problem is not surprisingly:

\(H\star F = \lambda F\)

For details please see here.

Now we can combine all this and past posts to introduce the high level overview of quantum mechanics in phase space and its relationship with quantum mechanics in Hilbert space.

Phase space:

observables = differentiable functions on phase space
generators = vector fields on phase space
1-to-1 map observable and generators: J with \(J^2 = -1\)

observable product: \(\sigma = cos(\frac{\hbar}{2} \overleftrightarrow{\nabla}) \)
generator product: \(\alpha = \frac{2}{\hbar} sin(\frac{\hbar}{2} \overleftrightarrow{\nabla})\)
associated product: \(\star = e^{J\frac{\hbar}{2}\overleftrightarrow{\nabla}}\)
state space (for the products \(\sigma\) or \(\star\)) = phase space

Hilbert space:

observables = hermitean operators
generators = anti-hermitean operators
1-to-1 map observable and generators: J with \(J^2 = -1\)

observable product: \(A\sigma B = \frac{1}{2}(AB + BA)\) - the Jordan product
generator product: \(A\alpha B = \frac{J}{\hbar}(AB-BA) = \frac{J}{\hbar} [A,B] \)
associated product: \(A\cdot B = A B\) regular operator multiplication

state space (for the products \(\sigma\) or \(\cdot\)) = Hilbert space

The states can be  defined for the observables product or for the associated product. In the first case one encounters the Jordan-GNS construction and in the second case the usual GNS construction.

One goes from the phase space formalism to the Hilbert space formalism by a quantization procedure, and the simplest one is Weyl quantization which directly constructs the operators in the Hilbert space. Many other quantization procedures are known.

The same thing is valid for classical mechanics with the main difference that \(J^2 = 0\)!!!! There the product \(\alpha\) is the Poisson bracket, and the product \(\sigma\) is regular function multiplication on phase space.

Deformation quantization is the process which transforms the products \(\alpha\) and \(\sigma\) from classical to quantum realizations. the dimension of the map \(J\) is \(\hbar\) and this observable to generator map (known as dynamic correspondence) corresponds to Noether's theorem.

So now what? How can we understand quantum mechanics? There are basically two camps, the ontic and the epistemic ones. If you are in the ontic camp you you may like the Bohmian interpretation where particles have a well define position at all times. But what prevents us to attach an ontic interpretation to the phase space formulation as well? After all phase space is well understood and classical mechanics does not have any interpretation issues. Sure, in the phase space formulation one encounters quasi-probabilities in the form of Wigner functions, but you have to compare their strangeness this with the strangeness of the quantum potential. Something has to be different than in the case of classical mechanics otherwise you don't get quantum effects.

If we attach an ontic meaning to quantum mechanics in phase space, now we have two distinct ontic interpretations. But an ontic interpretation must be unique, otherwise it cannot be taken seriously. 

The existence of the phase space formulation of quantum mechanics presents the greatest challenge to the Bohmian interpretation. This formalism has the same limitations as Bohmian quantum mechanics particularly in the treatment of spin which is a pure Hilbert space phenomena with no classical counterpart. Why should non-detectable violations of relativity (in the case of Bohmian quantum mechanics) be better than non-detectable violations of positive probabilities (in the case of quantum mechanics in phase space)?