# Is there a Schrödinger equation for phase space?

The Schrödinger equation is generally formulated in position space $$i \hbar \frac{\partial}{\partial t}\psi(x,t) = \hat H_x \psi(x,t) = \left [ \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x,t)\right ]\psi(x,t)$$ or in momentum space $$i \hbar \frac{\partial}{\partial t}\psi(p,t) = \hat H_p \psi(p,t) = \left[ \frac{p^2}{2m} + V\left(\frac{\hbar}{i}\frac{\partial}{\partial p},t\right)\right ]\psi(p,t).$$

But is it also possible to derive a Schrödinger equation for position $\times$ momentum $=$ phase space? I assume in that case it should be an equation acting on a wave function $\psi = \psi(x,p,t)$ . So I guess simply multiplying the two Schrödinger equations above is not enough. How could one then derive a Schrödinger equation for phase space, if it is possible at all?

• There are some complexities here, forget the wave function and consider the probaility distribution $P(r)$ where $r$ is a point in phase space. Basically over any probability distribution $D$ over the phase space characterized by $\sigma_x \sigma p \le \frac{\hbar}{2}$ We can apriori tell you by the uncertainty principle that $\int_{\text{all of phase space}} P(r) D = 0$ yet somehow for bigger probability distributions these integrals have to be not 0, so this is an unusual mathematical object to say the least. Oct 9 at 1:31

There is no "wavefunction in phase space" because the wavefunctions $\psi(x)$ and $\psi(p)$ are obtained from the abstract state vector $\lvert \psi\rangle$ by $\langle x\vert \psi\rangle$ and $\langle p\vert \psi\rangle$, respectively. Since position and momentum don't commute, there are no $\lvert x,p\rangle$ to get a naive $\psi(x,p)$.

However, from any wavefunction we may obtain the Wigner quasiprobability distribution $W(x,p,t)$ on the classical phase space by the Wigner-Weyl transform. It obeys the equation $$\partial_t W(x,p,t) = \{\{H(x,p,t),W(x,p,t)\}\}$$ with $H$ the classical Hamiltonian and the bracket on the r.h.s. as the Moyal bracket.

There actually is a formulation, due to Torres-Vega and Frederick, but let me hasten to add that you really don't want to go there, for about half a dozen good reasons, most of them very technical, and some quite conceptual.

As @ACuriousMind already outlined, the classic satisfactory and powerful formulation of QM in phase space involves the analog of the density matrix (which is traced with observables to yield expectation values in Hilbert space), the Wigner Function (which is integrated with c-number observables in phase space instead). You might consider this introduction Quantum mechanics in phase space, by Curtright, Fairlie & me, if you wished to know more about what this formulation gets you that the other two, Hilbert space, and path integrals, don't. (You might choose to focus on eqn (91) and Exercise 0.11 there for a gateway to the T-V & F reformulation. Remember, all phases needed for interference are already in off-diagonal Wigner Functions!)

The Torres-Vega/Frederick formulation I mentioned finds its meaning by painful projection/reference to the Wigner Function mentioned, but, once you stretch your conceptual horizons to appreciate that formulation, you might realize you really have no use for the Schroedinger equation you are envisioning, but its density matrix analog, the von Neumann equation, instead, much as you derive stat mech expectation values by integrating agains Liouville densities.

Added edit: The Ambiguity function connects to the WF via a 2d Fourier transform, as indicated; basically, all roads go through the WF. For instance, by multiplying the coordinate to the momentum space wave functions, you came close to defining the Mehta distribution function, cf my answer to 233353, Exercise 0.19, but by a phase... what Terletsky and Blokhintsev did in the 30s. The systematic theory of all of these connexions could be accessed in L Cohen's book.

• Your phase space function in your intro book looks like a signal processing ambiguity function. Good for analysis even if no different information than the original wavefunction. Anyway, does the magnitude of the function have a clear physical meaning, or does the phase and the negatives have some interpretation? Just thinking outloud, in comparison to signal processing. Have not read the online book yet but will look more carefully at it. Sep 19, 2016 at 0:36
• Yes, the ambiguity function and signal processing WF are invertible transforms of each other. The math of QM and signal processing are identical to each other, except the effective hbar in EE is 1. In QM the negative values are safe, necessary, and the hallmark of QM interference. They are protected from bservatin by the uncertainty principle. The magnitude of the (real) WF represents increased measure/contribution at a point in phase space. Sep 19, 2016 at 2:24
• Thanks. I've done more ambiguity functions than I ever wanted to, so I'll read your online book. Maybe I'll understand it. Sep 19, 2016 at 4:01
• Maurice de Gosson expands a little on the mathematics (see e. g. arxiv:math.0503709), but as you correctly point out, the whole point of view is a bit pointless. Mathematically speaking, what Gosson presents is equivalent to the usual point of view (the exact nature of the equivalency depends on your mathematical framework of choice), so there are no additional insights to be had. I'd much rather suggest you have a look at the Wigner-Weyl calculus, because this can be used to perform semiclassical limits and construct perturbation expansions. Sep 20, 2016 at 4:50

There is an obvious candidate for what you are looking for in the process of geometric quantisation (https://en.wikipedia.org/wiki/Geometric_quantization). This is mostly a mathematical game, and the only books I have ever encountered it in are of a strongly mathematical nature.

Geometric quantisation seeks to generate a quantum Hilbert space by considering wavefunctions on the phase space, together with position and momentum operators. The Schroedinger equation then has the same form, but the wavefunctions and operators are defined on phase space.

However, this approach runs into some serious trouble. Basically, a wavefunction on position or momentum space only already has the correct number of degrees of freedom, so using a wavefunction on phase space has too many. This problem is solved by only considering wavefunctions obeying certain symmetry requirements. The problem is that the operator content of the theory depends on a choice of multiple possible symmetry requirements, and usually the theory contains way too few operators (it is not uncommon for the Hamiltonian not to be part of the theory).

There is, however, a notable exeption. In Chern-Simons theory (https://en.wikipedia.org/wiki/Chern%E2%80%93Simons_theory), the classical phase space has a non-trivial geometry that makes geometric quantisation the only viable candidate. Luckily, this theory is a topological gauge theory, and therefore the operator content is extremely limited. Due to this, geometric quantisation is sufficient to quantise the theory in some simple cases, as Witten showed (Comm. Math. Phys. Volume 121, Number 3 (1989), 351-399). One then uses formal properties of the path integral to extend this to arbitrary closed spacetimes.

This last example might be a bit over the head of anyone not familiar with the topic already, but what one should take away from it is the following: Geometric quantisation is a mathematically heavy procedure that many physicists do not even know about, yet it has at least one useful application in theoretical physics, which makes it potentially useful despite the theoretical difficulties in learning it.

What you're asking for is the application to quantum theory of time-frequency analysis (along with time-frequency representations), which are common staples in Digital Signal Processing that haven't quite yet found their way into theoretical physics, with a few notable exceptions, such as: Kaiser and G.H. Battle. Usually, time-frequency analysis is applied to time series - hence the name - but it could also be applied to signal domains of two or more dimensions, like the two-dimensional (or even three-dimensional) signals in graphics and image processing. It can also be applied to developing state-space representations in quantum theory. I talked with Kaiser briefly a while back and he says he's working on a sequel to the work in his research, including the 2011 book that I linked to above, that's more geared toward quantum applications.

The Wigner Function is an example of a time-frequency distribution, but that's really only the tip of the iceberg. There are a lot more time-frequency distributions and transforms than that, and Quantum Physicists are significantly behind the curve on this matter and, as a rule, the theoretical Physics community has Dunning-Kruger on matters related to Fourier Analysis and its extension to Time-Frequency and Time-Scale Analysis. (To put it more directly: when they speak with authority on complementarity, they're outside their field and need to get back to their lane.)

More appropriate in place of time-frequency analysis, actually, is what would be considered "Time Scale" analysis - where in place of frequency on a linear scale, is the version of it on a logarithmic scale, which linked to what's called "scale". This is where octaves are equally-spaced. It is also more closely aligned to how rescaling is approached in renormalization theory. The basis functions in time-scale representations are called "wavelets", though the term is usually reserved for the special case of time-series transforms where the inverse transform has the same form as the forward transform. These are called the Wavelet Transforms - which can be done discretely or continuously. It has heavy application in multi-dimensional signal analysis, specifically image-processing and motion-processing. Battle and Kaiser work primarily with time-scale analysis, rather than time-frequency analysis, in the application of their works to quantum theory.

The Wigner function is the time-frequency transform that results if you apply the operator correspondence to the delta function, itself: $$δ\left(q - q_0, p - p_0\right) → δ\left(\hat{q} - q_0, \hat{p} - p_0\right),$$ the Wigner transform, itself, being the result of applying this to the wave function representation of a state, $$W_ψ\left(q_0, p_0\right) = ❬ψ|δ\left(\hat{q} - q_0, \hat{p} - p_0\right)|ψ❭.$$ The main advantage of the Wigner function is that it gives you the obvious results for reasons that are made clear by this way of defining it, e.g. \begin{align} \int W_ψ\left(q_0, p_0\right) F\left(q_0\right) dq_0 dp_0 &= \int ❬ψ|δ\left(\hat{q} - q_0, \hat{p} - p_0\right)|ψ❭ F\left(q_0\right) dq_0 dp_0\\ &= ❬ψ|\left(\int δ\left(\hat{q} - q_0, \hat{p} - p_0\right) F\left(q_0\right) dq_0 dp_0\right)|ψ❭\\ &= ❬ψ|F(\hat{q})|ψ❭. \end{align}. Similarly, $$\int W_ψ\left(q_0, p_0\right) G\left(p_0\right) dq_0 dp_0 = ❬ψ|G(\hat{p})|ψ❭.$$

A "quantized delta function" is ambiguous in that different operator ordering conventions may lead to different implementations of it. In fact, the quantized delta function could, itself, be considered as the very kernel of the operator-orderer that defines the operator ordering convention. The one that goes with the Wigner function is the "Weyl ordering", which symmetrizes the $$p$$'s and $$q$$'s, e.g. $$\widehat{(qp)} = ½\left(\hat{q}\hat{p} + \hat{p}\hat{q}\right)$$. Other variants are those corresponding to other conventions, such as putting all the $$p$$'s on one side, all the $$q$$'s on the other, which are instances of the time-frequency distributions known as the Rihaczek distributions (an example of an application).

They each have the disadvantage of being bi-linear, not linear. The more accurate way to write Wigner, for instance, would be as $$W_{ψψ'}\left(q_0, p_0\right) = ❬ψ|δ\left(\hat{q} - q_0, \hat{p} - p_0\right)|ψ'❭.$$ It's quadratic in $$ψ$$, with $$W_ψ$$ actually being $$W_{ψψ}$$. So, under addition, you get cross-terms: $$W_{ψ+ψ'} = W_ψ + W_ψ' + W_{ψψ'} + W_{ψ'ψ}.$$ Therefore, Wigner distributions are laden with interference.

Another disadvantage is that the distribution can be negative. In effect, it is an "over-sharpened" picture: it's what you get if you try to de-blur an already sharp image. In fact, if you smear a Wigner distribution with a Gaussian that has a one-SD ellipse sufficiently large, the result will be a strictly non-negative distribution. In turn, the result will be equivalent to a representation by "coherent states".

With either time-frequency or time-scale distributions it is possible to get most of the advantages of Wigner, without the interference or negativity issue; with the transforms being linear, not bi-linear. Wavelets already get you part of the way there, but you can actually go further by just simply dropping the requirement that the inverse transform have the same form as the forward transform.

Example
Here's an example of a Wigner-like time-scale transform constructed in my secret subterranean dungeon Scalographic demo. The mathematics for it are in the description, but I'll replicate the analysis here, to serve as a point of reference.

In the description, the functions $$f(t)$$ play the role of your wave function $$ψ(t)$$, while $$ψ$$ and its Fourier transform $$Ψ$$ are used to denote the wavelet functions. Sorry for any confusion of notation.

If $$f(t)$$ is the function for the waveform, $$f(t,p) = \int f\left(t + \frac{λ}{p}\right) ψ(λ)^* dλ$$ describes its scalogram, with $$t$$ as the horizontal coordinate, $$p$$ as the vertical coordinate (displayed on a logarithmic scale), and where $$(⋯)^*$$ denotes complex conjugate. The original waveform is recovered as $$f(t) = \int f(t,p) d \log p,$$ which requires that $$\int Ψ(γ) d \log γ = 1$$ (or more generally, that it be finite and non-zero), where $$Ψ(γ) = ∫ ψ(λ) \exp(-2πiγλ) dλ$$ is the Fourier transform of $$ψ(λ)$$.

The scalogram can be arbitrarily re-located as $$F(t,ν) = \int f(t,p) δ(ν - ν(t,p)) ν d \log p$$ where the part of the scalogram at $$(t,p)$$ is relocated to $$(t, ν(t,p))$$, and still produces the same sound $$f(t) = \int F(t,ν) d \log ν$$. The actual relocation carried out is that corresponding to the Instantaneous Frequency, given here by the identity $$4πi ν(t,p) |f(t,p)|^2 = f(t,p)^* \frac{∂}{∂t} f(t,p) - f(t,p) \frac{∂}{∂t} f(t,p)^*.$$

A simple, but naive, method (which was used in the video) is to just use the windowing $$ψ(λ) = \exp(2πiλ)$$ over one period $$λ ∈ [-½,+½]$$. It is sloppy, but gets the job done - mostly. However, with it, there will be effective interference, which registers as wiggling in the frequency lines in higher octaves. (That shows up in the video as the "wigglies" that are synched to frequencies below the displayed area - the sound is very deep. Use headphones!)

Windowing should actually be done in the frequency domain as $$Ψ(γ) = |φ(\log γ)|^2, \quad ψ(λ) = \int |φ(\log γ)|² \exp(2πiγλ) dγ,$$ where $$φ(z)$$ is any of the usual spectrographic windowing functions, with a cut-off, say, of one octave $$\log √½ ≤ z ≤ \log √2$$, and normalized with $$\int |φ(z)|^2 dz = 1$$.

• Actually this stuff on frequency analysis and wavelet analysis is pretty well known is theoretical physics. See for instance Ali ST, Antoine JP, Gazeau JP. Coherent states, wavelets and their generalizations. New York: Springer; 2000, or Cohen L 1995 Time-Frequency Analysis (Englewood Cliffs, NJ: Prentice Hall). Oct 9 at 1:46
• I will add that engineer typically use this formalism and try “clean up the interference” whereas physicist are particularly interested in those regions where interference effects are importants. Thus both discipline use the same tools, under different names, but in rather different regimes and with different applications in mind. Oct 9 at 1:48
• I think it needs to be better-known. It's still kind of sideshow in the theoretical community, though it has made headway on the strength of the applications of wavelet analysis in more recent times. Oct 10 at 3:31
• Mother Wavelet might be a little upset by the simplicity of analysis, compared to her treatments of the issue. :) Can the time-frequency and time-scale analyses be hybridized? That's what I've been trying to prepare for ALGLIB++. Also: what are the equations of motion under time-frequency or time-scale representations? Oct 11 at 0:21
• But, who's Mother Wavelet? I think Kaiser treats equations of motion in the final chapters of the book reference I linked to. Oct 11 at 16:23

The answers given so far are correct, but I don't think they hit the core of the issue. Quantum mechanics has either a position or a momentum representation where this or is an exclusive or. In general quantum physics is described according to a complete set of commuting operators, and in the case of momentum and space this means one must work in one representation, but not two at the same time.

Hamiltonian mechanics in phase space describes dynamics according to the energy surface in the $6n$ dimensional phase space, for $n$ particles in $3$ dimensional space, and $3$ dimensional momentum space. The condition $E~=~H$ on the energy surface reduces the space of motion to $6n-1$ dimensions. Lagrangian mechanics works in configuration variables that are $3n$ dimensional. This is then "half of phase space." Quantum mechanics then operates in this domain when working with configuration variables. This is even though QM and the Schroedinger equation uses a Hamiltonian.