Quantum phase space

Drag the blob by its ⊕ centre

W(x, p) −1/πħ

+1/πħ

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The picture is phase space: position x across, momentum p up. The red blob is a quantum state drawn as a cloud of "where, and how fast" (its Wigner function); the black dots are classical particles started in the same cloud. In this harmonic well the two circle together, once per period.

Drag the blob by its ⊕ centre (or click where you want it) and it circles from there. The strips above and at the right are the blob's shadows on the two axes: the probability distributions of x and of p, what a measurement would find.

Then press Quartic under Potential (or Set it up in experiment 2): the outer dots now run ahead of the inner ones and the cloud winds into a spiral. The quantum blob follows for a while, then breaks into fringes with blue patches, where its Wigner function is negative, which no cloud of classical particles can ever be.

1. Quantum and classical agree

A coherent state in a harmonic well, $V=\tfrac12\omega^2x^2$: a round Gaussian blob, as small as quantum mechanics allows, with 3000 classical points drawn from it. Watch it circle once per period $2\pi/\omega$ without changing shape, the classical points circling with it, exactly. Drag the blob somewhere else: a bigger circle, the same period. Under the plot, the least value of $W$ stays 0 (no blue anywhere), and in the readout the uncertainty $\sigma_x\sigma_p$ stays at its limit $\hbar/2$. Why: for a quadratic $H$ the Wigner function obeys the classical Liouville equation, so this $W$ flows exactly like a classical fluid.

2. Where quantum and classical part

The same coherent state in an anharmonic well, $V=\tfrac12x^2+\lambda x^4$, where bigger orbits take less time. Watch the dots: the outer ones run ahead of the inner ones and the cloud winds into a spiral, its arms ever longer and thinner. $W$ follows at first, then cannot: where two arms of the one wave function lie close together they interfere, and $W$ fills with fringes, negative (blue) between the positive ones; by $t=10$ the negative part of $W$ has a volume of 0.9 (under the plot). The classical cloud never does this. Why: $W$ moves by $$\frac{\partial W}{\partial t}=-\frac{p}{m}\frac{\partial W}{\partial x}+V'\frac{\partial W}{\partial p}-\frac{\hbar^2}{24}V'''\frac{\partial^3W}{\partial p^3}+\dots,$$ the classical Liouville equation plus corrections in powers of $\hbar^2$, with ever higher derivatives of $V$. For a quadratic $V$ they all vanish; for this quartic one only the term shown is left, and it is the whole difference between the two pictures. Make $\lambda$ smaller (the slider under $V$) and the parting comes later.

3. Free motion shears phase space

No force at all. Every point moves at its own speed: $x\to x+pt/m$, $p$ unchanged, so the blob is sheared into a tilted ellipse, ever longer and thinner. Watch its shadows: the one on $x$ widens (the packet spreads, $\sigma_x^2(t)=\sigma_x^2+\sigma_p^2t^2/m^2$ here), the one on $p$ never changes, and the blob's area never changes either. Again the classical points follow $W$ exactly. Drag the blob to a larger $p$: it travels faster but shears just the same, since the stretching comes from the spread in $p$, not from $p$ itself; squeeze $r$ harder (a narrower blob in $x$, so a wider spread in $p$) and it shears faster.

4. A squeezed state breathes

The ellipse is narrower than a coherent state in $x$ (by $e^{-r}$) and wider in $p$, with the same area. It turns rigidly, so its shadow on the $x$ axis, $|\psi(x)|^2$, narrows and widens twice per period: $\sigma_x$ goes from 0.32 up to 1.57 and back. $\sigma_x\sigma_p$ is $\hbar/2$ only when the ellipse lies along the axes. Try other angles and squeezes ($r<0$ squeezes $p$ instead).

5. Number states are rings, and negative

The oscillator's first excited state $|1\rangle$. Its Wigner function is $W_1=\frac{1}{\pi\hbar}\left(\frac{2(x^2+p^2)}{\hbar}-1\right)e^{-(x^2+p^2)/\hbar}$ (with $m\omega=1$): a positive ring around a negative centre, where $W_1(0,0)=-1/\pi\hbar$, as negative as any $W$ can be (the least $W$ under the plot: $-1/\pi\hbar$). Raise $n$: $|n\rangle$ has a disk and $n$ rings of alternating sign, with $W_n(0,0)=(-1)^n/\pi\hbar$, its outermost ring just inside the contour $H=E_n$. Nothing moves: a stationary state's $W$ is stationary. $W$ is not a probability here, so the classical points come from the Husimi $Q$ (experiment 11), which gives them about $\hbar\omega/2$ more energy on average (compare the two energies in the readout): they circle round the ring for ever.

6. A cat state's fringes

Two coherent states, at $\pm x_0$, added: $|\alpha\rangle+e^{i\varphi}|{-\alpha}\rangle$. Besides the two blobs, $W$ has fringes between them, positive and negative, spaced $\pi\hbar/x_0$ in $p$. They are the coherence between the blobs: a mixture (half a chance of each, no superposition) would have the blobs and no fringes. Change the phase $\varphi$: the fringes slide, and at $\varphi=180°$ the centre has $W(0,0)=-1/\pi\hbar$ (an odd state; in general $\pi\hbar\,W(0,0)$ is the mean value of the parity). The whole picture rotates rigidly, fringes and all. Drag one blob and the other goes opposite.

7. The marginals are what you measure

The same cat a quarter period on: the two blobs now have momenta $\pm p_0$ and sit on top of each other in $x$. $|\psi(x)|^2$ along the top has interference fringes: two waves with opposite momenta make a standing wave. Each point of it is the sum of $W$ over $p$ along a vertical line, positive and negative stripes together. The momentum distribution at the right has no fringes. Press Play and watch the fringes pass from one strip to the other as the picture turns. No measurement ever sees $W$ itself, only its shadows.

8. Stationary states sit on orbits

The Morse potential of a vibrating molecule, $V=D(1-e^{-ax})^2$, and its stationary state $n=4$. $W$ is concentrated along the classical orbit at the state's energy (the dashed contour), with fringes inside. The contours are drawn at the energies $E_n$, and the area inside each is $2\pi\hbar(n+\tfrac12)$: Bohr and Sommerfeld's rule, exact for this potential and close for most others. So each ring between two contours holds one state, and one state takes up an area $2\pi\hbar=h$ of phase space. Raise $n$ and the state moves out one ring. Above $E=D$ the orbits are open: the molecule comes apart.

9. Tunnelling in phase space

A double well, and $(|0\rangle+|1\rangle)/\sqrt2$: the particle in the left well, with a mean energy of 1.50, below the barrier at 3. Watch the classical points stay there, all but the 12% whose energy is above the barrier, which roll over the top and back. The quantum state goes through the barrier, where no classical orbit of its energy goes; halfway it is a cat state of the two wells, with fringes between them. It reaches the right well at $t=\pi\hbar/(E_1-E_0)=28.1$ and is back at 56.2. (Here the points come from $|\psi(x)|^2|\phi(p)|^2$, which has the state's own mean energy. The Husimi $Q$, the stand-in elsewhere, would raise their mean energy to 2.78 and send a third of them over the barrier: try it under Points, colours and grid.)

10. The uncertainty principle as an area

A squeezed state: squeeze it harder with $r$. The ellipse gets as thin as you like one way, but only by growing the other way: the area inside one standard deviation stays $\pi\hbar/2$, and $\sigma_x\sigma_p\ge\hbar/2$. Tilted, as here at 30°, the ellipse casts wider shadows on the axes than its own width, so the readout's $\sigma_x\sigma_p$ starts at 2.42, well above $\hbar/2$; what stays exactly $\hbar/2$ is $\sqrt{\sigma_x^2\sigma_p^2-\mathrm{cov}(x,p)^2}$, the area divided by $\pi$. (Set the angle to 0 and the readout shows $\hbar/2$ at $t=0$, and again every quarter period, when the turning ellipse lies along the axes.) No $W$ can be concentrated in a patch of phase space much smaller than $\hbar$: a Gaussian bump narrower than that in both $x$ and $p$ is not the Wigner function of any state, because a pure state has $\int\!\!\int W^2\,dx\,dp=1/2\pi\hbar$ exactly, and a mixed one less.

11. Husimi versus Wigner

The Husimi function $Q(x,p)=|\langle x,p|\psi\rangle|^2/2\pi\hbar$, built from the coherent state $|x,p\rangle$ centred at $(x,p)$, is $W$ smoothed by that coherent state's own Gaussian. It is never negative (its colours go to $1/2\pi\hbar$), but it pays for that: the cat's fringes are all but gone, and its marginals (dashed in the strips) are wider than $|\psi|^2$ and $|\phi|^2$. Switch between Wigner W and Husimi Q.

12. Tomography: shadows from every angle

The blue curve is the shadow of $W$ on the line at angle $\theta$: the distribution of the quadrature $X_\theta=x\cos\theta+(p/m\omega)\sin\theta$. In an oscillator $W$ turns, so the shadow at angle $\theta$ is what $|\psi(x)|^2$ will be a time $\theta/\omega$ later. Here $\theta=45°$: note the shadow's shape, press Play and watch the top strip. At $t=\pi/4=0.79$ it has that shape (Restart to compare again). Drag $\theta$ round: every shadow is a probability distribution that can be measured. A homodyne detector measures $X_\theta$ for a light field, with $\theta$ set by the phase of a reference laser; recording the shadows at all angles and inverting them gives $W$ itself. This is how the Wigner functions of cat states of light have been measured. The inversion is the one a CT scanner does with X-ray shadows: see the tomography demo (the projection-slice theorem and filtered back-projection), with $W$ in place of the patient.

13. Your own wave function

Type any $\psi_0(x)$: a flat-topped bump to start with. Its steep sides need high momenta, so $W$ reaches out in $p$, with fringes and negative regions, and the oscillator turns it all rigidly. Try exp(-x^2/2) (1 + x), exp(-(x-2)^2) + exp(-(x+2)^2) or exp(-x^2/2) cos(k x).

How it works

A classical particle is a point $(x,p)$ in phase space; an uncertain one is a cloud of points, a probability density that flows along the contours of $H(x,p)=p^2/2m+V(x)$. A quantum state has its own phase-space picture, the Wigner function $$W(x,p)=\frac{1}{\pi\hbar}\int\psi^*(x+y)\,\psi(x-y)\,e^{2ipy/\hbar}\,dy,$$ drawn here in colour: red where $W>0$, blue where $W<0$, white (black on a dark page) where it is zero. The colours are full at $\pm1/\pi\hbar$, the largest value $|W|$ can ever take. $W$ behaves like a probability density in one way: integrating it over $p$ gives $|\psi(x)|^2$ and over $x$ gives $|\phi(p)|^2$, the strips along the top and the right. But it can be negative, which no probability can; the line under the plot says how negative, and over how much of phase space.

The black dots are classical particles, each moving by Hamilton's equations under the same $H$; their histograms are the black lines in the strips. At $t=0$ they are drawn from $W$ when $W$ is positive; when it is not, from a positive stand-in (the note under the plot says which). The thin grey curves are contours of $H$ at the stationary states' energies; the dashed one is at $\langle E\rangle$. Units: $\hbar=m=1$.

How it is computed

$\psi$ evolves by the split-operator method, as in the 1D Schrödinger demo, on 1024 points (2048 for the free particle) between walls well outside the window. $W$ is computed from $\psi$ every frame, one FFT for every two columns of the picture: with $y=kh$ the integral is the sum $\frac{h}{\pi\hbar}\sum_k\psi^*(x-kh)\,\psi(x+kh)\,e^{-2ipkh/\hbar}$. Because of the 2 in the exponent, the momenta reach only $\pm\pi\hbar/2h$, half the usual range, so $\psi$ is first stripped of momenta beyond that (twice the window's), and then the sum is exact. $Q$ is a windowed Fourier transform, interpolated to the picture (to a few parts in a thousand of its peak). The stationary states come from the 1D demo's finite-difference eigenvalue solver; their energies are good to $10^{-4}$ or better here. Being the finite-difference states, not quite those of the exact kinetic energy, they do change a little as they evolve: by $10^{-4}$ to $3\times10^{-3}$ of their size by $t=10$. The classical points move by the leapfrog method, which keeps phase-space area exactly and energy to a few parts in a million. The quadrature shadow integrates $W$ along lines across the window. On a slow computer the picture of $W$ is computed on a coarser grid, to keep the page responsive.

New in 2026 for BrainFlux. Source: github.com/gallicchio/brainflux, MIT licence.