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I ran across a neat quantum mechanics question recently:

Consider a scattering gedankenexperiment in which spinless particles of a given energy are directed at a target. We wish to measure the wave function downstream from the scatter. We assume the beam is described by a pure state, and that the incident wave is a plane wave (over a sufficiently large spatial region). The beam is low density, so the particles do not interact with one another. To measure $|\psi(r)|^2$ over some volume of space, we just put a screen $S$ in a certain location, and gather enough statistics to get the probability density on this surface. We then move the surface and measure again.

Describe a modification by which the phase of the wave function $\psi(r)$ can be measured on the screen, apart from the overall phase, which is nonphysical and can never be measured.

I imagine it can be possible to infer the phase by seeing how the scattered wavefunction interferes with a "reference" wave, which can be done by, e.g. splitting off a piece of the incoming wave and routing it around the target to the desired point. But this feels a bit convoluted. Is there a nice and direct way to measure the phase of the scattered wave?

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knzhou
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  • In the simplest and the most direct method, the wave can be made to interfere with itself. For a sketch of such an experimental setup, see e.g., sec. 2.6 of Sakurai’s “Modern Quantum Mechanics” (1994), “Constant Potentials”; esp. Fig. 2.4. The following section on “Gravity in Quantum Mechanics” provides a real-world example of such an interference experiment; see Figs. 2.5-6. I do not see anything more basic than that setup. Everything else would “feel more convoluted” and less direct, since phases can be only measured via some form of the interference effect. – AlQuemist Jan 22 '18 at 08:48

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I'll describe here a conceptual method for phase measurement without going into the details of the experimental complications, the conditions of validity, or the accuracy.

Given a system of particles of mass $m$ described by the wave function in a stationary state. $$\psi(\mathbf{r}) = A(\mathbf{r}) e^{i\phi(\mathbf{r}) }$$ The probability current is given by: $$\mathbf{J} = \frac{\hbar}{2 mi } (\psi^{\dagger} \mathbf{\nabla} \psi - \psi \mathbf{\nabla} \psi^{\dagger} ) = \frac{\hbar}{ m } A(\mathbf{r}) \mathbf{\nabla} \phi(\mathbf{r})$$ Suppose that you have very small directional sensors, which can measure the intensity perpendicular to their surface, and a very precise timer. Then in principle you can put sensors along the coordinate axes in the vicinity of some point in space, and measure the currents which will be proportional to $\frac{\partial\phi}{\partial x}$, $\frac{\partial\phi}{\partial y}$, $\frac{\partial\phi}{\partial z}$. These measurements will allow the construction of the phase up to an additive constant.

The measurement results will be proportional to the square root of the intensity: $\sqrt{|A|^2}$; and in order to remove this factor, we will need an additional volumetric sensor to measure the intensity in a small volume around the same point in space.

This principle is the basis of the widely known noninterferrometric phase measurement (or phase retrieval\reconstruction) techniques.

David Bar Moshe
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  • nice answer; but, imo, it seems more "convoluted" than simple methods based on interference, although OP entails a rather subjective question. – AlQuemist Jan 22 '18 at 14:04
  • @PhilosophiaeNaturalis, please try to google the expressions I gave in the last paragraph to see in how many applications this technique has been used – David Bar Moshe Jan 22 '18 at 14:08
  • No doubt of importance and usage to the technique you mentioned. I noticed only its “complexity” compared to “simpler” interference-based experiments, because OP asks for simpler and more direct methods, afais. – AlQuemist Jan 22 '18 at 15:38
  • I like this, but I'm confused on one point. How exactly does a 'small directional sensor' measure a probability current? – knzhou Jan 23 '18 at 20:42
  • Under the same assumptions as in the question, suppose that our beam is composed of identical non-interacting particles, all in the same identical state given by the above wave function. Then what we are measuring is the second quantized version of the probability current, and its components can be estimated by the average number of counts in a certain direction per unit area per unit time. In order to achieve directionality, we will need not an omnidirectional screen as in the question, but a tiny screen that counts only the particles hitting it perpendicular to its surface. – David Bar Moshe Jan 24 '18 at 10:41
  • Cont. This is the directional sensor. Now, about how practically such a sensor can be realized. Suppose for a moment that our particles are photons, then our sensor can be conceptually realized as a tiny focusing lens and a counter located at an offset in the focusing plane. This counter will only count incoming photons reaching the lens from the direction diffracted to the offset position. By the way, in the book Quantum Mechanics on Phase Space by F.E. Schroeck you can find discussions and conceptual mathematical modeling of screens bubble chambers etc. – David Bar Moshe Jan 24 '18 at 10:41
  • @DavidBarMoshe Interesting, thanks! I'm slightly puzzled now, because I thought there was no unique way to define the probability current, i.e. you can always add a divergenceless function to it, but you seem to have a way to measure it directly. If you know the resolution there please look at this question too! – knzhou Jan 26 '18 at 13:49
  • I have read your previous question. In second quantization, (where we interpret $\psi$ as the annihilation operator and $\psi^{\dagger}$ as the creation operator; all three operators define self adjoint operators, therefore they should be measurable. In particular, $J_3$ has the meaning of the current in the center of mass system. They indeed differ by a divergenceless vector field, but they are measurable regardless. The first one $J_1$ is quartic in the annihilation\creation operators, thus it would need higher statistics in its measurements (not just counts). – David Bar Moshe Jan 29 '18 at 11:34