# Measuring photon momentum without annihilating it

Is it possible to build an apparatus that could do the following (at least ideally in principle)? If so, what would it be like?

A fairly localized light wave packet (with a corresponding spread in momentum) reaches the origin of our coordinate system from any direction in the XY plane (taking 2 dimensional case for simplicity).

It now interacts with the apparatus.

After the interaction, a pointer points along a certain direction which is the direction of the measured momentum. Another analog pointer with a scale shows the magnitude of the momentum. (Or maybe two pointers for the X and Y momentum respectively).

Meanwhile, the wave packet is instantly (?) converted into a highly extended, nearly plane wave -- ideally extending over all of space, since it has to be a momentum eigenstate. Of course, the pointer readings have to be consistent with the wave number of this extended wave.

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Simon Bridge
Homework Helper
The usual wisdom is that you cannot interact with a photon without destroying it...
Take care you are not conflating the QM wave-packet with an electromagnetic wave packet. In QM, the photon is a packet of energy ... the "wave-packet" is a mathematical construct encoding the statistical behaviour of photons in general in that situation.

You seem to be asking if one can, in practise, measure the momentum of a photon arbitrarily precisely, and still have a photon of some kind afterwards.
For instance - maybe you can do a scattering experiment and infer the (initial) momentum off the target recoil?
The scattered photon won't have the same momentum as the incoming photon - some went into the recoil.

What you ,may want to try is finding out how photon momenta may be measured ... how precisely can you measure it without having to absorb the photon... that boils down to how small a recoil you can measure.

Of course, in principle, the maths allows us to contemplate the ideal case...
The single narrow slit kinda measures position carefully, making (a component of) momentum very uncertain, resulting in a distribution of photons on a screen - are you looking for an analogous description where some ideal apparatus measures momentum precisely but leaves position uncertain? i.e. you know the momentum but you don't know where the measurement was taken?

are you looking for an analogous description where some ideal apparatus measures momentum precisely but leaves position uncertain? i.e. you know the momentum but you don't know where the measurement was taken?

Yes, that is pretty close to what I'm looking for. Only, I think that "where the measurement was taken" could be fairly localized if the incoming wave is prepared as a fairly compact wave function, approaching zero outside a reasonable sized volume. But after the measurement, the particle's probability of being found (by a subsequent position measurement) would be much more spread out if our momentum measurement is really precise. On the other hand, repeated momentum measurements should keep giving the same answer, preserving the particle in the same state. (Just like the well known behaviour of Stern Gerlach and polarizers etc.)

So I guess my question has two parts. Firstly, if we just assume that a "black box" momentum measuring device exists whose precision can be squeezed down as close as we like, then is my description valid according to QM. And secondly - if it is theoretically valid - then can we try to be more specific as to how this black box would work?

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Also, if it is not feasible to do this with photons without destroying them, we could consider anything else like an electron. The point is how to practically demonstrate the principle with actual apparatus for any particle.