Continuous spectra and measurement

Question

Suppose I have a particle whose momentum I measure to be $p$ with uncertainty $\delta p$. Right after the measurement we know that its wave function is given by $\psi(x)=\int g(p)e^{ipx/\hbar}dp$ (restricting to 1D for simplicity) with $g$ a certain function which is $0$ outside $p-\delta p$, $p+\delta p$ and with $\int |g(p)|^2dp=1$.

What more can we say about that function $g$?

I was ok with not being able to say anything more about it but I have to admit that, for instance, we know that $g$ is never much more concentrated than it has to be. For else if we'd take for instance the canon from the Stern-Gerlach experiment (which throws silver atoms at circa 500 m/s) and direct it at the middle of a door, we should see atoms bouncing back from time to time. Indeed if $g$ is extremely concentrated, the resulting wave can get quite spread out and one could see diffraction effects. But we never do.

Answer 1

It depends on the state before the measurement. If the pre-measurement wavefunction is given by some $\psi_i(x)$, then the post-measurement wavefunction $\psi_f(x)$ is the projection of $\psi_i(x)$ onto the subspace of the Hilbert space which is consistent with your measurement results.

More concretely, let $$\psi_i(x) = \frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^\infty A(p) e^{ipx/\hbar}dp$$

If you measure the momentum to be in the interval $p\in[p_0-\delta p, p_0 + \delta p]$ then the post-measurement state is simply $$\psi_f(x) = \frac{1}{\sqrt{2\pi\hbar}}\int_{p_0-\delta p}^{p_0 + \delta p} A(p) e^{ipx/\hbar}dp$$

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As a side note, the normalization is not automatically preserved by measurement, but that's okay because quantum states are elements of the projective Hilbert space and are only well-defined up to an overall multiplicative constant anyway. It's usually convenient to normalize the wavefunction, but note that you'll have to re-normalize it after a projective measurement.

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"If the interval in which you measure is small enough, then you can approximate as a constant function on that interval" Why?

Because as long as $A$ is at least continuous, then for a small interval this $g$

looks rather a lot like this one

That's if you assume, as my textbook very implicitly does, that one can in fact approximate with a constant, but I don't see this assumption anywhere in the postulates.

No, this is a straightforward application of the uncertainty principle to a wave packet. In order to observe diffraction effects, you need the spread of your wavepacket to be at least on the same order of magnitude as the size of the hole. For e.g. a Gaussian wave packet which saturates the uncertainty relation, the spread in $p$ is inversely proportional to the spread in $x$, implying that a large spatial delocalization corresponds to a fantastically localized momentum.

Answer 2

I think I finally found an answer to my question. The atoms in a Stern-Gerlach-type oven can't have too wide wave functions (space wise) because else they would interact a lot with each other (they would be 'very close everywhere') and this configuration is too energetic.