Finding the wave function when given the momentum eigenstate

[tryingtolearn1]

Homework Statement

A beam of particles is prepared in a momentum eigenstate $\ket{p_0}$. The beam is directed to a shutter that is open for a finite time $\tau$.

a) Find the wave function of the system immediately after passing through the shutter.
b) Find the momentum probability distribution of the beam after the shutter

Relevant Equations

$$\psi(x) = \frac{1}{\sqrt{2\pi\hbar}}\int\phi(p)e^{ipx/\hbar}dp$$

For part a, I have the following $$\ket{p_0} = \varphi_{p_0}(x)=\frac{1}{\sqrt{2\pi\hbar}}e^{ip_0x/\hbar}$$
but I am totally lost on how to proceed.

 

[PeroK]

Is there a diagram or any further information?

I'm confused by this too. $|p_0\rangle$, I assume, is a single particle wave function, yet it asks about the wave function for a system of an unspecified number of particles.

Where did you get this question?

 

[tryingtolearn1]

Is there a diagram or any further information?

I'm confused by this too. $|p_0\rangle$, I assume, is a single particle wave function, yet it asks about the wave function for a system of an unspecified number of particles.

Where did you get this question?

Unfortunately there is no diagram. This question is from the book "Quantum Mechanics", by David H. McIntyre on chapter 6 question 3.

I first thought to use wave packets equations $$\psi(x,t) = \int\phi(p)\varphi_p(x) e^{-iE_pt/\hbar}dp$$

 

[PeroK]

Okay, here's a possible interpretation of the problem. You have an initial (momentum space) wave function that evolves for time $\tau$. What is the wave function at this time?

 

[tryingtolearn1]

Will it just be the time evolution $$\therefore \psi(x,t) = \varphi(x)e^{-iE_pt/\hbar}?$$ where $\varphi(x)$ is the initial wave function.

 

[PeroK]

Will it just be the time evolution $$\therefore \psi(x,t) = \varphi(x)e^{-iE_pt/\hbar}?$$

Why use a position-space wave-function?

 

[tryingtolearn1]

Why use a position-space wave-function?

The momentum space will be

$$\phi(p) = \frac{1}{\sqrt{2\pi\hbar}}\int\psi(x)e^{-ipx/\hbar}dx$$

 

[PeroK]

The momentum space will be

$$\phi(p) = \frac{1}{\sqrt{2\pi\hbar}}\int\psi(x)e^{-ipx/\hbar}dx$$

That still involves a position-space wave function.

The momentum space wave-function will be the eigenfunction you were given $|p_0\rangle$.

Why not forget position space? The way you don't think about momentum space when working in position space.

 

[tryingtolearn1]

That still involves a position-space wave function.

The momentum space wave-function will be the eigenfunction you were given $|p_0\rangle$.

Why not forget position space? The way you don't think about momentum space when working in position space.

Hmm but my book says the following:

 

[PeroK]

Okay, do the problem in position space then!

If you really don't like momentum space, then don't use it!

 

[tryingtolearn1]

Okay, do the problem in position space then!

If you really don't like momentum space, then don't use it!

But I thought equation 6.29 and 6.30 in the picture is in momentum space? Also, I do want to use momentum space.

 

[PeroK]

But I thought equation 6.29 and 6.30 in the picture is in momentum space? Also, I do want to use momentum space.

What is you initial momentum space wave-function?

 

[tryingtolearn1]

What is you initial momentum space wave-function?

$$\ket{p_0} = \varphi_{p_0}(x)=\frac{1}{\sqrt{2\pi\hbar}}e^{ip_0x/\hbar}$$

 

[PeroK]

But I thought equation 6.29 and 6.30 in the picture is in momentum space? Also, I do want to use momentum space.

Ah! So, you want to find the momentum space eigenfunction by Fourier transforming the position-space momentum eigenfunction that you already know!

Okay, I assumed you'd know what a momentum eigenfunction in momentum-space must be.

 

[PeroK]

$$\ket{p_0} = \varphi_{p_0}(x)=\frac{1}{\sqrt{2\pi\hbar}}e^{ip_0x/\hbar}$$

That is a position-space wave-function! It's a function of $x$.

You want a function of $p$.

 

[tryingtolearn1]

That is a position-space wave-function! It's a function of $x$.

You want a function of $p$.

There is nothing in my book that provides a momentum space wave function with a function of only p. All the equations my books has, for the momentum space, has it with a function of x in it so in that respects I don't know what the momentum space wave function should be.

 

[PeroK]

There is nothing in my book that provides a momentum space wave function with a function of only p. All the equation my books has for the momentum space has a function of x in it so in that respects I don't know what the momentum space wave function should be.

That's not true. Equation 6.28 from the extract you quoted is:
$$\phi(p) = \langle p|\psi \rangle$$
That's the momentum-space wave-function for state $\psi$. And this function appears in the Fourier transforms.

PS McIntyre uses $\psi$ for the state and $\psi(x)$ for the position space wave-function, which suggests a more fundamental relationship between the state and position space. This is misleading, as $\phi(p)$ is just as valid a representation of the state $\psi$.

 

[tryingtolearn1]

That's not true. Equation 6.28 from the extract you quoted is:
$$\phi(p) = \langle p|\psi \rangle$$
That's the momentum-space wave-function for state $\psi$. And this function appears in the Fourier transforms.

PS McIntyre uses$\psi$ for the state and $\psi(x)$ for the position space wave-function, which suggests a more fundamental relationship between the state and position space. This is misleading, as $\phi(p)$ is just as valid a representation of the state $\psi$.

Hmm but I quoted earlier that $\phi(p) = \frac{1}{\sqrt{2\pi\hbar}}\int\psi(x)e^{-ipx/\hbar}dx$ which I know to be the Fourier transform between the momentum space and position space so why isn't that not a momentum space wave function but 6.28 is?

 

[PeroK]

Hmm but I quoted earlier that $\phi(p) = \frac{1}{\sqrt{2\pi\hbar}}\int\psi(x)e^{-ipx/\hbar}dx$ which I know to be the Fourier transform between the momentum space and position space so why isn't that not a momentum space wave function but 6.28 is?

Let's start again.

How do you propose to solve this problem?

 

[tryingtolearn1]

Let's start again.

How do you propose to solve this problem?

lol sorry.

The problem asks to solve the wave function for the particle that is prepared in a momentum eigenstate therefore I have to find the momentum space wave function which I initially thought was $\ket{p_0} = \varphi_{p_0}(x)=\frac{1}{\sqrt{2\pi\hbar}}e^{ip_0x/\hbar}$ but that is in the position space so now I am not sure how to proceed

 

[PeroK]

I've re-read the question and, to be honest, I don't understand it. We've had a discussion about momentum space and momentum eigenfunctions, but (useful as that may be in general) that may not help here.

Perhaps someone familiar with McIntyre's book (it gets recommended often enough on here) knows what he's driving at.

 

[vanhees71]

The question seems somehow flawed. I don't understand what the author wants to calculate. So here are some remarks:

As it seems to be a book, using the representation independent approach by Dirac, one should clearly distinguish the (abstract) Hilbert-space vectors $|\psi \rangle$ and wave functions in position or momentum space. Given a Hilbert-space vector $|\psi \rangle$ these wave functions are given by
$$\psi(x)=\langle x|\psi \rangle$$
and
$$\tilde{\psi}(p)=\langle p|\psi \rangle.$$
They are related by a Fourier transformation, using the well-known result for the generalized (!) momentum eigen functions in position representation
$$u_p(x)=\langle x|p \rangle=\frac{1}{\sqrt{2 \pi \hbar}} \exp(\mathrm{i} p x/\hbar),$$
by
$$\psi(x)=\int_{\mathbb{R}} \mathrm{d} p \langle x|p \rangle \langle p|\psi \rangle = \int_{\mathbb{R}} \mathrm{d} x \frac{1}{\sqrt{2 \pi \hbar}} \exp(\mathrm{i} p x/\hbar) \tilde{\psi}(p).$$
The momentum "eigenstate" $|p_0 \rangle$ cannot represent a physical particle state, because it's not square integrable. It's a generalized state. That you always have if you deal with self-adjoint operators with continuous "eigenvalues".

By the formulation of the problem one might guess that what's really meant refers to the following paper

M. Moshinsky, Diffraction in time, Phys. Rev. 88, 625 (1952)
https://doi.org/10.1103/PhysRev.88.625