Question

Particle in a infinite potential well, which has a width of L.

\[
E\psi (x)=\left[\frac{-\hbar ^2}{2m}\frac{d^2}{dx^2}+V(x)\right]\psi (x)\\
\psi _n (x)=\sqrt{\frac{2}{L}}\sin\left(\frac{\pi n x}{L}\right)\\
E_n=\frac{n^2\pi^2\hbar ^2}{2mL^2}
\]
The question is, what is the momentum of this particle?

Answer 1

I don't know whether it is valid to apply the momentum operator in this case. If I apply the momentum operator to the wavefunction, I get:\[
\begin{align*}
&\hspace{1.5em}\mathbf{p}\psi_n(x)\\
&=-i\hbar\frac{d}{dx}\psi_n(x)\\
&=-ih\sqrt{\frac{2}{L}}\frac{\pi n}{L}\cos\left(\frac{\pi n x}{L}\right)
\end{align*}
\]
Clearly you cannot have a imaginary momentum! Something must have gone wrong.

Answer 2

we can find the expection value of the momentum, or its value on the state represented by your wave function.
Now that value is given by the subsequent formula:
\[\Large p = \left\langle {{\psi ^*}} \right|\hat p\left| \psi \right\rangle = \int_0^L {{\psi ^*}} \left( { - i\hbar \frac{{d\psi }}{{dx}}} \right)dx\]
I assumed that your problem is described by the subsequent drawing:

Answer 3

I suggest to write your wave function as follows:

\[\Large \begin{gathered}
\psi \left( x \right) = A\left\{ {\exp \left( {i\Omega x} \right) - \exp \left( { - i\Omega x} \right)} \right\} = \hfill \\
\hfill \\
= 2iA\sin \left( {\Omega x} \right),\quad \left( {\Omega = \sqrt {\frac{{2mE}}{{{\hbar ^2}}}} } \right) \hfill \\
\end{gathered} \]

Answer 4

The drawing is correct. Is \(\Large p = \left\langle {{\psi ^*}} \right|\hat p\left| \psi \right\rangle\) the expectation value or the the momentum of the state? Furthermore, won't
\(\hspace{1.5em} \Large\int_0^L {{\psi ^*}} \left( { - i\hbar \frac{{d\psi }}{{dx}}} \right)dx\\
\Large=\int_0^L\sqrt{\frac{2}{L}}\sin\left(\frac{\pi n x}{L}\right)\left(-ih\sqrt{\frac{2}{L}}\frac{\pi n}{L}\cos\left(\frac{\pi n x}{L}\right)\right)dx\)
still be imaginary?

One of the multiple choice question in my high school examination paper, which I finished a few days ago, asks for the the momentum of which a particle in a infinite potential well cannot have. However, I read somewhere that the momentum is not quantised for a particle in a infinite potential well. So I am trying to derive the momentum of a particle in a potential well and learn basic quantum mechanics on the way. Grateful if you could give me some advice.