I have a problem with a spinless particle moving on the lateral surface of a cylinder of radius $r$.
If no Hamiltonian is given, is $H=\frac{p^2}{2m}$ only?
What are the Hamiltonian's eigenfunctions and eigenvalues?
Do I have to write down the Laplacian in cylindrical coordinates?
I've tried as follows:
$H=\frac{p^2}{2m}$ so in cylidrical coordinates the TISE is:
$\frac{-\hbar^2}{2m}\frac{\partial^2 }{\partial z^2}\psi(z,r)+\frac{L_z^2}{r^2}\psi(z,r)=E\psi(z,r)$ since $L_z^2=-\hbar^2\frac{\partial^2 }{\partial \phi^2}$ the Hamiltonian in separable and I can find a solution of the form $\psi(z,r)=F(z)R(r)$ and I don't know what to do next.
Best Answer
Perhaps I'm misunderstanding but it seems the question is simply 'what are the eigenfunctions of a quantum particle confined to the (lateral) surface of a cylinder?'. There is no need to add an infinite potential to the Hamiltonian in order to confine the particle to the surface. For instance, when you solve the problem of a free particle in a 'box' in 2D, one does not add an infinite potential to confine the particle along the 3rd dimension. For simplicity, I will take the radius of the cylinder to be $R=1$. I'll also be assuming that question is about the scenario in which the cylinder is infinitely long.
There is no motion along the radial direction $r$, but there is motion along $z$ and $\phi$. The Hamiltonian given by Salmone is almost correct. It is ($\hbar=1$) \begin{equation} H = -\frac{1}{2m} \frac{\partial^2}{\partial z^2} -\frac{1}{2m} \frac{\partial^2}{\partial \phi^2}, \end{equation} which is clearly separable. The eigenfunctions are then given by $\psi(z,\phi) = Z (z) \Phi(\phi)$, where $Z$ and $\Phi$ are the states of a free particle, i.e., \begin{align} Z(z) &= e^{ikz}\\ \Phi(\phi) &= e^{ik\phi}, \end{align} where I'll let you worry about the correct normalisation of these functions. For $Z$, the values of $k$ are any real number, while for $\Phi$ we have $\Phi(\phi+2 \pi) = \Phi (\phi)$, which implies that $k = 0, \pm1,\pm2, \dots$.
Then the corresponding eigenvalues for either $Z$ and $\Phi$ are simply $E = k^2/2m$ (where $Z$ and $\Phi$ have different constraints on the possible values of $k$)
Hope this helps.