Free particle in a ring with a periodic boundary condition

We consider a free particle restricted to a ring of length $L$ , with a complete lap around the ring taken to begin at position $-L/2$ and end at position $L/2$ . The general TISE is

$\big[\frac{\hat{p}^2}{2m} + V(x)\big]\psi = E\psi$

where

$\hat{p} = -i\hbar \frac{d}{dx}$

We take $V(x) = 0$ in the ring and we assume the periodic boundary condition $\psi(x + L) = \psi(x)$ .

The TISE becomes

$\psi^{\prime \prime}(x) + \frac{2mE}{\hbar^2} \psi = 0$

which has general solution

$\psi(x) = Ae^{i \sqrt{2mE}/\hbar\cdot x} + Be^{-i \sqrt{2mE}/\hbar\cdot x}$

In principle, this allows for two independent solutions superposed with coefficients $A$ and $B$ . However, the periodic boundary condition implies $\psi(-L/2) = \psi(L/2)$ which in turn implies $A = B$ , and periodic boundary conditions produce travelling waves, not standing waves. Therefore, the presence of both terms in the general solution above is a superposition of two running waves with the same amplitude but travelling in opposite directions. One of these running waves is superfluous for the purposes of developing the solution below, so we can set $B = 0$ and just focus on the forward travelling wave. The general solution then reduces to

$\psi(x) = Ae^{i \sqrt{2mE}/\hbar\cdot x}$

The periodic boundary condition requires $\frac{\sqrt{2mE}}{\hbar} L$ to be an integer multiple of $2\pi$ , so using $E_n = p_n^2/2m$ where $E_n$ and $p_n$ are the $n$ -th energy and momentum states respectively (and note that this expression for $E_n$ as $p_n^2/2m$ is allowed only because $V(x) = 0$ in the ring), we have

$p_n = \hbar \frac{2 \pi n}{L}$

and

$E_n = \frac{\hbar^2 2 \pi^2 n^2}{m L^2}$

Choosing units equivalent to setting $\hbar = 1$ , we can write the $n$ -th wave function as

$\psi_n(x) = Ae^{i p_n x}$

To find the normalising constant $A$ , we write

$\int_{-L/2}^{L/2} \psi_n(x) \cdot \psi_n^{\ast}(x) dx = A^2\int_{-L/2}^{L/2} dx = 1$

so $A = \frac{1}{\sqrt{L}}$ . The solutions for the particle in a ring are then travelling waves of the form

$\psi_n(x) = \frac{1}{\sqrt{L}} e^{i p_n x}$

with corresponding momentum and energy states

$p_n = \frac{2 \pi n}{L}$

and

$E_n = \frac{2 \pi^2 n^2}{mL^2}$

respectively.

Note that the solutions here are travelling waves, and are different from the standing wave solutions obtained for the more commonly encountered particle-in-a-box problem with left- and right-hand endpoints $0$ and $L$ respectively, and with non-periodic boundary conditions $\psi(0) = \psi(L) = 0$ . When no boundary conditions are specified at all, i.e., the particle is not confined to a box or a ring, then with $V(x) = 0$ the TISE and its general solution are the same as those initially obtained above, but energy and momentum are no longer quantized, i.e., they can take any values along a continuum.

Free particle in a ring with a periodic boundary condition

Published by Dr Christian P. H. Salas

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