String Waves and Matter Waves — Question 5

Question 5

Periodic confinement admits a constant mode that a hard-wall box excludes

A particle moves on a ring of circumference L. Let $\psi(s)=Ae^{iks}$ and require $\psi(s+L)=\psi(s)$. There is no magnetic flux, and $E=\hbar^2k^2/(2m)$.

1. Derive the allowed k values and normalize the traveling waves.

2. Identify the zero-energy state and the degeneracy of each positive energy.

Question 5 - Solution

Periodic confinement admits a constant mode that a hard-wall box excludes

(a) Method and calculation

$$e^{ikL}=1\Rightarrow\boxed{k_j=2\pi j/L},\quad j\in\mathbb Z.$$$$1=\int_0^L|A|^2ds=|A|^2L\Rightarrow\boxed{|A|=L^{-1/2}}.$$

(b) Interpretation and further calculation

$$j=0:\quad\boxed{\psi_0=L^{-1/2},\quad E_0=0}.$$$$E_j=\frac{2\pi^2\hbar^2j^2}{mL^2}=E_{-j}\Rightarrow\boxed{g_j=2\ (j\ne0)}.$$

Physical check

The two states at positive energy circulate in opposite directions. A ring has periodic continuity rather than zero-value endpoint constraints.