Wave Functions of a Trapped Electron — Question 1

Question 1

Normalization fixes a box amplitude while leaving its overall phase free

An electron has spatial wave function $\psi=A\sin(3\pi x/L)$ for $0<x<L$ and zero outside.

1. Find the magnitude of A and its physical units.

2. Find the probability in the left third and explain whether normalization fixes the phase of A.

Question 1 - Solution

Normalization fixes a box amplitude while leaving its overall phase free

(a) Method and calculation

$$1=|A|^2\int_0^L\sin^2(3\pi x/L)dx=|A|^2L/2.$$$$\boxed{|A|=\sqrt{2/L}},\qquad[A]=\mathrm{length}^{-1/2}.$$

(b) Interpretation and further calculation

$$P(0<x<L/3)=\frac2L\int_0^{L/3}\sin^2(3\pi x/L)dx=\boxed{1/3}.$$$$\boxed{A=\sqrt{2/L}\,e^{i\alpha}}\quad\text{for arbitrary real }\alpha.$$

Physical check

The left third contains one complete density lobe. A global phase leaves all position probabilities unchanged.