String Waves and Matter Waves — Question 4

Question 4

The apparent zero-index solution of a hard-wall well is the zero function

Inside $0<x<L$, a free stationary particle obeys $-\hbar^2\psi\prime\prime/(2m)=E\psi$ with $\psi(0)=\psi(L)=0$. A proposed state has $E=0$.

1. Solve the differential equation at exactly zero energy.

2. Apply both boundaries and decide whether the result can represent a normalized state.

Question 4 - Solution

The apparent zero-index solution of a hard-wall well is the zero function

(a) Method and calculation

$$E=0\Rightarrow\psi\prime\prime=0\Rightarrow\psi(x)=Ax+B.$$$$\psi(0)=0\Rightarrow B=0,\qquad\psi(L)=0\Rightarrow AL=0.$$

(b) Interpretation and further calculation

$$L>0\Rightarrow\boxed{A=B=0,\quad\psi\equiv0}.$$$$\int_0^L|\psi|^2dx=0\ne1\Rightarrow\boxed{\text{no normalized zero-energy state}}.$$

Physical check

The zero function solves the homogeneous equation but is not a physical state. This is why the hard-wall quantum number starts at one.