Question

Solve the Schrödinger equation for a particle in a box with sides at 𝑥 = −𝐿 and 𝑥 = 𝐿. Determine the eigenvalues and the normalized eigenfunctions. 

Answer 1

The Schrödinger equation for wave function of the system is as follows

$$-\hbar^2/2m\psi''(x)+V(x)\psi(x)=E\psi(x)$$

For particle inside a box $V(x)=0$, so

$$-\hbar^2/2m\psi''(x)=E\psi(x)$$

Solution

$$\psi(x)=A\cos(kx)+B\sin(kx), \quad k=\sqrt{2mE}/\hbar$$

The boundary conditions give

$$\psi(L)=A\cos(kL)+B\sin(kL)=0,\\ \psi(-L)=A\cos(kL)-B\sin(kL)=0.$$

So

$$A=0,\:\quad kL=\pi n,\quad E_n=\frac{\hbar^2\pi^2n^2}{2mL^2}$$

$$\psi_n(x)=B\sin(\pi nx/L)$$

Normalization condition gives

$$\int_{-L}^L\psi^2(x)dx=1$$

So

$$B^2\int_{-L}^L\sin^2(\pi n x/L)dx=1,\quad B=1/\sqrt{L}$$

Finally

$$\psi_n(x)=1/\sqrt{L}\sin(\pi nx/L)$$