Square Well Worksheet

Square Well Worksheet

A general space solution for a particle in a region where V(x)=0 is 𝜓(𝑥) = 𝐴 sin 𝑘𝑥 + 𝐵 cos 𝑘𝑥.

1. We will deduce the form of the solutions for a potential that is symmetric about x=0, that is, the potential is zero for -L/2<x<L/2 and infinite for |x|>L/2

a) What are the boundary conditions at $x=\pm \frac{L}{2}$ on $\psi (x)$ for a particle in the box?

$$\psi (x)=0@x\pm \frac{L}{2}$$

b) Rewrite the general solution above for each of the boundary conditions. (You will have two equations.)

$$\psi (x)=A\sin (kx)+B\cos (kx)$$

where $x=\pm \frac{L}{2}$ = 0

$$\psi \left(\frac{L}{2}\right)=A\sin \left(k\left(\frac{L}{2}\right)\right)+B\cos \left(k\left(\frac{L}{2}\right)\right)=0\text{\hspace{1em}(1)}$$ $$\psi \left(-\frac{L}{2}\right)=A\sin \left(k\left(-\frac{L}{2}\right)\right)+B\cos \left(k\left(-\frac{L}{2}\right)\right)=0$$

You can rewrite $\sin (-x)$ as $-\sin (x)$ and $\cos (-x)$ as $\cos (x)$

$$-A\sin \left(k\left(\frac{L}{2}\right)\right)+B\left(\cos \left(k\left(\frac{L}{2}\right)\right)\right)=0\text{\hspace{1em}(2)}$$

Adding and taking the difference of these two above functions 1 & 2

$$1)+2)=B\cos \left(k\left(\frac{L}{2}\right)\right)=0$$ $$1)-2)=0=2A\sin \left(k\left(\frac{L}{2}\right)\right)$$

c) What can you conclude about the allowed values of k for sin(kx) solutions and cos(kx) solutions? For $\sin (kx)$

$$k=\frac{n\pi }{L}\text{ where }n\text{ is an even integer}$$

For $\cos (kx)$

$$k=\frac{n\pi }{L}\text{ where }n\text{ is an odd integer}$$

d) What is the functional form of the ground state (longest wavelength) wavefunction for this well? For ground state $n=1$

$$k=\frac{\pi }{L}$$ $${\psi }_1=B\cos \left(\frac{\pi x}{L}\right)$$

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e) What is the wavefunction for the first excited state wavefunction for this well?

$$k=\frac{2\pi }{L}$$ $${\psi }_2=A\sin \left(\frac{2\pi x}{L}\right)$$

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