Particle in a finite well Questions
1. Solve the finite well problem for:
$$ |x| < \frac{L}{2} \space \text{for} \space V(x)=0$$
$$ |x| > \frac{L}{2} \space \text{for} \space V(x)=V_o $$
2. Wave function for infinite well is:
$$ \psi(x) = \frac{1}{\sqrt{14}} \psi_1 + \frac{3}{\sqrt{14}} \psi_2 +\frac{2}{\sqrt{14}} \psi_3 $$
Find the expectation value of Energy.
For an infinite well potential the energy eigen value is given by:
$$ \boxed{E_n = \frac{n^2 \hbar^2 \pi^2}{2mL^2}} $$
where n is the quantum numbers 1,2,3... $\\$
Let's take $E_1$ as energy of $\psi_1$, $E_2$ as energy of $\psi_2$ and $E_3$ as energy of $\psi_3$. So using the energy eigen value equation:
$$ E_1 = \frac{ \hbar^2 \pi^2}{2mL^2} , E_2 = \frac{ 4 \hbar^2 \pi^2}{2mL^2} , E_3 = \frac{ 9 \hbar^2 \pi^2}{2mL^2} $$
Expectation value of energy is given by:
$$ \langle E \rangle = \int_{-\infty}^{-\infty} \psi^* E \psi$$
$$ \langle E \rangle = \bigg(\frac{1}{\sqrt{14}} \bigg)^2 E_1 + \bigg(\frac{3}{\sqrt{14}}\bigg)^2 E_2 +\bigg(\frac{2}{\sqrt{14}}\bigg)^2 E_3 $$
$$ = \frac{73}{14} \frac{\pi^2 \hbar^2}{2mL^2}$$