PX156 - D3 - finite potential well

V (x) = {\begin{cases} V_{0} & f o r & x < 0, x > L \\ 0 & f o r & 0 \leq x \leq L \end{cases}

TISE:

- \frac{ℏ^{2}}{2 m} \frac{d^{2} ϕ}{d x^{2}} + V ϕ = E ϕ

it is no longer true that $ϕ (x) = 0$ for $x < 0, x > L$ , ie: particle can exist outside the well
inside the well, $V = 0$ :

\frac{d^{2} ϕ}{d x^{2}} + k^{2} ϕ = 0

where, $k^{2} = \frac{2 m E}{ℏ^{2}}$

outside the well:

\begin{aligned} - \frac{ℏ^{2}}{2 m} \frac{d^{2} ϕ}{d x^{2}} + V_{0} ϕ & = E ϕ \\ \frac{d^{2} ϕ}{d x^{2}} - \frac{2 m}{ℏ^{2}} (V_{0} - E) ϕ & = 0 \\ \frac{d^{2} ϕ}{d x^{2}} - α^{2} ϕ & = 0 \end{aligned}

where, $α^{2} = \frac{2 m}{ℏ^{2}} (V_{0} - E)$
- no analytical solution

the wavefunctions inside and outside need to match so that the boundary conditions are satisfied
$ϕ (x)$ and $\frac{d ϕ (x)}{d x}$ must be continuous at the boundary points, $x = 0, L$