PX156 - D2 - infinite potential well

V (x) = {\begin{cases} \infty & 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

to avoid infinite energies (to make sure the particle doesn't exist outside the well): $ϕ (x = 0)$ for $x < 0$ , $x > L$
for $0 \leq x \leq L$ , as $V (x) = 0 :$

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

where, $k^{2} = \frac{2 m E}{ℏ^{2}}$

trial solution: $ϕ = A \sin (k x) + B \cos (k x) :$

\frac{d^{2} ϕ}{d x^{2}} = - A k^{2} \sin (k x) - B k^{2} \cos (k x) = - k^{2} ϕ

using the boundary conditions to find the constants:
$ϕ = 0$ at $x = 0 ⟹ B = 0$
$ϕ = 0$ at $x = L ⟹ A \sin (k L) = 0 ⟹ k = \frac{n π}{L}, n \in Z$

ϕ (x) = A \sin (\frac{n π x}{L})

using normalizing conditions:

1 = \int_{0}^{L} A^{2} \sin^{2} (\frac{n π x}{L}) d x = \dots ⟹ A = {(\frac{2}{L})}^{\frac{1}{2}}

the normalized function:

ϕ = \sqrt{\frac{2}{L}} \sin (\frac{n π x}{L})

the energy levels:

E_{n} = \frac{ℏ^{2} k^{2}}{2 m} = \frac{ℏ^{2} n^{2} π^{2}}{2 m L^{2}}

examples:
- semiconductor quantum well
- carotenes (conjugated polyenes)
- conductance quantization in nanowires