# 411: Particle in a Box with Multiple Internal Barriers

- Page ID
- 137740

Integration limit: x_{max} = 1

Effective mass: \( \mu\) = 1

Barrier height: V_{0} = 100

Potential energy:

\[V(x) = \bigg|^{V_0~if~(x \geq .185)(x \leq .215) + (x \geq .385)(x \leq .415) + (x \geq .585) (x \leq .615) + (x \geq .785) (x \leq .815)}_{0~otherwise}\]

Numerical integration of Schrödinger's equation:

Given

\[ \frac{-1}{2 \mu} \frac{d^2}{dx^2} \psi (x) + V(x) \psi (x) = E \psi (x)\]

\[ \psi (0) = 0\]

\[ \psi ' (0) = 0.1\]

\[ \psi = Odesolve (x, x_{max}\]

Normalize wave function:

\[ \psi (x) = \frac{ \psi (x)}{ \sqrt{ \int_{0}^{x_{max}} \psi (x)^2 dx}}\]

Enter energy guess: E = 18.85

Calculate kinetic energy:

\[ T = \int_{0}^{1} \psi (x) \frac{-1}{2} \frac{d^2}{dx^2} \psi (x) dx = 5.926\]

Calculate potential energy:

\[ V = E - T = 12.924\]

Tunneling probability:

\[ \frac{V}{V_0} \times 100 = 12.924\]