# 422: Variation Method for a Particle in a Symmetric 1D Potential Well

Definite potential energy: $$V(x) := if[(x \geq -1) \cdot (x \leq 1), 0, \sqrt{ |x| - 1}]$$

Display potential energy:

Choose trial wave function: $$\Psi (x, \beta ) := ( \frac{ 2 \cdot \beta}{ \pi} )^{ \frac{1}{4}} \cdot exp (- \beta \cdot x^{2})$$

Evaluate the variational integral:

$$E ( \beta ) := \int_{- \infty}^{ \infty} \Psi (x, \beta) \cdot - \frac{1}{2} \cdot \frac{d^{2}}{dx^{2}} \Psi (x, \beta ) dx + \int_{- \infty}^{ \infty} V(x) \cdot \Psi (x, \beta )^{2} dx$$

Minimize the energy integral with respect to the variational parameter, $$\beta$$.

β := .2 β := Minimize(E, β) β := 0.363 E(β) = 0.313

Display wave function in the potential well.

Calculate the probability that the particle is in the potential barrier.

$$2 \cdot \int_{1}^{ \infty} \Psi(x, \beta )^{2} dx = 0.228$$

Define quantum mechanical tunneling.

Tunneling occurs when a quon (a quantum mchanical particle) has probability of being in a nonclassical region. In other words, a region in which the total energy is less than the potential energy.

Calculate the probability that tunneling is occurring.

$$|x| - 1 = 0.313^{2} |_{solve,~x}^{float,~4} \rightarrow {\begin{pmatrix} 1.098 \\ -1.098 \end{pmatrix}}$$

$$2 \cdot \int_{1.098}^{ \infty} \Psi (x, \beta )^{2} dx = 0.186$$

Calculate the kinetic and potential energy contributions to the total energy.

Kinetic energy:

$$\int_{- \infty}^{ \infty} \Psi (x, \beta ) \cdot - \frac{1}{2} \cdot \frac{d^{2}}{dx^{2}} \Psi (x, \beta ) dx = 0.182$$

Potential energy:

$$\int_{- \infty}^{ \infty} V(x) \cdot \Psi (x, \beta )^{2} dx = 0.131$$