# 454: Variation Method Using the Wigner Function: Finite Potential Well

Define potential energy:

$V(x) = if[( x \geq -1)(x \leq1), 0, 20$

Display potential energy:

Choose trial wave function:

$\psi (x, \beta ) = \left( \frac{2 \beta}{ \pi} \right)^2 exp( - \beta x^2)$

Calculate the Wigner distribution function:

$W(x, p, \beta ) = \frac{1}{2 \pi} \int_{- \infty}^{ \infty} \psi \left( x+ \frac{s}{2} , \beta \right) ds \big|_{assume,~ \beta > 0}^{simplify} \rightarrow \frac{1}{ \pi} e^{ \frac{-1}{2} \frac{ 4 \beta ^2 x^2 + p^2}{ \beta}}$

Evaluate the variational integral:

$E( \beta ) = \int_{- \infty}^{ \infty} \int_{- \infty}^{ \infty} (W, x, p, \beta ) \left( \frac{p^2}{2} V(x) \right) dx~dp$

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

$$\beta$$ = 1 $$\beta$$ = Minimize (E, $$\beta$$) $$\beta$$ = 0.678 E( $$\beta$$) = 0.538

Calculate and display the coordinate distribution function:

$P (x, \beta ) = \int_{- \infty}^{ \infty} W(x, p, \beta ) dp$

Probability that tunneling is occuring:

$2 \int_{1}^{ \infty} P (x, \beta ) dx = 0.1$

Calculate and display the momentum distribution function:

$Pp (p, \beta ) = \int_{- \infty}^{ \infty} W(x, p, \beta ) dx$

Display the Wigner distribution function:

N = 60 i = 0 .. N xi = $$-3 + \frac{6j}{N}$$ j = ) .. N pj = $$-5 + \frac{10j}{N}$$ Wigneri,j = W(xi, pj, $$\beta$$