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

- Page ID
- 137003

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 x_{i} = \( -3 + \frac{6j}{N}\) j = ) .. N p_{j} = \( -5 + \frac{10j}{N}\) Wigner_{i,j} = W(x_{i}, p_{j}, \( \beta\)