10.38: Variation Method Using the Wigner Function- Finite Potential Well

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Define potential energy:

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

Display potential energy:

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Choose trial wave function:

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

Calculate the Wigner distribution function:

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

Evaluate the variational integral:

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

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:

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

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Probability that tunneling is occuring:

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

Calculate and display the momentum distribution function:

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

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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, \( \beta\)

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