10.40: Variation Method Using the Wigner Function- The Harmonic Oscillator
- Page ID
- 137722
Define potential energy:
\[ V(x) = \frac{x^2}{2} \nonumber \]
Display potential energy:
Choose trial wave function:
\[ \psi (x, \beta = \left( \frac{2 \beta}{ \pi} \right)^{ \frac{1}{4}} exp( - \beta x^2) \nonumber \]
Calculate the Wigner distribution function:
\[ W(x, p, \beta ) = \frac{1}{2 \pi} \int_{- \infty}^{ \infty} \psi \left(x + \frac{s}{2}, \beta \right) exp(isp) \psi (\left( x- \frac{s}{2}, \beta \right) ds~ \bigg|_{assume,~ \beta > 0}^{simplify} \rightarrow \frac{1}{ \pi} e^{ \frac{-1}{2} \frac{4 \beta ^2 x^2 + p^2}{ \beta}} \nonumber \]
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, \( \beta\)) \( \beta\) = 0.5 E( \(\beta\)) = 0.5
Calculate and display the coordinate distribution function:
\[ Px(x, \beta ) = \int_{- \infty}^{ \infty} W(x, p, \beta ) dp \nonumber \]
Classical turning point: \( x_{cl} = 0.5^{ \frac{1}{2}}~~~ x_{cl} = 0.707\)
Probability that tunneling is occurring:
\[ 2 \int_{0.707}^{ \infty} Px (x, \beta ) dx = 0.317 \nonumber \]
Calculate and display the momentum distribution function:
\[ Pp(p, \beta ) = \int_{- \infty}^{ \infty} W(x, p, \beta ) dx \nonumber \]
Display the Wigner distribution function:
N = 60 i = 0 .. N xi = \( -3 + \frac{6i}{N}\) j = 0 .. N pj = \( -5 + \frac{10j}{N}\) Wigneri, j = W( xi, pj, \(\beta\))