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Chemistry LibreTexts

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:

    Screen Shot 2019-02-16 at 12.35.36 PM.png

    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:

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

    Screen Shot 2019-02-16 at 12.31.53 PM.png

    Probability that tunneling is occuring:

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

    Calculate and display the momentum distribution function:

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

    Screen Shot 2019-02-16 at 12.31.57 PM.png

    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, \( \beta\)

    Screen Shot 2019-02-16 at 12.32.02 PM.png