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4.14: The Harmonic Oscillator Quantum Jump

  • Page ID
    150726
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    This Mathcad worksheet determines whether an SHO spectroscopic transition is allowed assuming that the Bohr frequency condition is satisfied. It requires only the quantum numbers of the initial and final states.

    The \(v = 0\) to \(v = 1\) transition is allowed because the position distribution function, \(Y^{*}Y\), exhibits oscillating dipole character.

    \[ \begin{matrix} \text{Initial State:} & v_i = 0 & E_i = v_i + \frac{1}{2} & \text{Final state:} & v_f = 1 & E_f = v_f + \frac{1}{2} \end{matrix} \nonumber \]

    \[ \begin{matrix} \text{Set plot parameters:} & \text{Space} = 60 & \text{Time} = 10 & \text{Xmin} = 3 \\ j = 0 .. \text{Space} & x_j = - \text{Xmin} + \frac{2 \text{Xmin}j}{ \text{Space}} & k = 0 .. \text{Time} & t_k = \frac{k20}{ \text{Time}} \end{matrix} \nonumber \]

    Construct time-dependent superposition of the initial and final states:

    \[ \Psi (x,~t) = \text{exp} \left( - \frac{x^2}{2} \right) ( \text{Her} (v_i,~x) \text{exp} (-i E_i t) + \text{Her} (v_f,~x) \text{exp} (-i E_f t)) \nonumber \]

    Calculate and plot Y*Y:

    \[ \Psi \Psi_{(j,~k)} = \overline{ \Psi (x_j,~t_k)} \Psi (x_j,~t_k ) \nonumber \]

    In this contour plot the horizontal axis is the spatial axis. Time is graphed on the veritcal axis. If Y*Y asymmetric in time the transition is allowed.

    Screen Shot 2019-05-15 at 1.11.43 PM.png

    The v = 0 to v = 2 transition is allowed because the position distribution function, Y*Y, does not exhibit oscillating dipole character.

    \[ \begin{matrix} \text{Initial State:} & v_i = 0 & E_i = v_i + \frac{1}{2} & \text{Final state:} & v_f = 1 & E_f = v_f + \frac{1}{2} \end{matrix} \nonumber \]

    \[ \begin{matrix} \text{Set plot parameters:} & \text{Space} = 60 & \text{Time} = 150 & \text{Xmin} = 3 \\ j = 0 .. \text{Space} & x_j = - \text{Xmin} + \frac{2 \text{Xmin}j}{ \text{Space}} & k = 0 .. \text{Time} & t_k = \frac{k20}{ \text{Time}} \end{matrix} \nonumber \]

    Construct time-dependent superposition of the initial and final states:

    \[ \Psi (x,~t) = \text{exp} \left( - \frac{x^2}{2} \right) ( \text{Her} (v_i,~x) \text{exp} (-i E_i t) + \text{Her} (v_f,~x) \text{exp} (-i E_f t)) \nonumber \]

    Calculate and plot Y*Y:

    \[ \Psi \Psi_{(j,~k)} = \overline{ \Psi (x_j,~t_k)} \Psi (x_j,~t_k ) \nonumber \]

    In this contour plot the horizontal axis is the spatial axis. Time is graphed on the vertical axis. If Y*Y asymmetric in time the transition is allowed.

    Screen Shot 2019-05-15 at 1.16.03 PM.png


    This page titled 4.14: The Harmonic Oscillator Quantum Jump is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.