4.13: The Quantum Jump in Momentum Space
- Page ID
- 150725
This tutorial is a companion to "The Quantum Jump" which deals with the quantum jump from the perspective of the coordinate-space wave function. This tutorial accomplishes the same thing in momentum space.
The time-dependent momentum wave function for a particle in a one-dimensional box of width 1a0 is shown below.
\[ \psi (n,~p,~t) = n \sqrt{ \pi} \left[ \frac{1-(-1)^n \text{exp} (-i p)}{n^2 - \pi^2 - p^2} \right] \text{exp}(-i E_i t) \nonumber \]
The n = 1 to n = 2 Transition for the Particle in a Box is Allowed
This transition is allowed because it yields a momentum distribution that is asymmetric in time as is shown in the figure below. Consequently it allows for coupling with the perturbing electromagnetic field.
\[ \begin{matrix} \text{Momentum increment} & P = 100 & \text{Time Increment} & T = 100 & \text{Initial} & n_i = 1 & \text{Final state} & n_f = 2 \end{matrix} \nonumber \]
Initial and final energy states for the transition under study:
\[ \begin{matrix} E_i = \frac{n_i^2 \pi^2}{2} & E_f = \frac{n_f^2 \pi^2}{2} \end{matrix} \nonumber \]
Plot the wavefunction:
\[ \begin{matrix} j = 0 .. P & p_j = -10 + \frac{20j}{P} & k = 0 .. T & t_k = \frac{k}{T} \end{matrix} \nonumber \]
In the presence of electromagnetic radiation the particle in the box goes into a linear superposition of the stationary states. The linear superposition for the n = 1 and n = 2 states is given below.
\[ \Psi (p,~t) = n_i \sqrt{ \pi} \left[ \frac{1-(-1)^{n_i} \text{exp}(-ip)}{n_i^2 \pi^2 - p^2} \right] \text{exp} (-i E_i t) + n_f \sqrt{ \pi} \left[ \frac{1-(-1)^{n_f} \text{exp}(-ip)}{n_f^2 \pi^2 - p^2}\right] \text{exp} (-i E_f t) \nonumber \]
Calculate and plot the momentum distribution: Ψ*Ψ:
\[ \Psi \Psi_{(j,~k)} = \left( \left| \Psi (p_j,~t_k ) \right| \right)^2 \nonumber \]
The n = 1 to n = 3 Transition for the Particle in a Box is Not Allowed
This transition is allowed because it yields a momentum distribution that is symmetric in time as is shown in the figure below. Consequently it does not allow for coupling with the perturbing electromagnetic field.
\[ \begin{matrix} \text{Momentum increment} & P = 100 & \text{Time Increment} & T = 100 & \text{Initial} & n_i = 1 & \text{Final state} & n_f = 3 \end{matrix} \nonumber \]
Initial and final energy states for the transition under study:
\[ \begin{matrix} E_i = \frac{n_i^2 \pi^2}{2} & E_f = \frac{n_f^2 \pi^2}{2} \end{matrix} \nonumber \]
Plot the wavefunction:
\[ \begin{matrix} j = 0 .. P & p_j = -10 + \frac{20j}{P} & k = 0 .. T & t_k = \frac{k}{T} \end{matrix} \nonumber \]
In the presence of electromagnetic radiation the particle in the box goes into a linear superposition of the stationary states. The linear superposition for the n = 1 and n = 3 states is given below.
\[ \Psi (p,~t) = n_i \sqrt{ \pi} \left[ \frac{1-(-1)^{n_i} \text{exp}(-ip)}{n_i^2 \pi^2 - p^2} \right] \text{exp} (-i E_i t) + n_f \sqrt{ \pi} \left[ \frac{1-(-1)^{n_f} \text{exp}(-ip)}{n_f^2 \pi^2 - p^2}\right] \text{exp} (-i E_f t) \nonumber \]
Calculate and plot the momentum distribution: Ψ*Ψ:
\[ \Psi \Psi_{(j,~k)} = \left( \left| \Psi (p_j,~t_k ) \right| \right)^2 \nonumber \]
