The Quantum Jump in Momentum Space

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.

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.

Momentum Increment

Time Increment

Initial State

Final State

Initial and final energy states for the
transition under study

Plot the wavefunction:

In the presence of electromagnetic radiation the particle in the box goes into a linear superposition of the stationary states. The linear superpostion for the n = 1 and n = 2 states is given below.

Calculate and plot the momentum distribution: Y^*Y:

The n = 1 to n = 3 Transition for the Particle in a Box is Not Allowed

This transition is not 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.

Momentum Increment

Time Increment

Initial State

Final State

Initial and final energy states for the
transition under study

Plot the wavefunction:

In the presence of electromagnetic radiation the particle in the box goes into a linear superposition of the stationary states. The linear superpostion for the n = 1 and n = 3 states is given below.

Calculate and plot the momentum distribution: Y^*Y: