Wolfram Demonstrations Project

This Demonstration shows the evolution of the wavefunction

for a particle as it makes a transition between eigenstates of an infinite square well. Transitions are caused by a harmonic electric field

applied along the width of the well, where the driving frequency

is assumed to be equal to the natural frequency separation between the initial and final states (quantum numbers

and

, respectively), to within a "detuning frequency"

. The dimensionless electric field magnitude is

. Once an initial state is selected, only those final states for which the electric dipole matrix element is nonzero are shown. The time-dependent Schrödinger equation is solved by an expansion in square-well eigenstates. The initial and final states are included automatically, and additional eigenstates (with quantum number

) can be added.

The top figure shows

(red curve) along with the direction and relative strength of the electric field (blue arrow). Also shown is the potential energy function

(thick black line). The expectation value of the particle energy and the normalization of

are shown at the top of the plot. All quantities are dimensionless, with

, where

is the well width, and

and

are the particle's mass and charge, respectively.

The bottom figure shows the probability of being found in the initial or final states as a function of time (solid black and solid blue curves, respectively). Also shown is the probability of being in the final state, as calculated from first-order perturbation theory (dashed blue curve).

Contributed by: James A. Miller

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Consider a particle of mass

and charge

in a 1D infinite potential well between

and

. Suppose we perturb this system by applying an electric field

, so that the potential energy inside the well becomes

. In terms of the characteristic energy of the well

, the electric field magnitude can be written as

, for some dimensionless parameter

The particle is assumed to be initially in an eigenstate of the unperturbed well, with quantum number

. If we want to cause a transition to another unperturbed eigenstate, with quantum number

, the chance of this is maximized by having the driving frequency (the electric field frequency

in this case) equal to the natural frequency separation

between the initial and final eigenstates (which is equal to the energy difference divided by

). Of course, this fine tuning may not be possible, and the "detuning frequency"

is the difference between the optimal driving frequency and the actual one.

The time-dependent Schrödinger equation is solved by expanding the wavefunction

in terms of the unperturbed stationary states of the well

where

is the energy of level

. At a minimum, the expansion should include the initial and final states. However, additional states can be added to explore their effect (which is fairly minimal!). Since

is a separable function of space and time, the spatial integrations need only be performed once, and the matrix elements

are precalculated for the first 10 energy levels.

We also show the time-dependent probability of finding the particle in the initial or final states, along with the transition probability calculated from first-order perturbation theory. From perturbation theory, the probability of being found in the final state at time

is given by