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

of

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

,

where

is the dipole matrix element.

Snapshots show the

to

transition for

.

Snapshot 1: initial time, when the particle is in the

state

Snapshot 2: intermediate time, when the particle is about equally likely to be found in the

or

states upon measurement

Snapshot 3: at time

, where the probability is close to unity for being found in the

state