Consider a particle of mass
in a 1D infinite potential well between
. 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
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
is the dipole matrix element.
Snapshots show the
Snapshot 1: initial time, when the particle is in the
Snapshot 2: intermediate time, when the particle is about equally likely to be found in the
states upon measurement
Snapshot 3: at time
, where the probability is close to unity for being found in the