Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 10.9.
The one-dimensional driven harmonic oscillator can be solved exactly both classically and in quantum mechanics. If the oscillator’s natural frequency is , then with a driving force where the function has dimensions of length and can be any function of time (with the condition that for ), then the total force is
The classical solution for (subscript ‘c’ for ‘classical’) is given in Griffiths’s question as
I’m not sure how to solve the original ODE to get this solution, but to show that it is a solution, we can work backwards. To do this requires finding the derivative of the integral, which is complicated by the presence of the limit of integration inside the sine function in the integrand. We can see what the derivative is in the more general case by using the definition of a derivative:
In the second term, as , the integral becomes
Applying this to 2 we get
where in the last line, the function in 9 is
so occurs when , or
Thus 2 is actually a solution of 1. The initial conditions are
Griffiths now asks us to show that the exact quantum mechanical solution to the Schrödinger equation is
where is the eigenfunction for the unforced oscillator, with eigenvalue .
The Hamiltonian for the forced oscillator is
We can show this by applying and to and showing that they give the same result. We start with the time derivative, remembering that is a function of time. We’ll denote a derivative with respect to by a dot, and a derivative with respect to by a dash. We’ll also define
where we used 1 in the second line.
Now we apply to . However, the unforced eigenfunction in 17 is given as a function of , not , so the Hamiltonian that gives the standard harmonic oscillator eigenvalues is
so our forced Hamiltonian is
Applying this to 17 requires finding the second derivative:
Putting it together, we get
which is identical to 22, so 17 is indeed a solution of the Schrödinger equation.
Using a similar argument, we can find the eigenfunctions and eigenvalues of the full Hamiltonian. Griffiths gives the eigenfunctions as
where is the unforced eigenfunction evaluated at position . We can define an unforced Hamiltonian as above:
The full Hamiltonian becomes
Applying this to we get
which shows that is an eigenfunction and the eigenvalues are
Note that both the eigenfunctions and eigenvalues are time-dependent, through the parameter .
So far, everything has been exact, but we can now apply the adiabatic theorem to the case where the forcing function varies slowly with time. First, we can return to the classical result 2, and rewrite it as
where we used in the last line.
Now if varies slowly compared to the natural frequency we can take its derivative outside the integral to get an approximation:
then we neglect the second term to get the classical adiabatic approximation
We can use this approximation to get an adiabatic approximation for the quantum wave function 17:
Here the phase factors are (the dynamic phase) and (the geometric phase). From 39 we see that
which agrees with its earlier definition.
If the eigenfunctions are real, then the geometric phase should be zero. This isn’t strictly true here, but we’re assuming that is small, so should be close to zero.