Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.13.
As another example of the use of raising and lowering operators, consider a harmonic oscillator in the initial state of
Normalizing, we get
The time-dependent wave function is therefore
we can construct the probability function
The mean position is most easily calculated by expressing in terms of the raising and lowering operators:
Then as we did in the last post, we can apply the operators to the stationary states, and use the orthogonality of the stationary states to eliminate any integrals involving products of different states. We have
Note that if we replaced with we would get since all integrals would be over products of different states.
To check Ehrenfest’s theorem, we need which we can calculate by using the raising and lowering operators again, in the form
and we get, doing a similar integral to that for :
To find the mean of the gradient of the potential , we can use :
This satisfies Ehrenfest’s theorem.
The probabilities of the energies can be read off the original wave function, as they are just the squares of the coefficients of the stationary states. So we will measure with probability 9/25 and with probability 16/25.