In this video, we will discuss the dynamics of harmonic oscillator problem. In the previous video, we reconstructed the eigenstates of harmonic oscillator using annihilation and creation operator instead of solving Schrodinger's time-independent equation as we did before. Here we're going to build on it and solve Heisenberg equation of motion in order to discuss the time evolution of harmonic oscillator states. The Heisenberg equation, if you recall, is shown here. The time evolution of an operator is given by this commutator bracket of the operator with the Hamiltonian. We use the Hamiltonian of the harmonic oscillator problem. Then specifically, let's consider the time evolution of the momentum operator p, and the position operator x. You plug in p here for A and then using H here, and use the commutator bracket between p and x, you will obtain this. Then similarly, the derivative of x operator is given by the momentum operator. Now it's a coupled equation and you need to decouple to solve it. To do that, we use the annihilation and the creation operator once again and from the definition of the annihilation operator, we get this equation, and so the time derivative of the annihilation operator is negative i Omega times itself. Similarly, the creation operator time derivative is given by i Omega times the creation operator, which we can solve immediately to obtain this exponential time dependence. E to the negative i Omega t for annihilation operator a, e to the i Omega t for the creation operator, a dagger. Now, in terms of x and p going back to x and p using this solution, you can write the annihilation and creation operators in terms of x and p and write down their time evolution using these phase factor e to the i Omega t. Now we can solve for x and p as a function of time as this. X operator has a cosine Omega t time dependence multiplied to the initial value of x and then initial value of p, divide it by m Omega, multiply to sine Omega t. Similar behavior is found on operator p. Now, we have recovered the harmonic oscillation sine and cosine Omega t functions in our solution for the position x and momentum p. However, if you actually calculate the expectation values of operator x and operator p as a function of time for any number state, then this number state and the expectation value of x of t with respect to number state n, then, because the number state expectation value for x(0) and p(0) are both 0, you get 0. Say similarly, the expectation value of p with respect to the number state n is also zero. Why? Because number states are energy eigenstates and therefore they are stationary. Now in order to really observe oscillatory behavior as we should expect from a classical physics, one needs to construct a superposition state. For example, let's consider a superposition state Alpha, which is basically an equal superposition of n equals 0 and n equals 1, 2 lowest energy state. If you do that then the matrix elements for operator x and p with respect to the number states are given by this. This is something that we've seen in the previous lecture. Use that to find that the expectation value of x with respect to Alpha is given by this, and if we use a spatial wave function and calculate the Alpha absolute value square meaning, use the position representation and take the wave-function representation for Alpha and calculate the absolute value of Alpha squared. Then you get this. This here is the ground state wave function oscillating. This here is the first excited state wave function oscillating. This here is the expectation value of the superposition state oscillating. Now, the classical oscillation, the oscillation of a pendulum on the oscillation of a mass on a spring is recovered by constructing what we called a coherent state. Coherent state Lambda is a special superposition state given by this infinite sum of the number state coefficient c_n is given by this factor here. If we do that and sum over just the 10 state, then as you can see, you have an oscillation as shown here. You can of course, include more sum, and as you include more and more in the infinite sum, the Gaussian packet becomes narrow and narrower, and it converges into a point mass oscillating with a frequency Omega, which is what we expect from a classical harmonic oscillator. Now, we can go back to the Schrodinger equation and use the time-dependent Schrodinger equation. Because we already know all of these eigenstates of the time-independent Schrodinger equation, we've solved this already, and the solution is given by these Hermite polynomial, multiply to Gaussian envelope, and these are the energy eigenvalue. The energy eigenstate simply evolves over time with this simple face factor. We can construct the superposition state like this. This is the energy eigenstate and then the time evolution of each eigenstate, and this is the expansion coefficient c_n, and use the same coefficients that we have used to construct a coherent state as before. If you do this then you will recover exactly the same oscillatory behavior as we have found in the Heisenberg picture. Once again, the Schrodinger picture and Heisenberg picture gave us the identical results as they should. But if you recall in the Heisenberg picture, we simply calculate the time evolution of the operator x and p, and that allows us a very simply recover an oscillatory behavior that we should expect from an harmonic oscillator problem.
