## Harmonic Oscillator

There are a large number of different theoretical treatments of a particle bound to x=0 through a linear force. I will use this case to illustrate a couple simple points. You can obtain the exact solutions to Scrhodinger's equation when the t=0 wave function is proportional to a Gaussian. To do this use the same tricks as those discussed at the bottom of the page for an electron moving in a constant force field. There are three different movies that have links from this page. In this movie the wave function for an electron at t=0 is proportional to exp{-[x/10]^2} which represents a particle localized at 0 atomic units of distance with a velocity of 0 atomic units. One frame from the movie is given below. This is a very short yet boring movie. The probability distribution does not evolve with time. This is because the t=0 wave function was chosen to be an eigenstate (the ground state) of the Hamiltonian. Now you can see why eigenstates are also called stationary states. However, you must be careful not to be complacent and sloppy. Just because the probability distribution does not change with time does not mean that the wave function does not change with time. I've made a second movie from the wave function in this example. In the second movie, I show the temporal behavior of the real part of the wave function instead of showing the time dependence of the electron probability distribution. One frame from the movie is given below. There are a number of interesting features of this movie. The most important is that the Re[psi(x,t)]=psi(x,0) cos(wt) where w=E/hbar. Why is Re[psi] simply a product of a function of x and a function of t? What is E? What is the time behavior of the imaginary part of psi? Why is the wave function changing with time even though I've chosen a stationary state case? In this movie the wave function for an electron at t=0 is proportional to exp{-[x/10]^2}exp(ix) which represents a particle localized at 0 atomic units of distance with a velocity of 1 atomic units to positive x. One frame from the movie is given below. This movie can be quite surprising if viewed after the movies for a constant force and for a free particle. The aspect that leaps out at you is that the wave packet moves back and forth but does not change its shape at all. Part of the behavior can be predicted from the equal spacing of the energy levels. When you formally expand the time dependent wave function in terms of eigenstates you can show that |psi(x,t+T)|^2 = |psi(x,t)|^2 if T is the period of the classical harmonic oscillator. To show the precise behavior demonstrated in the movie, you need to find the exact wave function in terms of a time dependent Gaussian and show that there is a choice of initial conditions that will give a time dependent probability density that doesn't change shape with time. There are choices of initial condition for which the width and height of the packet will change with time but return to the original values after a time T; for example, if the initial width of the packet is changed, then the shape of the packet will oscillate. The packet in the movie moves as you should expect from classical mechanics; the particle moves harmonically back and forth with the particle initially moving to positive x. It isn't evident from the movie but the quantum probability distribution moves in exactly the same manner as the classical distribution (see the hints for the linear potential to derive this). You should ask yourself if the movie is roughly consistent with the information I've given; for example, is the extent of the motion consistent with the given information?