In this video, we will discuss harmonic oscillator. So classical harmonic oscillator is a mass m attached to an ideal spring, which exerts a restoring force proportional to the displacement from the equilibrium position. And there's this constant k here is often called the spring constant. Newton's equation of motion can be written like this. Mass times the acceleration is equal to the external force in this case, the restoring force exerted by the spring. The solution to this differential equation simply is a sinusoidal oscillating function as a function of time sine omega t, for example, where the angular frequency omega here is given by the root of k spring constant, divided by mass. Now, to write down the time independent Schrodinger equation, we first need to convert the force into potential energy because the Schrodinger's equation is given in terms of potential energy, and that can be done by taking the negative integration of the force. And so the force linearly proportional to the displacement z, gives you a quadratic dependence or parabolic shape of potential. And in terms of omega, instead of the spring constant k, the potential energy can be written like this. Now, we're ready to write down the Schrodinger equation, and time independent Schrodinger equation is given here. The E here on the right hand side is the energy eigenvalue, which we need to find by solving this equation. Now for mathematical convenience, we define a new dimension list variable psi here, and psi is defined like this. It's proportional to the displacement, but is multiplied by this quantity here to turn this whole thing as a dimension list quantity, then you can rewrite these Schrodinger's equation given here in terms of psi. And this new Schrodinger's equation, can now be solved and obtain function psi in terms of the new variables I. Now, we seek the solution of the form shown here. So A sub n here is a constant, and H sub n this is a polynomial. And these last term is a Gaussian envelope. So because the potential energy diverges as the Z displacement or Zai, the dimensional displacement variable goes to either plus or minus infinity. We expect that the wave function should approach 0, as psi or Z approaches positive and negative infinity. So this Gaussian envelope has the characteristics, and then the details of the wave function for various different quantum state is then given by this polynomial H sub n. So now we substitute dysfunctions, this solution general form of solution into the Schrodinger equation, and the Schrodinger's equation can now be written like this. An equation for polynomial H sub n. This equation is a known equation, and it's called Hermite differential equation. And the solution for this Hermite differential equations, can be found if this quantity here inside the parenthesis t over h bar omega minus 1 is uneven integer. Or the energy of the quantum state is given by this equation (n + 1/2) quantity times h bar omega here, the n is a non negative integer. So 0, 1, 2 and so on. The solutions of the Hermite differential equations are called the hermit polynomial, and the lowest value of n is 0. So the lowest order Hermite polynomial is H0, and that simply is 1. And as you increase the values of n, you can see that the order polynomial increases. And the remaining constant A sub n, can be found by the normalization condition. That is, you take the absolute value square of the wave function psi, and integrate that from negative infinity to positive infinity, set it equal to 1. And then you will find this rather complex looking normalization constant. Now we can put everything together, and write down the complete solution of the quantum harmonic oscillator problem. And here the first term here is the constant A sub n, Gaussian envelope and modulated by the Hermite polynomial. And if you look at the solutions of this quantum mechanical harmonic oscillator problem, two things we notice immediately won all these energy levels. There's different values of integer n or non negative integer n corresponding to a different energy, and they are all equally spaced. The spacing between 0 and 1 is H bar omega, 1 and 2 also H bar omega. All these energy levels are equally spaced by a unit of H bar omega. And the lowest energy level, the energy of the lowest energy state is not 0. So once again, this harmonic oscillator problem exhibits 0.0 energy just as the the infinite potential well problem, the lowest energy state has a non zero energy, which is a uniquely quantum mechanical phenomenon. And for the way function as I mentioned before, Gaussian function modulated by Hermite polynomial. And also, we noticed that all of these functions have a definite parody. In other words, they are either even or odd function with respect to Z equals 0. And the reason that all of these energy eigen states, solutions of the Schrodinger equation have definite parody is because the potential is symmetric about Z equal 0. Now before we close, we mentioned a few things about the solutions of the harmonic oscillator problem we just obtained, and our intuition about harmonic oscillator. We obtained the solution of harmonic oscillator problem by solving time independent Schrodinger equations. And therefore, there is no time dependence in our solution. We obtain a stationary state, there is no oscillation as a function of time. So these solutions of time independent Schrodinger's equation represents a non time varying stationary states, and these stationary solutions do not exhibit any actual oscillation of the classical oscillator. The classical oscillator can, however, be represented by a linear combination of the stationary solutions that we just obtained. And by combining them together, we can then recover the actual time dependent oscillation of the quantum system. This requires the solving time dependent Schrodinger equation, or some Heisenberg equation and other techniques that we will discuss later. But this stationary states that we just obtained now, will form the basis to construct the actual time varying states that may exhibit a harmonic oscillation
