Professor Pollock, I have a small question to ask you....I was looking at your final exam from last year, and was interested in the problem where you give a isotropic three dimensional haamonic oscillator potential. In that problem you ask for the theta and phi functions...By looking at our separation of variables from class, when we treated the three dimensional coulombic potential in the Schrodinger Equation, it would seem that independent of the potential the theta and phi dependence is given by the spherical harmonics IFF the potential is sphereically symmetric. Is this correct, or am I off somwhere? Please let me know. Thanks,And the answer: Absolutely correct! The spherical harmonics (Ylm's) are the universal solution, no matter what the potential, as long as the potential is spherically symmetric.
Hi there! I have been studying the harmonic oscillator and I have a few questions, some regarding how explicit we are going to need to be on the final (if you can answer them). 1. Will we need to show HOW to do a change of variables or HOW to convert SE to a unitless form? 2. I'm not very clear on what it means to normalize a wave function. Can you give me a quick and dirty explaination (and how/when it's used)? 3. Also, could you also tell me what the correspondence principal means? 4. Finally, when you measured the energy spectrum for molecular systems, I know how you got mu, but then I'm stuck. How did you "deduce the spring constant"? Thanks for your help.For all these questions, it would be easier in person (with a blackboard!) but I'll try to answer by email. If you don't get it, give me a call or come by my office hours this afternoon.
>1. Will we need to show HOW to do a change of variables or HOW to >convert SE to a unitless form?
Well, of course you *should* know how to do a change of variables! (but in fact I won't be asking you to do this on the final)
>2. I'm not very clear on what it means to normalize a wave function. >Can you give me a quick and dirty explaination (and how/when it's used)?
Quick and dirty: If I give you a wave function and ask you to normalize it, I want you to square it, integrate over *all* space, and ensure that that integral equals 1. (You do that by multiplying what you started with by one over the square root of the integral you did)
It's used whenever you care not just about the shape of the wave function, but really want to know about probabilities. A bit more detailed:
Once you find a solution to the Schrodinger Equation, call it psi, you can always multiply it by any constant you want, and it *still* solves the equation. (The equation is linear) Look at the SE and convince yourself that this is true if it doesn't make sense to you. Multiplying psi by a constant doesn't affect the boundary conditions either. (It still goes to zero in the same places, and it's still continuous, and its derivatives are still continous) So you're absolutely free to multiply the wave function by any constant you like.
But we want the wave function to tell us the probability density! That is, the square of the wave function should tell you the probability (per unit length, or volume) of finding a particle in a little region. Clearly, if you multiply this by a constant, you get a different number, but the probability of finding a particle is *physical*, it should have a definite answer. So, we appeal to the fact that the total probability of finding the particle *somewhere* must be one. You find this total prob by integrating psi^2 over all space. So, that integral must be one. If it isn't, you should figure out a constant to multiply it by so that it is! Make sense?
>3. Also, could you also tell me what the correspondence principal means?
This was Bohr's philosophical idea that when you have a quantum system, characterized by some "quantum number", as that number gets large, the system behaves more and more classically. That is, quantum mechanics should always smoothly and steadily merge with good old classical mechanics, when the size of the system starts getting macroscopic. (i.e,, not microscopic!)
Bohr used this to figure out a constant that appeared in the "Bohr model" of hydrogen - he looked what happened at large n, set it equal to what you expect classically, and in this way figured out what the constant had to be. Then, that constant is *fixed*, and can be used even for low n where things are decidedly *not* acting classically.
>4. Finally, when you measured the energy spectrum for molecular systems, >I know how you got mu, but then I'm stuck. How did you "deduce the spring >constant"?
A couple of ways are possible. The main relation for a harmonic oscillator is
E_n = (n+1/2) hbar omega_0
The n'th energy level is given by this simple formula, in terms of some fundamental quantity omega_0, where omega_0 (the "classical angular frequency") is just the usual (classical) frequency for a harmonic oscillator, (You can rederive omega_0 by looking back at your freshman physics book, in the chapter on simple harmonic oscillators)
omega_0 = Sqrt[C/mu]
So, if you know about the *energy spectrum*, then you know something about the E_n's. (Perhaps you know the ground state, or "zero point energy", which is when n=0. Or, perhaps you know the *difference* between two adjacent n levels, which is closely related) So, armed with this information, it's usually pretty simple to figure out what omega_0 is. Once you know that, if you know mu, the formula above for omega_0 gives you the "spring constant", which is C.
The reason its called the spring constant is that IF you have a spring with spring constant C, then F = -Cx (that's called "hooke's law"), and then you can integrate F.dx to find the potential energy, and discover it is just (1/2) C x^2, which is precisely the potential energy we put in the S.E. for a harmonic oscillator.
So that's another possible way you can figure out the spring constant, namely if you know the maximum amplitude A that the oscillator oscillates to, then E_n = (1/2) C A^2.
>Thanks for your help.
My pleasure!