Wednesday, April 6, 2011
Perturbation Theory and Tayor Expansions
My take on perturbation theory starts with the hydrogen atom. The ground state is a cloud of charge about the nucleus, and we ask: what is the simplest way we can distort the ground state? This is a vague question and it has at least two possible answers. One answer is that the simplest way do distort the ground state is to mix in a little bit of the first excited state, say for example the pz state. The effect of this is to more or less push the charge cloud a small distance along the z axis. The second way to answer this question is by suggesting we introduce a small, constant electric field to the region. Let's say we direct it along the z axis. The effect of this is to push the charge cloud a small distance along the z axis. These two answers suggest a third way to distort the ground state. Why not just push the charge cloud a small distance along the z axis? Isn't this exactly the same thing? Not quite. The first two methods do in fact, give exactly the same result. In the limit of a very small constant field in the z direction, the resulting perturbed state is in fact exactly equal to the original ground state plus a small amount of pz state. And it is approximately true that this perturbed field is similar to what you get if you just push the whole charge cloud a small distance north. But it is not exactly the same. You can see that is isn't by remembering that the wave function is "pointy" right at the origin, and you can't very well expect this "pointiness" to move away from the nucleus along with the rest of the charge cloud. So the third description can only be approximate at best. However, it's an interesting approximation and mathematically it's quite simple. If you have any function and you want to move it a small distance in the z direction, you can do so by taking the derivative of the function with respect to z, and adding a small component of this derivative to the original function. That's how a Taylor series starts out. For small displacements, you can ignore the higher order terms. Wouldn't it be nice if there was some kind of mathematical operation you could perform on the ground state of hydrogen that would give you the pz state? You can see that the simple operation of differentiation "comes close", but it clearly doesn't work exactly. I'm going to try and show how to construct an operator that works exactly. The idea is to go back to the second proposal for distorting the charge cloud: adding a small electric field. We know that the perturbation must consist of exactly the pz state, because that's the simplest possible distortion of the charge cloud, and in the limiting case it must be exact. I'm going to suggest putting the hydrogen atom between the plates of a capacitor, and adding a charge q to the capacitor. This gives us a small constant field. Let's begin with no charge on the plates, so the field is zero. Let's remember that in quantum mechanics there is something called the Hamiltonian; and for the hydrogen it is given by the equation H=-∇^2+V where V is the 1/r potential of the nucleus. Also remember that when we insert the wave function phi into the Hamiltonian, that the result is that the function returns a simple multiple of phi. We can scale it with an appropriate multiplicative constant so it returns phi itself. Now I'm going to add a charge q to the capacitor, and ask the question: how does H, the Hamiltonian, change? The change in H can obviously be written as dH/dq. What I want to show now is that dH/dq is itself an operator, and that when you operate on phi, the ground state of the hydrogen atom, the result of this operation is to return the first excited state. So dH/dq is what they call in quantum mechanics a "ladder operator" (up to at least an arbitrary multiplicative constant). My blogger program is acting up right now and not exactly permitting me to do paragraph breaks, so I'm going to end this post now and continue later.