## PHY356F: Quantum Mechanics I. Lecture 9 — Bound states.

Posted by peeterjoot on November 16, 2010

My notes from Lecture 9, November 16, 2010. Taught by Prof. Vatche Deyirmenjian.

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*Motivation.* Motivation for today’s physics is Solar Cell technology and quantum dots.

## Problem:

What ware the eigenvalues and eigenvectors for an electron trapped in a 1D potential well?

### MODEL.

Quantum state describes the particle. What should we choose? Try a quantum well with infinite barriers first.

These spherical quantum dots are like quantum wells. When you trap electrons in this scale you’ll get energy quantization.

### VISUALIZE.

Draw a picture for with infinite spikes at . (ie: figure 8.1 in the text).

### SOLVE.

First task is to solve the time independent Schr\”{o}dinger equation.

derivable from

In the position representation, we project onto and solve for . For the problems in Chapter 8,

where

We should be careful to be strict about the notation, and not interchange the operators and their specific representations (ie: not interchanging “little-x” and “big-x”) as we see in the text in this chapter.

Here the potential energy operator is time independent.

If and is time independent then implies

or

Here is the energy eigenvalue, and is the energy eigenstate. Our differential equation now becomes

where for . We won't find anything like this for real, but this is our first approximation to the quantum dot.

Our differential equation in the well is now

or with

Our solution for is then

and for we have since .

Setting we have

### Type I.

, . For we must have

or , where , so our solution is

### Type II.

, . For we must have

or , where , so our solution is

### Via determinant

We could also write

and then must have zero determinant, or

so we must have

or

regardless of and . We can then determine the solutions 5.106, and 5.107 simply by noting that this value for kills off either the sine or cosine terms of 5.105 depending on whether is even or odd.

## CHECK.

satisfy the time independent Schr\”{o}dinger equation, and the corresponding eigenvalues from from

or

for .

## On the derivative of at the boundaries

Integrating

over we have

which gives us

or

We can infer how the derivative behaves over the potential discontinuity, so in the limit where we must have wave function continuity at despite the potential discontinuity.

This sort of analysis, which is potential dependent, we see that for this infinite well potential, our derivative must be continuous at the boundary.

## Problem:

non-infinite step well potential.

Given a zero potential in the well

and outside of the well

Inside of the well, we have the solution worked previously, with

Then we have outside of the well the same form

With , this is

If , we have , and the states are “bound” or “localized” in the well.

Our solutions for this case are then

for , and respectively.

*Question:* Why can we not have

for ?

*Answer:* As we would then have

This is a non-physical solution, and we discard it based on our normalization requirement.

Our total solution, in regions respectively

To find the coefficients, set , , , and NORMALIZE .

Now, how about in region 2 (), implies that our equation is

We no longer have quantized energy for such a solution. These correspond to the “unbound” or “continuum” states. Even though we do not have quantized energy we still have quantum effects. Our solution becomes

*Question.* Why no , in the term?

Answer. We can, but this is not physically relevant. Why is because we associate with an incoming wave, with reflection in the interval, and both in the $latex {\left\lvert{x}\right\rvert} a$ region.

FIXME: scan picture: 9.1 in my notebook.

Observe that this is not normalizable as is. We require “delta-function” normalization. What we can do is ask about current densities. How much passes through the barrier, and so forth.

Note to self. We probably really we want to consider a wave packet of states, something like:

Then we’d have something that we can normalize. Play with this later.

# Setup for next week’s hydrogen atom lecture.

We’ll want to solve this using the formalism we’ve discussed. The general problem is a proton, positively charged, with a nearby negative charge (the electron).

Our equation to solve is

Here is the total kinetic energy term. For hydrogen we can consider the potential to be the Coulomb potential energy function that depends only on . We can transform this using a center of mass transformation. Introduce the centre of mass coordinate and relative coordinate vectors

With this transformation we can reduce the problem to a single coordinate PDE.

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