Using Probability Density

Given that there still seems to be confusion about squaring psi and multiplying by r (or r^2), EXPLAIN HOW TO SOLVE QUESTIONS A-D below and turn in your work on Monday as a problem set. The answers are given, I want you to show how to think through them to get these answers. Finish by saying a few words about the analogy to atomic orbitals in 3D space.

STUDENT QUESTION:

I still don't think I understand what the advantage is to maximize (or minimize) psi as opposed to maximizing psi^2 to find where the maximum electron density is. Is it just to avoid the trouble of going through and squaring all the variables or is there a different aspect of the probability density that we come to know by just looking at psi as opposed to psi^2?

Obviously the maximum of psi squared occurs at the same place as the maximum of psi (more properly at the maximum of the absolute value of psi). Thus to find the maximum of probability density you can work with whichever function you feel like (I suppose psi is generally easier).

A second kind of problem involves finding what the actual probability densities are (rather than where the maximum is). For this you have to use psi squared, obviously.

Why? Why would you want to know the probability densities (or at least the RELATIVE probability densities at various points - which you can get without all the normalizing constants)?

Answer: In order to calculate relative probabilities or average values of some quantity.

This should be clearer if you work out answers to the following problems.

PROBLEMS FOR MONDAY

Suppose a ball must be in one of two buckets, one of them 5 feet away and the other 10 feed away from where you are standing. Suppose it is three times as likely to be in the closer bucket (this you get from the relative values of psi squared).

Over a large number of trials,
A) What is the distance to the highest-probability bucket? (5 feet)

B) What is the average distance? (6.25 feet)

Suppose a number of buckets are set rim to rim in two concentric circles, one of radius 5 feet with 20 buckets, the other of radius 10 feet with 40 buckets. Again suppose that psi squared shows that a bucket 5 feet away is three times as likely to contain the ball as a bucket 10 feet away.

Over a large number of trials,
C) What is the distance to the highest-probability bucket? (5 feet)

D) What is the average distance? (7 feet)

E) How is this situation analogous to molecular orbitals in 3D space?