The Probability Density of an Electron

Question

The following is a graph of |psi(x)|^2

http://www.physicsforums.com/attachment.php?attachmentid=14887&d=1217540634

Question is.

What is the probability that an electron will be detected in a 0.0010-cm-wide region at x = 0.50 cm? At x= 0.999 cm?

b) If 10^4 electrons are detected, how many are expected to land in the interval -0.30 cm - 0.30 cm.

Also, lets suppose that this was a graph for the |(x)|^2 of a neutron.

How you you find the value 'a' if it wasn't given?

I know that Probability of landing at x = (probability density at x )* (length)

Also know that expected value = number of electrons * probability.

But I am having trouble putting these two together. Can some please show me how to do this question?

Answer 1

I assume that this is linear and not 3d.

First, what is the area of a triangle? 1/2 b * h

A = 1/2 cm^2 (ignoring the negative value side)

What is the height at .5cm? .5cm

Prob = .5cm *.001 cm / A = .0005cm^2/.5 cm^2 = .0001

Now at .999 cm.

Prob = .999cm *.001 cm /.5 cm = .0001998 ~.0002

For the neutron it is the same thing.  It is just the probability area/ total area.