The probability of finding an electron in dx at a point x is

The probability of finding an electron in dx at a point x is given by P(x)dx where P(x) = * or ||2. The wave function describing a state of an electron confined to move along the x axis is given at time zero by (x, 0) = Aexp(-x2/42). Find the probability of finding the electron in a region dx centered at
a.   x = 0,
b.   x = , and
c.   x = 2.
d.   Where is the electron most likely to be found?

Solution

The probability to find the electron in a region dx around x is:

P(x)dx=* or ||² dx= Aexp(-x2/42) dx

a) for x=0 , P(x)dx=|(0,0)|² dx=A²dx

b) for x= , P(x)dx=A²e^-1/2dx =.607A²dx

c) for x=2 , P(x)dx=A²e^-4/2dx =.135A²dx