Problems for problem set 6
For questions 1 and 2: Consider a potential step 10eV high so that V(x)=0 for
for x<0 and V(x) = 10eV for x>0. You might want to look at the document
Step potential summary, which summarizes the class discussion and also has a brief
comment of fluxes (needed in 1b below).
1. Electrons with a kinetic energy of 15eV are beamed at
the step from the left.
a) Use the Schroedinger equation to get the momentum of the electrons
for x>0. How does your result compare with what you would
get from classical reasoning?
b) What fraction of the electrons travel to right past the
step? In doing this, think in terms of quanton fluxes.
2. Electrons with a kinetic energy of 5eV are beamed at
the step from the left. Roughly what is the distance to the
right of x=0 that there is a reasonably large probability to
observe an electron?
3. Hydrogen atom estimate.
Consider the hydrogen atom. Classical physics cannot account for the size of the atom
or the energy that it takes to remove the electron. With the Schroedinger equation in its
3-dimensional version, one can do a detailed calculation that gives the size and ionization
energy. However, the calculation is a bit involved, and it is easy to lose sight of what is
really going on. Here is a “back-of-the-envelop” calculation that can only be expected to
be roughly correct (to within factors of 2 and π and such) that makes it easier to see
what is really happening. However, apparently by accident, it actually gives the correct
answer with all the numerical factors right!
Work through the following steps explicitly:
Assume that ψE(x) for the electron is large in a volume around the proton with radius R
and is very small outside that volume. Then the kinetic energy term in the Schroedinger
equation is something like
−2
2m
∂
2
ψ
E
∂
x2+
∂
2
ψ
E
∂
y2+
∂
2
ψ
E
∂
z2
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟ →2
2m
1
R2
ψ
E
.
(This is a given; don’t try to derive it. But you should be able to see why it is a plausible
estimate.) Put this into the Schroedinger equation with the potential as given in the text
discussion of the Bohr model
V x,y,z
( )
→ − ke2
R
.
