PH437 - Nuclear Physics
Spring 2008
Dr. Daniel K. Marble
Quantum Mechanics
Homework Set #3 : 60 marks
This homework set contains material that you have covered in your previous physics classes and
is due at the beginning of next class (lesson #4). These problems have been assigned to ensure
that you have necessary skills and solution techniques for material that will be covered during
PH437 thus all work must be shown to obtain credit. Expect a major writ on the review material
from the first three homework assignments on lesson #5.
1.
The ability of a particle beam or x-ray beam to probe the structure of a target medium
is limited by the probe’s wavelength. Thus, Dr. Coyote has asked you to calculate the
deBroglie wavelengths of the following particles:
a) 8 MeV proton
b) 1 keV electron
c) 8 MeV electron
d) 120 MeV electron
2.
In PH365, we showed that a circular electron orbit of radius r in the Bohr atom was
actually an example of constructive interference of the electron wave with wavelength 
following the condition:
n  2r
where n is the principle quantum number (n=1,2,3,.....).
Using this relation and the experimental fact that the largest nucleus has a maximum
radius of about 7 fm, determine which of the particles in part (a) could exist in the
nucleus according to Bohr’s theory (Note: The least restrictive condition is for n=1.)
3.
Modern particle physics describes the interaction (force) between two charged particles
as
the emission and absorption of virtual photons instead of the electric field used in
classical E&M. A quantum system creates a virtual particle when it violates conservation
of energy by borrowing a small amount energy, E, for such a short time, t b, that the
Heisenberg uncertainity principle prevents us from directly detecting the particle.
a)
Assuming that the virtual particle travels at approximately the speed of light, use
the Heisenberg uncertainty principle, E t  h/2, and the fact that t b  t to
show that the maximum range of the force transmited by a virtual particle is
inversely proportional to the particle’s mass by the relationship:
r
b)


 
mc p
Find the approximate mass of the virtual particle that carries the strong nuclear
force given that the strong nuclear force has a range of approximately 1.4 fm.
4.
One of the first quantum models applied to the nucleus was a 1-D square well with finite
walls as shown in the figure below with a well depth of Vo and a width of 2R:
Vo
0 eV
-R
+R
For a bound nucleon of energy E, answer the following questions:
a)
write the Schrodinger equation for each of the three potential energy regions
b)
write the wavenumber for each of the three potential regions
c)
write the general solution of the Schrodinger equation for each region
d)
apply boundary conditions at x= to simplify your expressions and explain the
physical justification of this process.
e)
draw a graph of the probability distribution of the nucleon for any one of the
allowed nuclear states.
f)
if the depth of the potential well is 20 MeV and the energy of the nucleon is 10
MeV, what is the binding energy of the nucleon?