5.8 Sample Test Problems
- A nucleus has a radius of 4 Fermis. Use the uncertainty principle toestimate the kinetic energy
for a neutron localized inside the nucleus. Do the same for an electron.
Answer
∆p∆x≈ ̄h
pr≈ ̄h
Try non-relativistic formula first and verify approximation when we have the energy.
E=
p^2
2 m
=
̄h^2
2 mr^2
=
( ̄hc)^2
2 mc^2 r^2
=
(197. 3 MeVF)^2
2(940MeV)(4F)^2
≈ 1. 3 MeV
This is much less than 940 MeV so the non-relativistic approximation is very good.
The electron energy will be higher and its rest mass is only 0.51 MeV so itWILL be relativistic.
This makes it easier.
pr≈ ̄h
E=pc=
̄hc
r
=
197. 3 MeVF
4 F
≈ 50 MeV
2.*Assume that the potential for a neutron near a heavy nucleus is given byV(r) = 0 for
r >5 Fermis andV(r) =−V 0 forr <5 Fermis. Use the uncertainty principle to estimate the
minimum value ofV 0 needed for the neutron to be bound to the nucleus.
- Use the uncertainty principle to estimate the ground state energy of Hydrogen.
Answer
∆p∆x≈ ̄h
pr≈ ̄h
E=
p^2
2 m
−
e^2
r
=
p^2
2 m
−
e^2
̄h
p
(We could have replacedpequally well.) Minimize.
dE
dp
=
p
m
−
e^2
̄h
= 0
p=
me^2
̄h
E=
p^2
2 m
−
e^2
̄h
p=
m^2 e^4
2 m ̄h^2
−
me^4
̄h^2
=
me^4
2 ̄h^2
−
me^4
̄h^2
=−
me^4
2 ̄h^2
α=
e^2
̄hc
e^2 =α ̄hc
E=−
1
2
α^2 mc^2
4.*Given the following one dimensional probability amplitudes in the positionvariable x, com-
pute the probability distribution in momentum space. Show that the uncertainty principle is
roughly satisfied.
