Problem Set #10

Physics 322: Modern Physics
Spring 2015
Problem Set #10
(Toward the Point [Particle] where Spin Gets
Weird)
Due Friday, April 17 in Lecture
ASSUMED READING: Before starting this homework, you should have read
Chapter 7 and Chapter 8, Section 1 through 3 of Harris’ Modern Physics.
1. [Harris 7.12 tweaked] A particle is trapped in a spherical infinite well.
The potential energy is 0 for r<a and infinite for r>a. Which, if any,
quantization conditions would you expect it to share with hydrogen, and
why? HINT: The force here is still going to be a central force, since the
potential depends only on r, although it is very different from Coulomb’s
law. What parts of the wave function !ψ (r,θ , φ ) = R(r)Θ(θ )Φ(φ ) that solves
the Schrödinger’s equation will be the same and what will be different?
What are the implications?
2. [Harris 7.56 extended] For a hydrogen atom in the ground state,
determine
a. the most probable location at which to find the electron and
b. the most probable radius at which to find the electron (yes, it’s
different than (a)).
c. Comment on the relationship between your answers in parts (a)
and (b). Specifically why are these results different? Don’t just cite
the equations, explain in words a non-physics major [but maybe a
math major] could understand. HINT: How does the area of a
spherical shell vary with radius? Why does that matter?
d. Recall that the potential energy U(r) in the hydrogen atom is just
the Coulomb potential
1 e2
! U(r) = −
.
4π ε0 r
If we consider that in the ground state of a hydrogen atom, the
electron has energy:
me4
! E1 = −
2
2 ( 4π ε0 ) ! 2
Then the classical limit on the radius of the electron’s orbit occurs
when E1 – U(r) = KE = 0, such that E1 = U(r), and thus we can
compute the maximum classically allowed radius of a ground state
electron:
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Physics 322: Modern Physics
Spring 2015
E1 = U(r)
−
me4
2 ( 4π ε0 ) ! 2
2
=−
1 e2
4π ε0 r
2
e2 2 ( 4π ε0 ) !
r=
4π ε0
me4
2
!
2 ( 4π ε0 ) ! 2
me2
∴ r = 2a0
=
But, as I noted, this is the classical limit. What is the probability
that a ground state electron will lie outside this limit? Comment on
the magnitude of the probability. NOTE: The integral you will
need to tackle is best tackled by repeated integration by parts.
3. [Harris 7.58 extended] Let’s repeat some of the work done in Example
7.7, but for an electron in a different state.
a. WITHOUT redoing the derivation that lead to equation 7-38,
explain why it makes sense that to turn the probability density R2(r)
[probability per volume] into the radial probability P(r) [probability
per unit radial distance], we need to multiply it by r2
b. What is the radial probability P(r) [probability per unit radial
distance] of an electron in the hydrogen atom in the 3p state?
c. What is the expectation value of the distance from the proton for
the electron in part (a)? HINT: The inside cover of your textbook
∞
m!
has the solution to integrals in the form! ∫ x m e−bx dx = m+1 , use it!
0
b
d. How does it compare with the expectation value in the 3d state
calculated in Example 7.7? Discuss the differences in light of the
plots of the radial probabilities shown in Figure 7.17. Does your
result make sense?
4. [Harris 8.5 tweaked] The neutron comprises multiple charged quarks.
Can a particle that is electrically neutral but really composed of charged
constituents have a magnetic dipole moment? Clearly explain your
answer. HINT: What combination of electric charges would be neutral?
What sort of magnetic dipole moment exists when you have positive and
negative charges counter circulating (by which I mean the positive charges
are circulating in the opposite direction as the negative ones)?
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Physics 322: Modern Physics
Spring 2015
5. [Harris 8.25 tweaked] Just to convince you of the non-classical
situation we have with the electron’s intrinsic spin, consider the following
question. The electron is known to have a radius no larger than 10-18 m. If
actually produced by circulating mass, its intrinsic angular momentum of
roughly ! ! would imply a very high speed, even if all the mass were as far
from the axis as possible.
! !
a. Using simply rp (from ! r × p ) for the angular momentum of a mass
at radius r, obtain a rough value of p and show that it would imply
highly relativistic speed.
b. At such speeds, ! E = γ mc 2 and ! p = γ mu combine to give ! E ≈ pc (just
as for the speedy photon or the LHC-produced protons from the
first midterm). How does this energy compare with the known
internal energy of the electron? What does that imply about how
“classical” the electron’s intrinsic angular momentum is?
6. [Harris 8.32 tweaked] Is intrinsic angular momentum “real” angular
momentum? The famous Einstein-de Haas effect demonstrates it is!
Suppose you have a cylinder 2 cm in diameter hanging motionless from a
threat connected at the very center of its circular top. A representative
atom in the cylinder has atomic mass 60 and one electron free to respond
to an external field. Initially, spin orientations are as likely to be up as
down, but a strong magnetic field in the upward direction is suddenly
applied, causing the magnetic moments of all free electrons to align with
the field.
a. Viewed from above, which way would the cylinder rotate?
b. What would be the initial rotation rate?
c. While the actual experiment is constructed a bit differently, the
equivalent rotation has been seen. Should you be impressed the
physicists made the measurement and that the measurement
matched our predictions? What does this mean about the nature of
electron spin?
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