Download Week 10: Space quantization

Spring 2009
Week 10: Space quantization
When a light source is brought into a magnetic field, each emitted spectral line is split
into a number of components.
-Herzberg
Reading Assignment
Herzberg, G., Atomic Spectra and Atomic Structure, Dover Publications, 1944. Ch. II, 3.
Scientific terms and concepts
1.
2.
3.
magnetic moment
gyroscopic force
Stern-Gerlach experiment
4.
5.
6.
inhomogeneous
larmor precession
Bohr magneton
Homework exercises
(1) Precessing tops and Larmor precession: Consider a top consisting of a
z
wheel set atop a (nearly) massless post of length d. The wheel’s mass, M, is
concentrated at its radius, R. The axle of the wheel is aligned at an angle of Θ
degrees from the z-axis, as shown to the right. It is set spinning counterclockwise (when viewed from above), at a rate of N turns per second, about its
y
axle.
x
(A) What is the magnitude, and direction, of the angular velocity vector, ω, of
the wheel about its axis (in radians per second)?
(B) What is the rotational inertia, I, of the wheel?
(C) What is the magnitude, and direction, of the wheel’s angular momentum vector, L?
(D) Suppose that at time t=0 the axle of the wheel lies in the x-z plane. What is the magnitude, and
direction, of the torque, τ, acting on the center of gravity of the wheel due to the force of gravity.
(Hint: the torque produced by a force is the cross product of the lever arm and the force itself; the
lever arm is a projection of the wheel’s axle on the x-y plane.)
!Lh= " !t
(E) The gravitational torque causes a change in the angular momentum vector
with time, according to (the angular form) of Newton’s second law: τ =dL/dt.
Lh,i
Consequently, the angular momentum vector precesses in a circle, when
Lh,f
viewed from above, as shown in the figure to the right. Here are shown the
(horizontal component of the) initial and the final angular momentum vectors over a small time interval Δt. In which direction does the angular momentum vector precess? (i.e. counterclockwise, as shown in the figure, or
clockwise?) Also, does the magnitude of L change with time as a result of the
gravitational torque?
(F) Looking at the figure above, we can say that the tip of the vector representing the horizontal
component of the angular momentum precesses through a distance 2 π Lh (the whole circumference of the circle) at a speed given by the magnitude of torque during a time interval T. Then we
have T = 2 π Lh / τ. Solve for the precessional frequency Ω = 2π/T, in terms of the size of the
Quantum Mechanics- Week 10, Spring 2009
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Unit 1: The vacuum
wheel and the axle, R and d, the magnitude of the angular frequency of the spinning wheel, ω,
and the magnitude of the acceleration of gravity, g.
(G) Just as a spinning top in a gravitational field experiences precession about the gravitational field
direction, a particle of charge, e, mass, m, and angular momentum, L, placed in a magnetic field,
H, also experiences precession about the magnetic field direction. The magnetic moment, μ,
which characterizes the particle, has a magnitude given by μ=-eL/2mc. Now, the torque acting on
the particle is caused by the magnetic field: τ = μ x B. As you did in sections (E) and (F) above,
find an expression for the precessional frequency, Ω, of the particle immersed in a magnetic field.
(H) This precessional frequency, Ω, is called the Larmor frequency. When a pulse of radiation of this
exact frequency strikes a particle immersed in a magnetic field, it flips the particle over. This flipping of nuclear spins is the basis for nuclear magnetic resonance spectroscopy. Determine the
larmor frequency for a proton (i.e. a hydrogen nucleus) immersed in a uniform 5000 gauss magnetic field (typical for a modern MRI machine). In what range of the electromagnetic spectrum
does a radiation pulse of this frequency lie?
(2) Anomalous Zeeman effect of the Sodium D lines: Consider a gas of sodium atoms placed in a uniform magnetic field whose strength is 1.0 gauss.
(A) What is the value of S for the sodium atoms? Consider only the optically active electron(s).
(B) Enumerate all the possible values of J and M for the atoms in the state having L=1.
(C) Draw a “naive” and an “exact” vector model corresponding to each possible value of J and M, as in
Fig. 43 of Herzberg.
(D) Why can’t the component M of the angular momentum vector J ever equal the magnitude of J?
In particular, to what principle does Herzberg appeal to explain this?
(E) Enumerate all of the allowed values of the energy of this atom (relative to the value of the energy
in the absence of a magnetic field). Be sure to indicate which values of J and M they are associated with. (Hint: you will need to use the correct formula for the magnetic moment of the sodium
atom, which includes the Landé g-factor 1.)
(F) Now enumerate all the possible values of J and M for atoms in the state having L=0.
(G) How many emission lines occur due to transitions between states with L=1 and L=0?
(H) What are their respective wavelengths? Calculate them. (Hint: you can use Fig. 29 in Herzberg to
get the wavelengths in the absence of a magnetic field.)
(I) How is the light for each emission line polarized?
(J) What selection rule(s) governs the allowed transitions?
(K) What happens to these emission lines when the magnetic field is reduced to zero strength?
(L) What about if the magnetic field strength is increased to the point where the magnetic splitting
becomes greater than the multiplet splitting?
(M) Finally, if the magnetic field approaches, but does not exactly equal, zero, what is the relative
probability of finding atoms having the same value of L, but belonging to different J-multiplets?
1 The
Landé g-factor accounts for the double magnetism of the electron: that the magnetic moment associated with electron spin has a different physical origin than the magnetic moment associated with electron
orbital motion
Quantum Mechanics- Week 10, Spring 2009
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