1) Do electrons really spin? Although electrons act as if they are "spinning" and they definitely have a
magnetic dipole moment, they aren't actually tiny spinning objects.
a) As a simple model for a spinning charged object, let' is consider a circular ring of charge with
radius r and total charge 9. If the charged ring is rotating around its axis with speed v,
prove that the current is given by
What is the magnetic dipole moment un for
b) The magnetic moment of an electron is where y is
"gyromagnetic ratio". Set this expression equal to your answer from part (a) and solve for the speed of
the "ring" electron in terms of r, y, 9. and fundamental physical constants.
c) The "classical" radius of the electron, also known as the Compton radius, can be derived by
setting the electrostatic potential energy of a sphere of charge e and radius l' equal to the relativistic rest
energy of the electron,
m_c2 Using this and the fact that the gyromagnetic ratio is equal to
for an electron, calculate the speed at which the "ring" electron model must spin to
obtain the observed angular momentum.
d) Comment on the validity of the "spinning" electron model, in light of your answer to part (c).
2) NMR Spectroscopy, Part I. Like an electron, a proton is a spin-1/2 particle. The energy eigenvalues
of the orthonormal spin-up and spin-down B) eigenstates in a magnetic field B = BJk are given by
where y - 26.7522 10
(the gyromagnetic ratio) for 1H. For reference, a 400-MHz NMR, which is
what our department has, contains a magnet with a field strength of 9.41
a) What is true about the energy of the two states in the absence of a magnetic field?
b) Calculate the population ratio of the two states at 298 K in a 300-MHz NMR. You should set
up the ratio in order to get a number less than 1. Write your answer to the first nonrepeating digit.
c) What does your answer to part (b) mean physically?
3) NMR Spectroscopy, Part II. Just like orbital angular momentum L, we can define operators for
spin angular momentum S:
S =S, -S,
c) Show that S/1)/00/a). What does this operator accomplish physically?
Show that S -1/2/00/B). What does this operator accomplish physically?
e) Evaluate S a). What does this result mean physically?
f) The intensity of a spin transition from (i)-(f) -> is proportional to (F/S,/i). From this,
derive the selection rule for NMR spectroscopy.
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