Physics 424: Quantum Mechanics, Fall 2015
Reading/Homework Assignment 6
Due in Class
Monday
, 10/26/15
1
Reading
•
Townsend Sec. 4.4 – Magnetic Resonance
•
(Townsend Secs. 4.5-4.6 will not be covered in class at this time and
are optional reading)
•
Townsend Sec. 4.7 – Summary
•
Townsend Sec. 5.1 – Basis States for Two Spin 1/2’s
•
Townsend Sec. 5.2 – The Hyperfine Splitting
•
Townsend Sec. 5.3 – The Addition of AM for Two Spin 1/2’s
You don’t need the following sections for these HW problems, but we will
start discussing these topics in class:
•
Townsend Sec. 5.4 – The EPR Paradox
•
Townsend Sec. 5.5 – Bell Inequalities
Problems to work out and turn in (75 pts total)
1. (15 pts total) Note: This entire problem is
classical
(no quantum me-
chanics). In class we showed that the classical equation of motion for
the spin vector
S
in an applied field
B
is
d
S
dt
=
γ
S
×
B
where
γ
is the gyromagnetic ratio.
1
2015-10-20 P424 HW6.tex c 2015 Donald Candela
1

(a) (5 pts) AssumeB=B0kwherekis the unit vector in the +zdirection. Show that the equations of motion for the three com-ponents ofSaredSxdt=γB0Sy,dSydt=-γB0Sx,dSzdt= 0.(b) (10 pts) Solve these equations of motion to show thati. The magnitude ofSremains constant.ii. The angle betweenSand thez-axis remains constant.iii.Srotates around thez-axis (clockwise looking down the axisifγ >0) at angular frequencyω0=γB0(the Larmor fre-quency).2. (10 pts) This is a simple problem, but it will require you to look upsome numbers and keep your units straight (both useful skills). Thegyromagnetic ratio for the proton isγp= +2.68×108s-1T-1, while

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- Fall '15
- mechanics, Work, Nuclear magnetic resonance, magnetic resonance