PH425 Spins Homework 4
Due 1/25/13
REQUIRED:
1. Using the Spins simulation, nd the probabilities for the projection of the spin onto all
three standard axes for unknowns |1 and |3 . Use these probabilities to write the
unknowns in the z -basis.
Solution:
PH425 Spins Homework 5
Due 1/30/13
REQUIRED:
1. A beam of spin-1 particles is prepared in the state
2
3
4
| = |1 + i |0 | 1 .
29
29
29
(a) What are the possible results of a measurement of the spin component Sz , and
with what probabilities would they occ
PH425 Spins Homework 6
Due 2/1/13 at 4 pm
REQUIRED:
1. Consider a spin-1/2 particle with a magnetic moment. At time t = 0, the state of the
particle is | (t = 0) = |+ .
(a) If the observable Sx is measured at time t = 0, what are the possible results and
PH425 Spins Homework 3
Due 1/23/13
REQUIRED:
1. With the Spins simulation set for a spin 1/2 system, measure the probabilities of all
the possible spin components for each of the unknown initial states |i (i = 1, 2, 3, 4).
(a) Use your measured probabilit
PH425 Spins Homework 2
Due 1/18/13
REQUIRED:
Note: In problem 1, I have changed the name of the angle in the problem and in my
solutions from the one that was originally posed to conform to the angle name we have been
using in class. Be careful as you rea
PH425 Spins Homework 1
Due 1/16/13
REQUIRED:
1. (a) Use Eulers formula ei = cos + i sin and its complex conjugate to nd formulas
for sin and cos . In your physics career, you will often need to read these formula
backwards, i.e. notice one of these combin
3.2.2 Magnetic Field in general direction
B = B0 z + B1 x
B
B0
B1
tan =
B0
B1
Larmor frequencies:
eB0
eB1
0 =
, 1 =
me c
me c
Hamiltonian:
e
e
H = B =
S B =
( S x B1 + S z B0 ) = 0 S z + 1S x
me c
me c
In matrix representation
0 1 0 1 0 1 0
H=
0 1 + 2 1
Group work
Show that the Sx, Sy and Sz matrices can be
written as a linear combination of projection
operators. (Projection operators are outer
products of the eigenvectors with
themselves.) The coefficient of each term is
the eigenvalue associated with
Sz is the operator that corresponds to the physical observable of spin
along the z-axis.
Write the eigenvalue equation for Sz using the above notation
Write the matrix Sz, and the kets |+> and |-> in matrix notation
In the z-basis, use what we know
about
Expectation Value of Spin Angular Momentum:
Sz
= + P(+) + P () = (t ) S z (t )
2
2
=e
t
i0
2
cos
2
= cos
2
2
e
i ( + 0 t )
1 0
sin
0 1 e
2 2
t
i 0
2
cos
i ( + 0 t )
2
e
sin
2 e i ( +0 t ) sin
2
2
2
= cos sin = cos
2
2
2 2
cos
2
e i ( +