### Materials covered in the chapter 1

```Intermediate Quantum Mechanics I
Midterm exam 1,Fall 2014
n  cos

 Z  e i sin

 Z ,in
2
2




correspondence with the unit vector n  sin  cos i  sin  sin j  cos k ,which specifies a
The state of a spin-1/2 particle may be parametrized as
point on a unit sphere in terms of polar angle  and azimuthal angle  .
z

y

x
Problem1
The state is given as
 
1
i 3
Z 
Z .
2
2

(1) Determine a particular axis vector n (namely  and  ),by comparing

 with n .
(2) If one measures the spin-1/2 along the n direction,what will be the outcome(s)?
(3) Calculate the expectation value S Z and its uncertainty S Z
(4) Calculate the expectation value S X and its uncertainty S X
Problem2
A beam of spin-1/2 particles is sent through 3 sequential Stern-Gerlach measurement devices as
SZ 
N

2
Sn 
SGn
SGZ

2
SGZ
SZ  


2

(1) When the middle axis vector n  j (namely the y axis),what fraction of the N particles
transmitted through the first device will survive the third measurement?

(2) For a general axis n ,what is the state  n which represents 

outcome of the second
2

measurement along the n direction?

(3) For a general axis n ,calculate the fraction of the particles which survives the third
measurement in terms of  and  .
(4) What is the maximum of the fraction calculated in (3),and when is it achieved?
(5) What is the minimum of the fraction calculated in (3),and when does it happen?
```