Question

atoms with angular momentum s=1 is sent through a Stern-Gerlach apparatus oriented in the z (represented as SGz) direction and only the mz = 0 atoms are allowed to go through other sides.

1. Calculate the output at SGx

2. suppose one state (I 1 -1 > )x emerges . What happens if these beams are sent through SGz

Answer 1

1. The state with , ie, the state is the only one that's allowed to go in.

Therefore we need to calculate the output at when the state is incident on it.

Note that this is equivalent to making a measurement in the basis.

The S_x basis states are given by , and with corresponding spin values , 0 and respectively.

These states can be written in terms of the basis as:

With this information, let us calculate the output of the state passing through :

a) The probability of emerging (with spin-x value ) is:

Given

this becomes

b) The probability of emerging (with spin-x value 0) is:

Given ,

this becomes

c) The probability of emerging (with spin-x value ) is:

Given

this becomes

Therefore, the output is either the state with or , each with probability 1/2.

2)

We are told that emerges.

When this is sent through SGZ, it is equivalent to measuring in the z basis and collapses into one of the three basis states

a) The probability of   emerging (with spin-z value ) is:

Given

b) The probability of   emerging (with spin-z value 0) is:

Given

c)

The probability of   emerging (with spin-z value ) is:

Given

Therefore if is sent through , then we obtain atoms with either m_z=1 or m_z=-1 with probability 1/4 each or m_Z=1 with probability 1/2