1 Suppose we have N assume N is an even number distinguishab

1) Suppose we have N (assume N is an even number) distinguishable magnetic moments that can each point up or down along a fixed direction, let us call that direction \"z.\" The magnetic moment of each particle is \"u.\" There is no magnetic field to interact with the magnetic moments.

a) What is the total number of accessible states with zero energy?

b) What is the probability that there is a net magnetic moment of M*u along z (where M can be positive or negative)? Justify your answer by showing that when you sum up all of the possible values for M that you get 1.

Solution

Ans a: total number of acessible states

Step 1: Underlying physics

When magnetic field B is zero, energy =\\mu.b=0

So all particles are at zero enrgy

Step2 : Number of particles

There are N partcles. Each particle can be eith in up or in down state i.e 2 available states.

Total number of states- 2X 2 X... N times= 2^N

Hence , 2^N states are possible.

Ans b: Probability

Step1 : Estimating the number of Particles

Let n₁ particles be in state 1 . Then n₂=N-n₁ particles are in state 2.

Then, the system to have net magnetic moment m*u along z we should have:

n₁-n₂=M

Step2: Probability of number of states with n1 spin up

2^-N [ \\sum ^N C_n1] = 2 ^-NX 2^N=1