# Magnetic Moment- Spin System - Probability

#### VVS

1. Homework Statement

Spin-system:
A spin-system has N = 30 localized particles with spin quantum number s = ½ and magnetic moment µ0. m = M/µ0 is the magnetic moment M in vertical direction (ms = +½) in units of µ0.
Calculate the probability PN(m) for the total magnetic moment M to achieve in vertical direction the value m
a) Without magnetic field (p = q =1/2) and
b) With magnetic field (p = 0,7; q = 0,3),
under the assumption that p und q = 1-p are the probabilities for ms = +1/2 and ms = -1/2, respectively.
Draw the values for PN(m) in both cases.

2. Homework Equations

all given in 1.

3. The Attempt at a Solution

View attachment Problem_Sheet_1_Exercise_3.pdf

I just don't know how the spin is related to the magnetic moment.
Is m=1/2 or -1/2 or M=1/2 or -1/2

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#### sankalpmittal

1. Homework Statement

Spin-system:
A spin-system has N = 30 localized particles with spin quantum number s = ½ and magnetic moment µ0. m = M/µ0 is the magnetic moment M in vertical direction (ms = +½) in units of µ0.
Calculate the probability PN(m) for the total magnetic moment M to achieve in vertical direction the value m
a) Without magnetic field (p = q =1/2) and
b) With magnetic field (p = 0,7; q = 0,3),
under the assumption that p und q = 1-p are the probabilities for ms = +1/2 and ms = -1/2, respectively.
Draw the values for PN(m) in both cases.

2. Homework Equations

all given in 1.

3. The Attempt at a Solution

View attachment 56430

I just don't know how the spin is related to the magnetic moment.
Is m=1/2 or -1/2 or M=1/2 or -1/2
Hint: What is the formula for "spin angular momentum" ?
What is the formula for "magnetic moment" ?
Once you know the two, you can see the obvious relation.

#### VVS

Hi,

Thanks for the hint. I checked online, but I can't find any formulae for sping angular momentum and magnetic moment.

Could you please give me another hint?

thank you

#### sankalpmittal

Hi,

Thanks for the hint. I checked online, but I can't find any formulae for sping angular momentum and magnetic moment.

Could you please give me another hint?

thank you
Spin angular momentum = [h√{S*(S+1)}]/2π

Magnetic moment = √{4S*(S+1)}

Now can you carry out further ?

S is spin quantum number.

#### VVS

Thanks,

From those two formulae it is clear that Spin Angular Momentum = h/(4*pi)*magnetic moment.

But I am not sure how this is supposed to help me with the problem. The sentence I didn't understand was:

m = M/µ0 is the magnetic moment M in vertical direction (ms = +½) in units of µ0. Calculate the probability PN(m) for the total magnetic moment M to achieve in vertical direction the value m under the assumption that p und q = 1-p are the probabilities for ms = +1/2 and ms = -1/2, respectively.

Is M=±1/2 or M/μ0 = ±1/2?

#### sankalpmittal

Thanks,

From those two formulae it is clear that Spin Angular Momentum = h/(4*pi)*magnetic moment.

But I am not sure how this is supposed to help me with the problem. The sentence I didn't understand was:

m = M/µ0 is the magnetic moment M in vertical direction (ms = +½) in units of µ0. Calculate the probability PN(m) for the total magnetic moment M to achieve in vertical direction the value m under the assumption that p und q = 1-p are the probabilities for ms = +1/2 and ms = -1/2, respectively.

Is M=±1/2 or M/μ0 = ±1/2?
I cannot understand this:

A spin-system has N = 30 localized particles with spin quantum number s = ½ and magnetic moment µ0. m = M/µ0 is the magnetic moment M in vertical direction (ms = +½) in units of µ0.
µ0 is magnetic moment of 30 localized particles, fine. But what is m?
I guess that M being vertical component, we have µ0 sinθ=M. This means m=sinθ. What is ms? In my textbook, ms is spin quantum number, but here already its denoted by S. Might be that you should contact other HHs.

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