Help with atomic polarization question

[justinbaker]

Studying for finals and this is one of the reveiw questions our teacher wanted us to take a look at. A little help please, i got started somewhat as you will see below

"Suppose that we have a lot of noninteracting atoms (a gas) in an external magnetic field. You may take as given the fact that each atom can be in one of two states, whose energies differ by an amount ΔE = 2µB, depending on the strength of the magnetization is taken to be +1 if it's in the lower energy state or -1 if it's in the higher state

a.) Find the average magnetization of hte entire sample as a function of the applied magnetic field B [Remark: Your answer can be expressed in terms of ΔE by using a hyperbolic trigonmetric function; if you know these, then write it this way.]

b.)Discuss how your solution behaves when B--> ∞ and when B--> 0, and why your results make sense."A.) so i am a little confused but this is what i have so far
E1=+µB
E2=-µB

and Probability1=e^(-E1/(KT))
Probability2=e^(-E2/(KT))

also P1 + P2= 1

so is there numbers that i need to plug into find the answer? B.) figure this out when i solve A

 

[lightgrav]

So, with these occupation fractions of "North up" and "South up" atoms,
what is the total (net) magnetization?
If T=infinite, you have the same number of Norths as Souths pointing upward,
so the total (and fractional) magnetization is zero.

It is important that the positive Energy is opposite the negative Energy,
or else the sum (and difference) is not a hyperbolic trig function.

 

[justinbaker]

ok so i am still really confused here, so is the avg magnetization zero?

 

[nrqed]

Studying for finals and this is one of the reveiw questions our teacher wanted us to take a look at. A little help please, i got started somewhat as you will see below

"Suppose that we have a lot of noninteracting atoms (a gas) in an external magnetic field. You may take as given the fact that each atom can be in one of two states, whose energies differ by an amount ΔE = 2µB, depending on the strength of the magnetization is taken to be +1 if it's in the lower energy state or -1 if it's in the higher state

a.) Find the average magnetization of hte entire sample as a function of the applied magnetic field B [Remark: Your answer can be expressed in terms of ΔE by using a hyperbolic trigonmetric function; if you know these, then write it this way.]

b.)Discuss how your solution behaves when B--> ∞ and when B--> 0, and why your results make sense."


A.) so i am a little confused but this is what i have so far
E1=+µB
E2=-µB

and Probability1=e^(-E1/(KT))
Probability2=e^(-E2/(KT))

also P1 + P2= 1

so is there numbers that i need to plug into find the answer?


B.) figure this out when i solve A

You have two possible values of magnetaziation. Just calculate (first value times the probability of having the first value) plus (second value times the probability of the second value) all that divided by the sum of the two porbabilities. It will give an hyperbolic function.

 

[superwolf]

The probabiilities are

$$\frac{1}{1+exp(- \Delta E/k_BT)}$$

and

$$\frac{1}{1+exp(\Delta E/k_BT)}$$,

but how do I know which probability represents wihch magnetization?

c)

If I am right, the expected value approaches zero both when B -> infinity and when B -> 0, but I can't see why that makes sense...