Elementary particle Pauli’s exclusion principle

[8614smith]

Homework Statement

An elementary particle called the omega minus has spin 3/2. Calculate the magnitude of the spin angular momentum for this particle and the possible angles the spin angular momentum vector makes with the z-axis. Does this particle obey Pauli’s exclusion principle?

Homework Equations

$$\bar{S}=\sqrt{s(s+1)}\hbar$$
$$cos\frac{S_Z}{\bar{S_Z}}$$

The Attempt at a Solution

using the eqn. i get the right answer for S, $$\frac{\sqrt{15}}{2}$$

But using the 2nd eqn. i can't get the right angles, the right angles are: 39.2, 75.0, 105.0, 140.8. And why would the particle obey the pauli exclusion principle??

 

[Matterwave]

Since cosine is adjacent over hypotenuse, it should be: Cos(phi)=Sz/S

You can get different values of phi for different Sz

As far as the Pauli exclusion principle, do you know the difference between Fermions and Bosons, and what spins each has?

 

[8614smith]

Since cosine is adjacent over hypotenuse, it should be: Cos(phi)=Sz/S

You can get different values of phi for different Sz

As far as the Pauli exclusion principle, do you know the difference between Fermions and Bosons, and what spins each has?

Sorry, yes that's what i meant to write,

$$cos(\theta)=\frac{S_Z}{\bar{S}}$$

Fermions have half integer spin, bosons have integer spin or is it the other way round?

so as the particle has half integer spin its a fermion and no two fermions can occupy the same quantum state simultaneously...i don't see how i can tell if it obeys the principle or not

 

[Matterwave]

The statement "no two fermions can occupy the same quantum state simultaneously" is precisely the Pauli exclusion principle.

You can tell whether particles obey the principle or not by whether they are fermions or bosons. Fermions obey the principle, and bosons do not. Fermions have half-integer spins, and bosons have integer spins.

The real reason the two kinds of particles behave this way has to do with the set-up of the combined wave function for 2 indistinguishable particles. Fermions have wave functions which are anti-symmetric, while Bosons have symmetric wave functions. If 2 Fermions were occupying the same state, their wave functions would disappear.

 

[8614smith]

The real reason the two kinds of particles behave this way has to do with the set-up of the combined wave function for 2 indistinguishable particles. Fermions have wave functions which are anti-symmetric, while Bosons have symmetric wave functions. If 2 Fermions were occupying the same state, their wave functions would disappear.

Are you talking about even and odd functions here? as in even about the y axis? and that is why their wave functions disappear - because of destructive interference?

 

[Matterwave]

Yes, even and odd functions.

For example, 2 fermion's wave function may be described as such (not worrying about normalization atm): (Assume 2 states, A and B, in which 2 particles 1 and 2 are)

PsiA(x1)PsiB(x2)-PsiA(x2)PsiB(x1) (notice the negative sign, it's what defines the 2 particles as fermions)

If I assume x1 and x2 are interchangeable (that is, indistinguishable), and that the two particles are both in state A, then my total wavefunction becomes:

PsiA(x1)PsiA(x2)-PsiA(x1)PsiA(x2)=0

For a boson, the sign is a plus sign instead of a minus sign. In which case you just get 2PisA(x1)PsiB(x2) instead of 0 (again, neglecting normalization).