Quantum Numbers for Electron Spin

[Gavroy]

Hi

afaik S² does commute with an arbitrary component of the Spin, so this should give 2 quantum numbers, but they always say that there is just this 1/2 and -1/2 for the hydrogen atom. So my question is: Why is there only one quantum number to characterize the spin of an electron?

 

[Vanadium 50]

In hydrogen, there is only one electron, so it has only one possible value of spin.

 

[Polyrhythmic]

Spin +1/2 or -1/2 refers to the spin component in the z-direction, the spin quantization axis.

 

[mathfeel]

Hi

afaik S² does commute with an arbitrary component of the Spin, so this should give 2 quantum numbers, but they always say that there is just this 1/2 and -1/2 for the hydrogen atom. So my question is: Why is there only one quantum number to characterize the spin of an electron?

1/2 and -1/2 are eigenvalues of $S_z$ (or in fact anyone component).

The eigenvalues for $S^2$ is, in this case, $\frac{3}{4} = \frac12 (\frac12 + 1)$. Physically $S^2$, which is the magnitude of the spin square, can only take positive values.

 

[Gavroy]

http://en.wikipedia.org/wiki/Spin_(physics)#Mathematical_formulation_of_spin"

I don't know if you really answered my question, cause for example this link above shows two eigenvalue equations for S² and Sz, where they use different eigenvalues.

When I think about the angular momentum, there are also two different eigenvalues for L² and Lz and here it is necessary to know both.

Apparently for the spin operator, this is not so.

I don't really know why.

 

[mathfeel]

http://en.wikipedia.org/wiki/Spin_(physics)#Mathematical_formulation_of_spin"

I don't know if you really answered my question, cause for example this link above shows two eigenvalue equations for S² and Sz, where they use different eigenvalues.

When I think about the angular momentum, there are also two different eigenvalues for L² and Lz and here it is necessary to know both.

Apparently for the spin operator, this is not so.

I don't really know why.

Spin operator is identical algebraically to an angular momentum operator. There is no difference: spin is just an internal kind of angular momentum, as oppose to orbital angular momentum, which comes from motion.

With orbital angular momentum, you usually need to know both to know which orbit the electron is occupying. For example, electron orbit p_z corresponds to l=1, l_z = 1. You can always move (excite) electron to an orbit with different total angular momentum. So you have to specify that.

With spin, the fact that electron is a 1/2 spin particle is an intrinsic property. So when you say electron is in a spin-up states, you are saying s=1/2, s_z = 1/2. You can take an electron out of the total spin 1/2 state. So its total spin is always implied.

There are other particle with spin 0, 1/2, 1, 3/2 etc. Nuclear spin, for example. So you cannot say S always equals 1/2.

 

[Ken G]

Indeed, you can change the nature of a particle, thus changing its spin. Hence the analog of changing orbital angular momentum, in the case of spin, is like changing the label we will attach to the particle itself. So the difference is almost sociological-- we don't attach the same importance to a "p-shell electron" as we do to a "particle with spin 1/2."
ETA: I suppose it is overstating the issue to call it sociology-- there are certainly more fundamental differences between particles of different spin than particles of different orbital angular momentum, I merely think that is the analogy that helps answer the OP, in agreement with mathfeel.

 

[The_Duck]

When I think about the angular momentum, there are also two different eigenvalues for L² and Lz and here it is necessary to know both.

Apparently for the spin operator, this is not so.

I don't really know why.

All electrons have the same value of S², namely hbar*3/4 = hbar*(1/2)*(1 + 1/2). This is what we mean when we say that the electron is a "spin 1/2 particle." So we don't bother to mention this when describing the state of any given electron.