What does spin quantum number mean?

[Eve Litman]

A spin quantum number can have a value of 1/2 or -1/2. I assume that the negative means that these two spins are in opposite directions. However, what does the 1/2 mean? Why 1/2, rather than a whole integer? Are there some sort of units associated with this value?

 

[eys_physics]

To be more precise, there are two quantum numbers associated with spin, i.e. the magnitude and the projection on the ##z##-axis. There are two kinds of particles, namely fermions having half-integer spin and bosons which have integer spin. The unit of spin is ##\hbar##.

 

[Eve Litman]

To be more precise, there are two quantum numbers associated with spin, i.e. the magnitude and the projection on the ##z##-axis. There are two kinds of particles, namely fermions having half-integer spin and bosons which have integer spin. The unit of spin is ##\hbar##.

Are there SI base units associated with that unit? And what does an integer versus half integer spin truly mean?
Also, thanks for the reply!

 

[PeterDonis]

A spin quantum number can have a value of 1/2 or -1/2. I assume that the negative means that these two spins are in opposite directions.

You're not talking about two spins. You're talking about one spin. The two quantum numbers are the two possible measurement results when you measure the spin of a spin-1/2 particle (like an electron) along a particular direction.

Why 1/2, rather than a whole integer?

Now you've opened a real can of worms. The short answer is that, in quantum mechanics, if we rotate a particle with spin, its quantum state changes, and the "spin" of the particle describes how it changes. For a 360 degree rotation, the phase of the particle's wave function changes by ##2 \pi## times the spin.

When QM was first discovered, physicists thought that spins could only be integers, which would mean that a complete 360 degree rotation would change the phase by an integer multiple of ##2 \pi##. Since phases are angles, that is the same as not changing the phase at all. But in the 1920s, as more experiments were done to explore the properties of quantum particles like electrons, it became apparent that electrons had a peculiar property: a complete 360 degree rotation multiplied the wave function by ##-1## (instead of ##1##). That corresponds to the phase changing by only ##\pi## for a 360 degree rotation, which means a spin of 1/2.

There is a lot more here, as I implied above; this subject is really too much for a PF discussion. But the above is the gist of it.

Are there SI base units associated with that unit?

The SI units of spin are the SI units of Planck's constant.

 

[eys_physics]

The SI unit of ##\hbar## is ##J\cdot s##. Well, if you have a quantum-mechanical state ##|s m>## which is an eigenstate of the spin operator ##\hat{s}##, you have the relations
$$\hat{s}^2|s m>= s(s+1)\hbar^2 |s m>,$$
and
$$\hat{s}_z |s m>= m\hbar |s m>$$.

The wave function of a two-fermion state is antisymmetric under exchange of the of the two particles, whereas a two-boson is symmetric.

 

[Eve Litman]

Thank you for the thorough response. I get the gist of this, but I don't know enough about quantum mechanics to fully understand. Could you refer me to a good resource to learn more about these concepts?

 

[jtbell]

Perhaps Hyperphysics?

http://hyperphysics.phy-astr.gsu.edu/hbase/spin.html

Follow links from there to explore related concepts such as quantized angular momentum in general:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qangm.html#c1

 

[eys_physics]

I would recommend to start with an introductory book on QM, e.g. Griffiths "Introduction to Quantum Mechanics", especially chapters 4 and 5 of that book could be useful.