Exploring Spin Functions: What Are They?

[Repetit]

Hey!

Can someone explain to me what the spin functions are? I understand that a spin up is described by a function which is often called $$\alpha$$ and spin down is described by a similar function called $$\beta$$. But what are these functions? What do they look like and what parameters do they take? Are they orthonormal like spatial wave functions?

If you have a good source for information on this I'd be grateful. I've tried searching Google but that didn't really give me any good information.

Thanks

 

[smallphi]

The spatial wave functions can be considered vectors (points) in an infinite dimensional vector space (the functional space of square integrable functions).

Analogously, the spin wave functions are vectors in the spin vector space which is finite dimensional. For electrons of spin 1/2, the spin space is two dimensional and can be spanned by two base vectors which you called alpha and beta.

The total Hilbert space of the system is the direct sum of the spatial space and the spin space i. e. the full wavefunction has space part and spin part.

The spin part of the wave function looks very contrived to beginners and most textbook do not put it in proper hystorical perspective to show that mathematical machinery was created to explain the experiment. The atomic spectra, the Zeeman effect, the Pauli exclusion principle all suggested that electrons in atoms have another quantum number. Later it was realized that number corresponded to a new degree of freedom unrelated to spatial degrees but associated with the internal angular momentum state of the electron. That is how the spin part of the wave function was introduced into the picture.

 

[Gza]

The spatial wave functions can be considered vectors (points) in an infinite dimensional vector space (the functional space of square integrable functions).

Analogously, the spin wave functions are vectors in the spin vector space which is finite dimensional. For electrons of spin 1/2, the spin space is two dimensional and can be spanned by two base vectors which you called alpha and beta.

The total Hilbert space of the system is the direct sum of the spatial space and the spin space i. e. the full wavefunction has space part and spin part.

The spin part of the wave function looks very contrived to beginners and most textbook do not put it in proper hystorical perspective to show that mathematical machinery was created to explain the experiment. The atomic spectra, the Zeeman effect, the Pauli exclusion principle all suggested that electrons in atoms have another quantum number. Later it was realized that number corresponded to a new degree of freedom unrelated to spatial degrees but associated with the internal angular momentum state of the electron. That is how the spin part of the wave function was introduced into the picture.

This is an excellent response, thank you.

 

[Repetit]

Thanks alot, that cleared things up for me!