Spinors and spatial coordinates

[nobody101]

I was reading a web page (http://electron6.phys.utk.edu/qm1/modules/m12/spinor.htm) that claims that the state vector of a spin-1/2 particle is completely specified by a two-component spinor, just as the state vector of a spinless particle is completely specified by its components in position space.

This confused me greatly - it seems to me that the complete description of a spin-1/2 particle would be a state vector with components in the tensor product of both the "position space" and the "spin space" (the latter being two-dimensional). How do you get a "complete description" by collapsing all this information into a spinor with two complex components, and how do you extract the spatial coordinates of the particle from the spinor? By "spatial coordinates," I mean the components of the particle's state vector in that tensor product space I mentioned.

I'm trying to understand this as a step toward understanding the Dirac and Pauli equations. It seems very straightforward that the Schrodinger equation generates a time evolution of the position-dependent wavefunction for a spinless particle, but it's not at all clear to me how these former two equations generate such a time evolution for a particle with spin when they seem to concern only the spin components of the wavefunction, and not the spatial components.

 

[George Jones]

I was reading a web page (http://electron6.phys.utk.edu/qm1/modules/m12/spinor.htm) that claims that the state vector of a spin-1/2 particle is completely specified by a two-component spinor, just as the state vector of a spinless particle is completely specified by its components in position space.

I don't see where the web page says this.

This confused me greatly - it seems to me that the complete description of a spin-1/2 particle would be a state vector with components in the tensor product of both the "position space" and the "spin space" (the latter being two-dimensional).

The web page says this, too.

 

[nobody101]

I don't see where the web page says this.

It says that in the "notation" section, right after the sentence with the green words "two component spinor."

So I'm still wondering about my original questions.

 

[George Jones]

It says that in the "notation" section, right after the sentence with the green words "two component spinor."

So I'm still wondering about my original questions.

The author has just been a bit careless; the page should read

"For a spin ½ particle $| \psi >$ is completely specified by the position-dependent two component spinor, just as for a spinless particle $| \psi >$ is specified by the wave function $\psi (r)$."

From what is written above this, this is clearly what the author means. Note that in the first paragraph, state space is defined as a tensor product.

 

[nobody101]

I see. Are the components of the spinor just complex numbers, or are they themselves vectors? (the latter would make more sense to me, given the state space)

 

[Ben Niehoff]

The components of the spinor are functions.

Essentially, the wavefunction is a spinor-valued function of position.

 

[nobody101]

Ah, OK - thanks, Ben and George.