Electron spin state representation

[ModusPwnd]

Homework Statement

The spin of an electron points along the +x direction.

a)
What is this state in the representation where
[tex]
+z=
\left( {\begin{array}{c}
1 \\
0 \\
\end{array} } \right)
[/tex]
[tex]

-z=
\left( {\begin{array}{c}
0 \\
1 \\
\end{array} } \right)
[/tex]


b)
What is this spin state after its spin has been rotated by an angle [itex] \phi [/itex] about an axis [itex] \hat{n} [/itex]? (The components of [itex]\hat{n}[/itex] are the usual spherical coordinates)

Homework Equations

??

The Attempt at a Solution

a)
I presume it is something along the lines of ,
[tex] \frac{1}{\sqrt{2}} \left[\left( {\begin{array}{c}
1 \\
0 \\
\end{array} } \right)
\pm i
\left( {\begin{array}{c}
0 \\
1 \\
\end{array} } \right)\right]
[/tex]

But I do not know how to logically come to this conclusion. I don't know whether it is the plus case or the minus case, nor do I know which vector should get the 'i'.

Any ideas?

Thanks!

 

[vela]

Hint: Pauli matrices.

 

[ModusPwnd]

Ok... What about them? lolI have read through the brief section on them in griffiths, and also googled them for a bit. I still don't get it though.

If I apply the z spin matrix to my x state, I just get one state... But I should get the superposition of two states right?

 

[vela]

How are the Pauli matrices related to the spin operators S_x, S_y, and S_z?

 

[ModusPwnd]

They are the spin operators, for spin 1/2 with a constant factored out.

 

[Dickfore]

How can you rotate the unit vector along the z-axis [itex]\hat{\mathbf{z}}[/itex] to the unit vector [itex]\hat{\mathbf{n}}[/itex]?

 

[ModusPwnd]

How can you rotate the unit vector along the z-axis [itex]\hat{\mathbf{z}}[/itex] to the unit vector [itex]\hat{\mathbf{n}}[/itex]?

Euler angles?

 

[Dickfore]

I wanted you to tell me what rotations about the fixed axes and by what angle would produce the necessary transition.

 

[ModusPwnd]

I don't know... I know you guys are big on the socratic method and all... But I am more confused now that ever...

?

How do I take a state vector in the x representation and put it into the z representation?

If I act a spin matrix/operator on a state, I get another state - not a superposition of states. But I should get a superposition of states, right?@Dickfore - I don't know if you are trying to help me with pt a or pt b. ?

 

[Dickfore]

I'm trying to help you with part b.

Take the [itex]\hat{\mathbf{z}}[/itex]. What does it transform to when you do a rotation around the y-axis by an angle [itex]\theta[/itex] followed by a rotation around the z-axis by an angle [itex]\phi[/itex]?

 

[ModusPwnd]

A unit vector in a different direction. The direction depending on theta and phi.

 

[vela]

I don't know... I know you guys are big on the socratic method and all... But I am more confused now that ever...

Frankly, I'm looking for a bit more effort on your part. It just seems like you're tossing out guesses rather than trying to analyze the problem. If you really have no clue on even where to start, you need to read your book because your questions are the kind that you can probably find the answers to in your textbook.


What does "The spin of an electron points along the +x direction" mean in terms of eigenstates and eigenvalues of which operator?

You know the Pauli matrices are essentially matrix representations of the spin operators with respect to some basis. Which basis is this?

 

[ModusPwnd]

Well, yea... If I knew how to analyze the problem I wouldn't be asking the questions. lol

I have read Griffiths, Sakurai and Zettili. They all mention that the pauli matrices give me the eigenvalue of the spin. They all operate in the z basis. There is no mention in any of them about changing the basis.

I've even asked some grad students, none of them could help me either...

 

[ModusPwnd]

What does "The spin of an electron points along the +x direction" mean in terms of eigenstates and eigenvalues of which operator?

It means the eigenvalue of the state with the x spin operator acting on it is 1/2.

 

[ModusPwnd]

Sorry to keep bumping my own post...

I asked another student who think I just need to find the eigen vectors of [itex] \sigma_z[/itex], but we are not sure how the original condition of being in the +x direction plays into this.

 

[vela]

It means the eigenvalue of the state with the x spin operator acting on it is 1/2.

In other words, the state is an eigenstate of S_x, so how might you find what that state is from [itex]\sigma_x[/itex]?