Quantum Mechanics - Unitary Operators and Spin 1/2

[Tangent87]

Hi, I'm doing question 2/II/32D at the top of page 68 here (http://www.maths.cam.ac.uk/undergrad/pastpapers/2005/Part_2/list_II.pdf ). I have done everything except for the last sentence of the question.

This is what I have attempted so far:

$$|\chi\rangle=|\uparrow\rangle=\left( \begin{array}{c}<br /> 1 \\<br /> 0 \end{array} \right)$$

Then $$U|\chi\rangle=\left( \begin{array}{c}<br /> cos(\theta /2) \\<br /> 0 \end{array} \right)-(\boldsymbol{n}.\boldsymbol{\sigma})\left( \begin{array}{c}<br /> isin(\theta /2) \\<br /> 0 \end{array} \right)$$

Now I need to choose n. If I want the spin up state measured along the direction (sin@,0,cos@) am I correct in thinking I need the eigenvector corresponding to the +1 eigenvalue of this matrix?:

$$\sigma_1 sin\theta+\sigma_3 cos\theta=\left( \begin{array}{cc}<br /> cos\theta & sin\theta \\<br /> sin\theta & -cos\theta \end{array} \right)$$

In which case this the desired state is $$U|\chi\rangle=\left( \begin{array}{c}<br /> sin\theta \\<br /> 1-cos\theta \end{array} \right)$$

But I don't think it's possible to choose n such that this is the case, so where have I gone wrong? Also do I need to worry about any $$\hbar /2$$ since $$\boldsymbol{S}=\frac{\hbar}{2}\boldsymbol{\sigma}$$?

Thanks.

 

[diazona]

Hi, I'm doing question 2/II/32D at the top of page 68 here (http://www.maths.cam.ac.uk/undergrad/pastpapers/2005/Part_2/list_II.pdf ). I have done everything except for the last sentence of the question.

This is what I have attempted so far:

$$|\chi\rangle=|\uparrow\rangle=\left( \begin{array}{c}<br /> 1 \\<br /> 0 \end{array} \right)$$

Then $$U|\chi\rangle=\left( \begin{array}{c}<br /> cos(\theta /2) \\<br /> 0 \end{array} \right)-(\boldsymbol{n}.\boldsymbol{\sigma})\left( \begin{array}{c}<br /> isin(\theta /2) \\<br /> 0 \end{array} \right)$$

Now I need to choose n. If I want the spin up state measured along the direction (sin@,0,cos@) am I correct in thinking I need the eigenvector corresponding to the +1 eigenvalue of this matrix?:

$$\sigma_1 sin\theta+\sigma_3 cos\theta=\left( \begin{array}{cc}<br /> cos\theta & sin\theta \\<br /> sin\theta & -cos\theta \end{array} \right)$$

In which case this the desired state is $$U|\chi\rangle=\left( \begin{array}{c}<br /> sin\theta \\<br /> 1-cos\theta \end{array} \right)$$

But I don't think it's possible to choose n such that this is the case, so where have I gone wrong?

Right idea, wrong matrix. Think about it like this: you are trying to arrange for the component of spin along a particular axis to be $+\hbar/2$, so you need to find the +1 eigenvalue of the matrix that measures spin along that axis. If the axis were the z axis, you'd use $S_z$. If it were the x axis, you'd use $S_x$. For an arbitrary axis, though, you don't have a precomputed spin matrix, so you'll need to calculate it by applying a rotation to one of the known spin matrices.

Now, do you know how to use $U$ to implement a rotation? Specifically, remember what the meanings of the vector $\hat{n}$ and the parameter $\theta$ are, in the context of rotations.

Also do I need to worry about any $$\hbar /2$$ since $$\boldsymbol{S}=\frac{\hbar}{2}\boldsymbol{\sigma}$$?

Keep in mind that $\sigma$ is unitless, and just make sure the units are consistent

 

[Tangent87]

Right idea, wrong matrix. Think about it like this: you are trying to arrange for the component of spin along a particular axis to be $+\hbar/2$, so you need to find the +1 eigenvalue of the matrix that measures spin along that axis. If the axis were the z axis, you'd use $S_z$. If it were the x axis, you'd use $S_x$. For an arbitrary axis, though, you don't have a precomputed spin matrix, so you'll need to calculate it by applying a rotation to one of the known spin matrices.

Now, do you know how to use $U$ to implement a rotation? Specifically, remember what the meanings of the vector $\hat{n}$ and the parameter $\theta$ are, in the context of rotations.

Keep in mind that $\sigma$ is unitless, and just make sure the units are consistent

Can you tell me what I've done wrong in using the matrix $$\boldsymbol{m}.\boldsymbol{\sigma}=<br /> \sigma_1 sin\theta+\sigma_3 cos\theta=\left( \begin{array}{cc}<br /> cos\theta & sin\theta \\<br /> sin\theta & -cos\theta \end{array} \right)$$ where m=(sin@,0,cos@) (the direction for the new spin up state). Now we have to choose an n such that $$U=exp(-i\boldsymbol{n}.\boldsymbol{\sigma}\theta/2)$$ rotates our original spin up state into this new one? As you say, U here is the rotation operator which will rotate the state by an angle theta about n.

 

[Tangent87]

Actually I think I might have just figured it out. I think I've got the RIGHT matrix, just the WRONG eigenvectors. My spin up eigenvector can be simplified to (cos(@/2),sin(@/2)) and if I choose n to be n=(0,1,0) I think that will work?

 

[vela]

Yes, that's correct.

 

[diazona]

Can you tell me what I've done wrong in using the matrix $$\boldsymbol{m}.\boldsymbol{\sigma}=<br /> \sigma_1 sin\theta+\sigma_3 cos\theta=\left( \begin{array}{cc}<br /> cos\theta & sin\theta \\<br /> sin\theta & -cos\theta \end{array} \right)$$ where m=(sin@,0,cos@) (the direction for the new spin up state). Now we have to choose an n such that $$U=exp(-i\boldsymbol{n}.\boldsymbol{\sigma}\theta/2)$$ rotates our original spin up state into this new one? As you say, U here is the rotation operator which will rotate the state by an angle theta about n.

Ah, sorry about that. I didn't go all the way to the end of the calculation so I didn't realize that the matrix you have is actually the right one for this case.