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 &amp; sin\theta \\<br /> sin\theta &amp; -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 &amp; sin\theta \\<br /> sin\theta &amp; -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 &amp; sin\theta \\<br /> sin\theta &amp; -cos\theta \end{array} \right)<br /> 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 &amp; sin\theta \\<br /> sin\theta &amp; -cos\theta \end{array} \right)<br /> 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.