- 148

- 0

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 [Broken]). I have done everything except for the last sentence of the question.

This is what I have attempted so far:

[tex]|\chi\rangle=|\uparrow\rangle=\left( \begin{array}{c}

1 \\

0 \end{array} \right)[/tex]

Then [tex]U|\chi\rangle=\left( \begin{array}{c}

cos(\theta /2) \\

0 \end{array} \right)-(\boldsymbol{n}.\boldsymbol{\sigma})\left( \begin{array}{c}

isin(\theta /2) \\

0 \end{array} \right)[/tex]

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?:

[tex]\sigma_1 sin\theta+\sigma_3 cos\theta=\left( \begin{array}{cc}

cos\theta & sin\theta \\

sin\theta & -cos\theta \end{array} \right)[/tex]

In which case this the desired state is [tex]U|\chi\rangle=\left( \begin{array}{c}

sin\theta \\

1-cos\theta \end{array} \right)[/tex]

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 [tex]\hbar /2[/tex] since [tex]\boldsymbol{S}=\frac{\hbar}{2}\boldsymbol{\sigma}[/tex]?

Thanks.

This is what I have attempted so far:

[tex]|\chi\rangle=|\uparrow\rangle=\left( \begin{array}{c}

1 \\

0 \end{array} \right)[/tex]

Then [tex]U|\chi\rangle=\left( \begin{array}{c}

cos(\theta /2) \\

0 \end{array} \right)-(\boldsymbol{n}.\boldsymbol{\sigma})\left( \begin{array}{c}

isin(\theta /2) \\

0 \end{array} \right)[/tex]

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?:

[tex]\sigma_1 sin\theta+\sigma_3 cos\theta=\left( \begin{array}{cc}

cos\theta & sin\theta \\

sin\theta & -cos\theta \end{array} \right)[/tex]

In which case this the desired state is [tex]U|\chi\rangle=\left( \begin{array}{c}

sin\theta \\

1-cos\theta \end{array} \right)[/tex]

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 [tex]\hbar /2[/tex] since [tex]\boldsymbol{S}=\frac{\hbar}{2}\boldsymbol{\sigma}[/tex]?

Thanks.

Last edited by a moderator: