- #1
Markus Kahn
- 112
- 14
Homework Statement
Let ##\vec{e}\in\mathbb{R}^3## be any unit vector. A spin ##1/2## particle is in state ##|\chi \rangle## for which
$$\langle\vec{\sigma}\rangle =\vec{e},$$
where ##\vec{\sigma}## are the Pauli-Matrices. Find the state ##|\chi\rangle##
Homework Equations
:[/B] are all given above.The Attempt at a Solution
Well, I honestly have trouble even understanding what exactly the exercise is about. First of all, I'm really confused about the fact that the expectation value of an operator is supposed to be a vector. The only explanation I have for this kind of notation is
$$\langle\vec{\sigma}\rangle =\vec{e} \Leftrightarrow \sigma_i|\chi\rangle = e_i|\chi\rangle \hspace{0.5cm} i=1,2,3$$
Is this the right interpretation of the notation?
If this was true my approach was as follows: I assumed a basis of the corresponding Hilber space to be ##\mathcal{H}=\operatorname{span}\left(|0\rangle ,|1\rangle \right)## and therefore ##a,b\in\mathbb{C}^2## exist such that ##|\chi\rangle = a|0\rangle + b |1\rangle##. Now applying ##\sigma_1## to ##|\chi\rangle## results in the equation ##e_i \cdot (a|0\rangle + b|1\rangle )= b|0\rangle + a|1\rangle \Leftrightarrow a=e_i b## and ##b = e_i a##, which doesn't make any sense at all considering that I want to find ##a,b## as functions of ##e_1,e_2,e_3##.
I really don't know how to approach this problem and would be thankful for any suggestions.