Rahmuss
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[SOLVED] Electron Spin State and Values
An electron is in the spin state:
X = A\begin{pmatrix} 1-2i \\ 2 \end{pmatrix}
(a) Determine the constant A by normalizing X
(b) If you measured S_{z} on this electron, what values could you get, and what is the probability of each? What is the expectation value of S_{z}?
S_{z} = \frac{\hbar}{2}\begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}
\left\langle S_{z}\right\rangle = \left\langle X | S_{z}X\right\rangle
Part a):
A^{2}\left[ |1-2i|^{2} + |2|^{2}\right] = 1 \Rightarrow
A^{2}\left[ 1-4i+4+4\right] = 1 \Rightarrow
A^{2}\left[ 9-4i\right] = 1 \Rightarrow
A^{2} = \frac{1}{9-4i} \Rightarrow
A = \sqrt{\frac{1}{9-4i}} \Rightarrow
Part b):
For this part I'm a bit confused (thus the posting). I'm not sure what they mean when they talk about measuring S_{a} on the electron. Are they just saying, if you measured the z-component of the spin of the electron? And if so would I have something like:
\frac{\hbar}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix}1-2i \\ 2 \end{pmatrix} \Rightarrow
\frac{\hbar}{2}\begin{pmatrix} 1-2i \\ -2 \end{pmatrix}
And where do I go from there? And as far as the probabilities I may be able to get that if I know the normalization constant. And I think I can get the expectation value from the normalization constant as well.
Homework Statement
An electron is in the spin state:
X = A\begin{pmatrix} 1-2i \\ 2 \end{pmatrix}
(a) Determine the constant A by normalizing X
(b) If you measured S_{z} on this electron, what values could you get, and what is the probability of each? What is the expectation value of S_{z}?
Homework Equations
S_{z} = \frac{\hbar}{2}\begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}
\left\langle S_{z}\right\rangle = \left\langle X | S_{z}X\right\rangle
The Attempt at a Solution
Part a):
A^{2}\left[ |1-2i|^{2} + |2|^{2}\right] = 1 \Rightarrow
A^{2}\left[ 1-4i+4+4\right] = 1 \Rightarrow
A^{2}\left[ 9-4i\right] = 1 \Rightarrow
A^{2} = \frac{1}{9-4i} \Rightarrow
A = \sqrt{\frac{1}{9-4i}} \Rightarrow
Part b):
For this part I'm a bit confused (thus the posting). I'm not sure what they mean when they talk about measuring S_{a} on the electron. Are they just saying, if you measured the z-component of the spin of the electron? And if so would I have something like:
\frac{\hbar}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix}1-2i \\ 2 \end{pmatrix} \Rightarrow
\frac{\hbar}{2}\begin{pmatrix} 1-2i \\ -2 \end{pmatrix}
And where do I go from there? And as far as the probabilities I may be able to get that if I know the normalization constant. And I think I can get the expectation value from the normalization constant as well.