- #1
Rahmuss
- 222
- 0
[SOLVED] Electron Spin State and Values
An electron is in the spin state:
[tex]X = A\begin{pmatrix} 1-2i \\ 2 \end{pmatrix}[/tex]
(a) Determine the constant A by normalizing [tex]X[/tex]
(b) If you measured [tex]S_{z}[/tex] on this electron, what values could you get, and what is the probability of each? What is the expectation value of [tex]S_{z}[/tex]?
[tex]S_{z} = \frac{\hbar}{2}\begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}[/tex]
[tex]\left\langle S_{z}\right\rangle = \left\langle X | S_{z}X\right\rangle[/tex]
Part a):
[tex]A^{2}\left[ |1-2i|^{2} + |2|^{2}\right] = 1 \Rightarrow[/tex]
[tex]A^{2}\left[ 1-4i+4+4\right] = 1 \Rightarrow[/tex]
[tex]A^{2}\left[ 9-4i\right] = 1 \Rightarrow[/tex]
[tex]A^{2} = \frac{1}{9-4i} \Rightarrow[/tex]
[tex]A = \sqrt{\frac{1}{9-4i}} \Rightarrow[/tex]
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 [tex]S_{a}[/tex] 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:
[tex]\frac{\hbar}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix}1-2i \\ 2 \end{pmatrix} \Rightarrow[/tex]
[tex]\frac{\hbar}{2}\begin{pmatrix} 1-2i \\ -2 \end{pmatrix}[/tex]
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:
[tex]X = A\begin{pmatrix} 1-2i \\ 2 \end{pmatrix}[/tex]
(a) Determine the constant A by normalizing [tex]X[/tex]
(b) If you measured [tex]S_{z}[/tex] on this electron, what values could you get, and what is the probability of each? What is the expectation value of [tex]S_{z}[/tex]?
Homework Equations
[tex]S_{z} = \frac{\hbar}{2}\begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}[/tex]
[tex]\left\langle S_{z}\right\rangle = \left\langle X | S_{z}X\right\rangle[/tex]
The Attempt at a Solution
Part a):
[tex]A^{2}\left[ |1-2i|^{2} + |2|^{2}\right] = 1 \Rightarrow[/tex]
[tex]A^{2}\left[ 1-4i+4+4\right] = 1 \Rightarrow[/tex]
[tex]A^{2}\left[ 9-4i\right] = 1 \Rightarrow[/tex]
[tex]A^{2} = \frac{1}{9-4i} \Rightarrow[/tex]
[tex]A = \sqrt{\frac{1}{9-4i}} \Rightarrow[/tex]
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 [tex]S_{a}[/tex] 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:
[tex]\frac{\hbar}{2}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix}1-2i \\ 2 \end{pmatrix} \Rightarrow[/tex]
[tex]\frac{\hbar}{2}\begin{pmatrix} 1-2i \\ -2 \end{pmatrix}[/tex]
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.