Homework Statement
A two-level system is spanned by the orthonormal basis states |a_{1}> and |a_{2}> . The operators representing two particular observable quantities A and B are:
\hat{A} = α(|a_{1}> <a_{1}| - |a_{2}> <a_{2}|)
and \hat{B} = β(|a_{1}> <a_{2}| + |a_{2}> <a_{1}|)
a) The state...
Homework Statement
What is the expectation value of \hat{S}_{x} with respect to the state \chi = \begin{pmatrix}
1\\
0
\end{pmatrix}?
\hat{S}_{x} = \frac{\bar{h}}{2}\begin{pmatrix}
0&1\\
1&0
\end{pmatrix}Homework Equations
<\hat{S}_{x}> = ∫^{\infty}_{-\infty}(\chi^{T})^{*}\hat{S}_{x}\chi...
Ok, so I've followed this through and I end up with \frac{1}{2}\frac{\bar{h}}{2}\int^{\infty}_{-\infty} \begin{pmatrix}
e^{-i\frac{ω}{2}t}t&e^{i\frac{ω}{2}t}
\end{pmatrix}\begin{pmatrix}
e^{i\frac{ω}{2}t}\\
e^{-i\frac{ω}{2}t}
\end{pmatrix} which then works out to be \frac{\bar{h}}{2} ?
The first part all makes sense but now I'm stuck on part b) - calculate <\hat{S}_{x}>.
\hat{S}_{x} = \frac{\bar{h}}{2}\begin{pmatrix}
0&1\\
1&0
\end{pmatrix}
So the expectation value = \int^{\infty}_{-\infty} \psi*<\hat{S}_{x}>\psi dx so I get to \int^{\infty}_{-\infty}...
Ah ok, the normalisation makes sense. Forgot about that. Also makes sense about the extra i - I stuffed up typing in the equation, it should have read:
\begin{pmatrix}
\frac{iω}{2}a(t)\\
-\frac{iω}{2}b(t)\\
\end{pmatrix} = \frac{\delta\psi}{\delta t}
Still struggling to work out how I get...
Okay so, from there I got to \hat{H} = ω\hat{S}_{z} so then ω\hat{S}_{z} \begin{pmatrix}
a(t)\\
b(t)
\end{pmatrix} = i\bar{h}\frac{d\psi}{dt} so then
\begin{pmatrix}
\frac{ω \bar{h}}{2}&0\\
0&-\frac{ω \bar{h}}{2}
\end{pmatrix} \begin{pmatrix}
a(t)\\
b(t)
\end{pmatrix} =...
Homework Statement
The evolution of a particular spin-half particle is given by the Hamiltonian \hat{H} = \omega\hat{S}_{z}, where \hat{S}_{z} is the spin projection operator.
a) Show that \upsilon = \frac{1}{\sqrt{2}}\begin{pmatrix}
e^{-i\frac{\omega}{2}t}\\
e^{i\frac{\omega}{2}t}...