Recent content by crowlma

Sorry to bump but is anyone able to provide any assistance at all?

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}...