A solved problem for spins

An electron is at rest in an oscillating magnetic field

$B=B_{0}Cos\left(\omega t\right){\hat {k}}$

where $B_{0}$ and $omega$ are constants.

(a) Construct the Hamiltonian matrix for this system.

(b) The electron starts out (at t = 0) in the spin-up state with respect to the x-axis [that is,$\chi ^{0}=\chi _{+}^{(x)})$]. Determine $\chi (t)$ at any subsequent time. Beware.' This is a time-dependent Hamiltonian, so you cannot get $\chi (t)$ in the usual way from stationary states. Fortunately, in this case you can solve the time-dependent Schr/Sdinger equation directly.

(c) Find the probability of getting $-\hbar /2$ if you measure $S_{x}$

(d) What is the minimum field $(B_{0})$ required to force a complete flip in $S_{x}$?

Solution:

(a)

$H=-\mu \mathbf {B} .\mathbf {S} =-\mu B_{0}Cos(\omega t)S_{z}=-{\frac {\mu B_{0}\hbar }{2}}Cos(\omega t){\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$

(b)

$\chi (t)={\begin{pmatrix}\alpha (t)\\\beta (t))\end{pmatrix}}$ with $\alpha (0)=\beta (0)={\frac {1}{\sqrt {2}}}$

$i\hbar {\frac {\partial \chi }{\partial t}}=i\hbar {\begin{pmatrix}{\dot {\alpha }}\\{\dot {\beta }}\end{pmatrix}}=\mathbf {H} \chi =-{\frac {\mu B_{0}\hbar }{2}}Cos(\omega t){\begin{pmatrix}1&0\\0&-1\end{pmatrix}}{\begin{pmatrix}\alpha \\\beta \end{pmatrix}}=-{\frac {\mu B_{0}\hbar }{2}}Cos(\omega t){\begin{pmatrix}\alpha \\-\beta \end{pmatrix}}$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{\alpha }=\frac{i\mu B_{0}}{2}Cos(\omega t)\alpha \Rightarrow \frac{\mathrm{d} \alpha }{ \alpha }=\frac{i\mu B_{0}}{2}Cos(\omega t)dt\Rightarrow Ln\alpha =\frac{i\mu B_{0}}{2}\frac{Sin(\omega t)}{\omega }+constant.}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha (t)=Ae^{\frac{i\mu B_{0}}{2}\frac{Sin(\omega t)}{\omega }};\alpha (0)=A=\frac{1}{\sqrt{2}}} , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha (t)=\frac{1}{\sqrt{2}}e^{\frac{i\mu B_{0}}{2}\frac{Sin(\omega t)}{\omega }}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{\beta }=-i\frac{\mu B_{0}}{2}Cos(\omega t)\beta \Rightarrow \beta (t)=\frac{1}{\sqrt{2}}e^{-i\frac{\mu B_{0}}{2}\frac{Sin(\omega t)}{\omega }}\Rightarrow \chi (t)=\frac{1}{\sqrt{2}}\begin{pmatrix} e^{i\frac{\mu B_{0}}{2}\frac{Sin(\omega t)}{\omega }}\\ e^{-i\frac{\mu B_{0}}{2}\frac{Sin(\omega t)}{\omega }} \end{pmatrix}}