A solved problem for spins
An electron is at rest in an oscillating magnetic field
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 B=B_{0}Cos\left ( \omega t \right )\hat{k}}
where 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 B_{0}} and 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 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,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 \chi ^{0}=\chi _{+}^{(x)} )} ]. Determine 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 \chi (t)} at any subsequent time. Beware.' This is a time-dependent Hamiltonian, so you cannot get 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 \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 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 -\hbar/2} if you measure 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 S_{x}}
(d) What is the minimum field 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 (B_{0})} required to force a complete flip in 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 S_{x}} ?
Solution:
(a)
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 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)
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 \chi (t)=\begin{pmatrix} \alpha (t)\\\beta (t)) \end{pmatrix}} with 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 (0)=\beta (0)=\frac{1}{\sqrt{2}}}
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 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}}