A solved problem for spins

From PhyWiki
Revision as of 12:33, 26 April 2010 by AnthonyKuchera (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

This problem is added by team 8; Source: Introduction to Quantum Mechanics,D. Griffiths,Problem 4-34.

Problem: 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}}

(c)

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 c_{-}^{(x)}=\chi _{-}^{(x)T}\chi =\frac{1}{2}\begin{pmatrix} 1 & -1 \end{pmatrix}\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}=\frac{1}{2}[e^{i\frac{\mu B_{0}}{2}\frac{Sin(\omega t)}{\omega }}-e^{-i\frac{\mu B_{0}}{2}\frac{Sin(\omega t)}{\omega }}]=iSin[\frac{\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 P_{-}^{(x)}(t)=\left |c_{-}^{(x)} \right |^{2}=Sin^{2}(\frac{\mu B_{0}}{2}\frac{Sin(\omega t)}{\omega })}

(d)

The argument of 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 Sin^{2}} must reach 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 \frac{\pi }{2}} (so P=1)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 \Rightarrow \frac{\mu B_{0}}{2\omega }=\frac{\pi }{2}} , or 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}=\frac{\pi \omega }{\mu }.}

Retrieved from "http://wiki.physics.fsu.edu/index.php?title=A_solved_problem_for_spins&oldid=8064"