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(c) Determine the coefficients <math>C_{1}</math>, <math>C_{2}</math>, <math>C_{3}</math> and <math>D_{4}</math> using the initial conditions spedified above. Note that the coefficients are not all independent(<math>a_{1}</math> and <math>a_{2}</math> satisfy differential equations).
(c) Determine the coefficients <math>C_{1}</math>, <math>C_{2}</math>, <math>C_{3}</math> and <math>D_{4}</math> using the initial conditions spedified above. Note that the coefficients are not all independent(<math>a_{1}</math> and <math>a_{2}</math> satisfy differential equations).
(d) Obtain the time-dependent probabilities of finding the system in level 1 and in level 2.
(e) Consider the amplitude of the probability of finding the system in state 2 as a function of <math>\omega</math>. What is the resonance condition? Obtain the full width at half maximum of the resonance.


<math>i\hbar\frac{\partial\Psi}{\partial t}=H\Psi</math>
<math>i\hbar\frac{\partial\Psi}{\partial t}=H\Psi</math>

Revision as of 21:41, 29 April 2010

Problem in Time Dependent Perturbation theory: Magnetic Resonance

Consider the HamiltonianFailed 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=H_{0}+H_{1}(t)}

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_{0}=E_{1}|1\rangle\langle1|+E_{2}|2\rangle\langle2|}

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_{1}(t)=\gamma e^{i\omega t}|1\rangle\langle2|+\gamma e^{-i\omega t}|2\rangle\langle1|}

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 E_{2}>E_{1}} , 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 \gamma} 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 real and positive. At the time 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 t=0} assume thatthe lower energy level is populated, i.e. the probability for the level 1 is one and the one for level 2 is zero.

(a) Assuming that the wavefuction of the system is given by

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 \psi=a_{1}(t)e^{-iE_{1}t/\hbar}|1\rangle+a_{2}(t)e^{-iE_{2}t/\hbar}|2\rangle}

(b) Solve the coupled differential equations obtained in (a). For this purporse reduce the coupled equations to a single second order differential equation for 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 a_{1}} . The solutions are of the form 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 a_{j}(t)=C_{j}e^{i\lambda_{+}t}+D_{j}e^{i\lambda_{-}t},\ j=1,2} . Obtain the frequenciesFailed 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 \lambda_{+}} 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 \lambda_{-}} .

(c) Determine the coefficients 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_{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 C_{2}} , and using the initial conditions spedified above. Note that the coefficients are not all independent( and satisfy differential equations).

(d) Obtain the time-dependent probabilities of finding the system in level 1 and in level 2.

(e) Consider the amplitude of the probability of finding the system in state 2 as a function of . What is the resonance condition? Obtain the full width at half maximum of the resonance.

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 |a_{2}(t)|^{2}=\frac{(\gamma/\hbar)^{2}}{\frac{1}{4}(\omega-\omega_{21})^{2}+\frac{\gamma^{2}}{\hbar^{2}}}\sin^{2}[t\sqrt{\frac{(\omega-\omega_{21})^{2}}{4}+\frac{\gamma^{2}}{\hbar^{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 |a_{1}(t)|^{2}=\frac{1}{\frac{1}{4}(\omega-\omega_{21})^{2}+\frac{\gamma^{2}}{\hbar^{2}}}\{[\frac{(\omega-\omega_{21})^{2}}{4}+\frac{\gamma^{2}}{\hbar^{2}}]\cos^{2}[t\sqrt{\frac{(\omega-\omega_{21})^{2}}{4}+\frac{\gamma^{2}}{\hbar^{2}}}]+\frac{(\omega-\omega_{21})^{2}}{4}\sin^{2}[t\sqrt{\frac{(\omega-\omega_{21})^{2}}{4}+\frac{\gamma^{2}}{\hbar^{2}}}]\}=1-\frac{\gamma^{2}/\hbar^{2}}{\frac{(\omega-\omega_{21})^{2}}{4}+\frac{\gamma^{2}}{\hbar^{2}}}\sin^{2}[t\sqrt{\frac{(\omega-\omega_{21})^{2}}{4}+\frac{\gamma^{2}}{\hbar^{2}}}]=1-|a_{2}(t)|^{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 F(\omega)=\frac{(2\gamma/\hbar)^{2}}{(\omega-\omega_{21})^{2}+(2\gamma/\hbar)^{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 \omega=\omega_{21}=\frac{E_{2}-E_{1}}{\hbar}}

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 4\gamma/\hbar}