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Problem in Time Dependent Perturbation theory: Magnetic Resonance | Problem in Time Dependent Perturbation theory: Magnetic Resonance | ||
Consider the Hamiltonian<math>H=H_{0}+H_{1}(t)</math> | Consider the Hamiltonian<math>H=H_{0}+H_{1}(t)</math> | ||
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<math>H_{1}(t)=\gamma e^{i\omega t}|1\rangle\langle2|+\gamma e^{-i\omega t}|2\rangle\langle1|</math> | <math>H_{1}(t)=\gamma e^{i\omega t}|1\rangle\langle2|+\gamma e^{-i\omega t}|2\rangle\langle1|</math> | ||
where <math>E_{2}>E_{1}</math>, and <math>\gamma</math> and <math>\omega</math> are real and positive. At the time <math>t=0</math> assume that the 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 | |||
<math>\psi=a_{1}(t)e^{-iE_{1}t/\hbar}|1\rangle+a_{2}(t)e^{-iE_{2}t/\hbar}|2\rangle</math> | <math>\psi=a_{1}(t)e^{-iE_{1}t/\hbar}|1\rangle+a_{2}(t)e^{-iE_{2}t/\hbar}|2\rangle</math> | ||
<math>a_{j}(t)=C_{j}e^{i\lambda_{+}t}+D_{j}e^{i\lambda_{-}t},\ j=1,2</math> | (b) Solve the coupled differential equations obtained in (a). For this purporse reduce the coupled equations to a single second order differential equation for <math>a_{1}</math>. The solutions are of the form <math>a_{j}(t)=C_{j}e^{i\lambda_{+}t}+D_{j}e^{i\lambda_{-}t},\ j=1,2</math>. Obtain the frequencies<math>\lambda_{+}</math> and <math>\lambda_{-}</math>. | ||
(c) Determine the coefficients <math>C_{1}</math>, <math>C_{2}</math>, <math>D_{1}</math> and <math>D_{2}</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. | |||
Solution: | |||
(a) | |||
<math>i\hbar\frac{\partial\Psi}{\partial t}=H\Psi</math> | <math>i\hbar\frac{\partial\Psi}{\partial t}=H\Psi</math> | ||
<math>(i\hbar\dot{a}_{1}e^{-iE_{1}t/\hbar}+E_{1}a_{1}e^{-iE_{1}t/\hbar})|1\rangle+(i\hbar\dot{a}_{2}e^{-iE_{2}t/\hbar}+E_{2}a_{2}e^{-iE_{2}t/\hbar})|2\rangle=a_{1}e^{-iE_{1}t/\hbar}(E_{1}|1\rangle+\gamma e^{-i\omega t}|2\rangle)+a_{2}e^{-iE_{2}t/\hbar}(E_{2}|2\rangle+\gamma e^{i\omega t}|1\rangle)</math> | <math>(i\hbar\dot{a}_{1}e^{-iE_{1}t/\hbar}+E_{1}a_{1}e^{-iE_{1}t/\hbar})|1\rangle+(i\hbar\dot{a}_{2}e^{-iE_{2}t/\hbar}+E_{2}a_{2}e^{-iE_{2}t/\hbar})|2\rangle=a_{1}e^{-iE_{1}t/\hbar}(E_{1}|1\rangle+\gamma e^{-i\omega t}|2\rangle)+a_{2}e^{-iE_{2}t/\hbar}(E_{2}|2\rangle+\gamma e^{i\omega t}|1\rangle)</math> | ||
<math>|1\rangle</math> and <math>|2\rangle</math> are orthogonal | |||
<math>i\hbar\dot{a}_{1}=a_{2}e^{i(E_{1}-E_{2})t/\hbar}\gamma e^{i\omega t}=\gamma e^{i(\omega-\omega_{21})t}a_{2}</math> | <math>i\hbar\dot{a}_{1}=a_{2}e^{i(E_{1}-E_{2})t/\hbar}\gamma e^{i\omega t}=\gamma e^{i(\omega-\omega_{21})t}a_{2}</math> | ||
<math>i\hbar\dot{a}_{2}=a_{1}e^{i(E_{1}-E_{2})t/\hbar}\gamma e^{-i\omega t}=\gamma e^{-i(\omega-\omega_{21})t}a_{1}</math> | <math>i\hbar\dot{a}_{2}=a_{1}e^{i(E_{1}-E_{2})t/\hbar}\gamma e^{-i\omega t}=\gamma e^{-i(\omega-\omega_{21})t}a_{1}</math> | ||
(b) | |||
<math>i\hbar e^{-i(\omega-\omega_{21})t}\dot{a}_{1}=\gamma a_{2}</math> | <math>i\hbar e^{-i(\omega-\omega_{21})t}\dot{a}_{1}=\gamma a_{2}</math> | ||
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<math>(\omega-\omega_{21})\hbar e^{-i(\omega-\omega_{21})t}\dot{a}_{1}+i\hbar e^{-i(\omega-\omega_{21})t}\ddot{a}_{1}=\gamma\dot{a}_{2}=\frac{\gamma^{2}}{i\hbar}e^{-i(\omega-\omega_{21})t}a_{1}</math> | <math>(\omega-\omega_{21})\hbar e^{-i(\omega-\omega_{21})t}\dot{a}_{1}+i\hbar e^{-i(\omega-\omega_{21})t}\ddot{a}_{1}=\gamma\dot{a}_{2}=\frac{\gamma^{2}}{i\hbar}e^{-i(\omega-\omega_{21})t}a_{1}</math> | ||
<math>-\hbar^{2}\ddot{a}_{1}+i\hbar^{2}(\omega-\omega_{21})\dot{a}_{1}-\gamma^{2}a_{1}=0<math> | Hence | ||
<math>-\hbar^{2}\ddot{a}_{1}+i\hbar^{2}(\omega-\omega_{21})\dot{a}_{1}-\gamma^{2}a_{1}=0</math> | |||
<math>a\sim e^{i\lambda t}</math> | Assume<math>a\sim e^{i\lambda t}</math> | ||
<math>\hbar^{2}\lambda^{2}-\hbar^{2}\lambda(\omega-\omega_{21})-\gamma^{2}=0</math> | <math>\hbar^{2}\lambda^{2}-\hbar^{2}\lambda(\omega-\omega_{21})-\gamma^{2}=0</math> | ||
<math>\lambda_{\pm}=\frac{1}{2}(\omega-\omega_{21})\pm\frac{1}{2}\sqrt{(\omega-\omega_{21})^{2}+4\frac{\gamma^{2}}{\hbar^{2}}}</math> | <math>\lambda_{\pm}=\frac{1}{2}(\omega-\omega_{21})\pm\frac{1}{2}\sqrt{(\omega-\omega_{21})^{2}+4\frac{\gamma^{2}}{\hbar^{2}}}</math> | ||
Now | |||
<math>a_{1}(t)=C_{1}e^{i\lambda_{+}t}+B_{1}e^{i\lambda_{-}t}</math> | <math>a_{1}(t)=C_{1}e^{i\lambda_{+}t}+B_{1}e^{i\lambda_{-}t}</math> | ||
<math>a_{2}(t)=C_{2}e^{i\lambda_{+}t}+B_{2}e^{i\lambda_{-}t}</math> | <math>a_{2}(t)=C_{2}e^{i\lambda_{+}t}+B_{2}e^{i\lambda_{-}t}</math> | ||
(c) Initial conditions: | |||
<math>a_{2}(0)=0</math> | <math>a_{2}(0)=0</math> | ||
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<math>C_{2}=-\frac{\gamma}{2\hbar\sqrt{\frac{(\omega-\omega_{21})^{2}}{4}+\frac{\gamma^{2}}{\hbar^{2}}}}</math> | <math>C_{2}=-\frac{\gamma}{2\hbar\sqrt{\frac{(\omega-\omega_{21})^{2}}{4}+\frac{\gamma^{2}}{\hbar^{2}}}}</math> | ||
(d) | |||
<math>|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}}}]</math> | <math>|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}}}]</math> | ||
<math>|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}</math> | <math>|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}</math> | ||
(e) | |||
<math>F(\omega)=\frac{(2\gamma/\hbar)^{2}}{(\omega-\omega_{21})^{2}+(2\gamma/\hbar)^{2}}</math> | <math>F(\omega)=\frac{(2\gamma/\hbar)^{2}}{(\omega-\omega_{21})^{2}+(2\gamma/\hbar)^{2}}</math> | ||
<math>\omega=\omega_{21}=\frac{E_{2}-E_{1}}{\hbar}</math> | Resonance condition: <math>\omega=\omega_{21}=\frac{E_{2}-E_{1}}{\hbar}</math> | ||
<math>4\gamma/\hbar</math> | Full width at half-maximum: <math>4\gamma/\hbar</math> | ||
Latest revision as of 22:13, 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 that the 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}} , 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 D_{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 D_{2}} using the initial conditions spedified above. Note that the coefficients are not all independent(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}} 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 a_{2}} 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 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} . What is the resonance condition? Obtain the full width at half maximum of the resonance.
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 i\hbar\frac{\partial\Psi}{\partial t}=H\Psi}
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\dot{a}_{1}e^{-iE_{1}t/\hbar}+E_{1}a_{1}e^{-iE_{1}t/\hbar})|1\rangle+(i\hbar\dot{a}_{2}e^{-iE_{2}t/\hbar}+E_{2}a_{2}e^{-iE_{2}t/\hbar})|2\rangle=a_{1}e^{-iE_{1}t/\hbar}(E_{1}|1\rangle+\gamma e^{-i\omega t}|2\rangle)+a_{2}e^{-iE_{2}t/\hbar}(E_{2}|2\rangle+\gamma e^{i\omega t}|1\rangle)}
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 |1\rangle} 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 |2\rangle} are orthogonal
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\dot{a}_{1}=a_{2}e^{i(E_{1}-E_{2})t/\hbar}\gamma e^{i\omega t}=\gamma e^{i(\omega-\omega_{21})t}a_{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\dot{a}_{2}=a_{1}e^{i(E_{1}-E_{2})t/\hbar}\gamma e^{-i\omega t}=\gamma e^{-i(\omega-\omega_{21})t}a_{1}}
(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 i\hbar e^{-i(\omega-\omega_{21})t}\dot{a}_{1}=\gamma a_{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})\hbar e^{-i(\omega-\omega_{21})t}\dot{a}_{1}+i\hbar e^{-i(\omega-\omega_{21})t}\ddot{a}_{1}=\gamma\dot{a}_{2}=\frac{\gamma^{2}}{i\hbar}e^{-i(\omega-\omega_{21})t}a_{1}}
Hence
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}\ddot{a}_{1}+i\hbar^{2}(\omega-\omega_{21})\dot{a}_{1}-\gamma^{2}a_{1}=0}
AssumeFailed 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\sim e^{i\lambda 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 \hbar^{2}\lambda^{2}-\hbar^{2}\lambda(\omega-\omega_{21})-\gamma^{2}=0}
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_{\pm}=\frac{1}{2}(\omega-\omega_{21})\pm\frac{1}{2}\sqrt{(\omega-\omega_{21})^{2}+4\frac{\gamma^{2}}{\hbar^{2}}}}
Now
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)=C_{1}e^{i\lambda_{+}t}+B_{1}e^{i\lambda_{-}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 a_{2}(t)=C_{2}e^{i\lambda_{+}t}+B_{2}e^{i\lambda_{-}t}}
(c) Initial conditions:
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}(0)=0}
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}=-B_{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_{2}(t)=2iC_{2}e^{i(\omega-\omega_{21})t/2}\sin[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)=\frac{i\hbar}{\gamma}e^{i(\omega-\omega_{21})t}\dot{a}_{2}=-\frac{2\hbar}{\gamma}C_{2}\{i\frac{\omega-\omega_{21}}{2}\sin[t\sqrt{\frac{(\omega-\omega_{21})^{2}}{4}+\frac{\gamma^{2}}{\hbar^{2}}}]+\sqrt{\frac{(\omega-\omega_{21})^{2}}{4}+\frac{\gamma^{2}}{\hbar^{2}}}\cos[t\sqrt{\frac{(\omega-\omega_{21})^{2}}{4}+\frac{\gamma^{2}}{\hbar^{2}}}]\}e^{i\frac{3}{2}(\omega-\omega_{21})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 a_{1}(0)=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}=-\frac{\gamma}{2\hbar\sqrt{\frac{(\omega-\omega_{21})^{2}}{4}+\frac{\gamma^{2}}{\hbar^{2}}}}}
(d)
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}}
(e)
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}}}
Resonance condition: 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}}
Full width at half-maximum: 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}