Phy5646/homeworkintimeperturbation

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|}

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}

$a_{j}(t)=C_{j}e^{i\lambda _{+}t}+D_{j}e^{i\lambda _{-}t},\ j=1,2$

$i\hbar {\frac {\partial \Psi }{\partial t}}=H\Psi$

$(i\hbar {\dot {a}}_{1}e^{-iE_{1}t/\hbar }+E_{1}a_{1}e^{-iE_{1}t/\hbar })|1\rangle +(i\hbar {\dot {a}}_{2}e^{-iE_{2}t/\hbar }+E_{2}a_{2}e^{-iE_{2}t/\hbar })|2\rangle =a_{1}e^{-iE_{1}t/\hbar }(E_{1}|1\rangle +\gamma e^{-i\omega t}|2\rangle )+a_{2}e^{-iE_{2}t/\hbar }(E_{2}|2\rangle +\gamma e^{i\omega t}|1\rangle )$

$i\hbar {\dot {a}}_{1}=a_{2}e^{i(E_{1}-E_{2})t/\hbar }\gamma e^{i\omega t}=\gamma e^{i(\omega -\omega _{21})t}a_{2}$

$i\hbar {\dot {a}}_{2}=a_{1}e^{i(E_{1}-E_{2})t/\hbar }\gamma e^{-i\omega t}=\gamma e^{-i(\omega -\omega _{21})t}a_{1}$

$i\hbar e^{-i(\omega -\omega _{21})t}{\dot {a}}_{1}=\gamma a_{2}$

$(\omega -\omega _{21})\hbar e^{-i(\omega -\omega _{21})t}{\dot {a}}_{1}+i\hbar e^{-i(\omega -\omega _{21})t}{\ddot {a}}_{1}=\gamma {\dot {a}}_{2}={\frac {\gamma ^{2}}{i\hbar }}e^{-i(\omega -\omega _{21})t}a_{1}$

$-\hbar ^{2}{\ddot {a}}_{1}+i\hbar ^{2}(\omega -\omega _{21}){\dot {a}}_{1}-\gamma ^{2}a_{1}=0<math><math>a\sim e^{i\lambda t}$

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}}}}

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}}

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}}}}}

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}