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|}

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 $\gamma$ and $\omega$ are real and positive. At the time $t=0$ 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

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

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

(c) Determine the coefficients $C_{1}$, $C_{2}$, $C_{3}$ and $D_{4}$ using the initial conditions spedified above. Note that the coefficients are not all independent($a_{1}$ and $a_{2}$ 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 $\omega$. 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}}}}}

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}