Problem in Time Dependent Perturbation theory: Magnetic Resonance
Consider the Hamiltonian H = H 0 + H 1 ( t ) {\displaystyle H=H_{0}+H_{1}(t)}
H 0 = E 1 | 1 ⟩ ⟨ 1 | + E 2 | 2 ⟩ ⟨ 2 | {\displaystyle H_{0}=E_{1}|1\rangle \langle 1|+E_{2}|2\rangle \langle 2|}
H 1 ( t ) = γ e i ω t | 1 ⟩ ⟨ 2 | + γ e − i ω t | 2 ⟩ ⟨ 1 | {\displaystyle H_{1}(t)=\gamma e^{i\omega t}|1\rangle \langle 2|+\gamma e^{-i\omega t}|2\rangle \langle 1|}
ψ = a 1 ( t ) e − i E 1 t / ℏ | 1 ⟩ + a 2 ( t ) e − i E 2 t / ℏ | 2 ⟩ {\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 λ + t + D j e i λ − t , j = 1 , 2 {\displaystyle a_{j}(t)=C_{j}e^{i\lambda _{+}t}+D_{j}e^{i\lambda _{-}t},\ j=1,2}
i ℏ ∂ Ψ ∂ t = H Ψ {\displaystyle i\hbar {\frac {\partial \Psi }{\partial t}}=H\Psi }
( i ℏ a ˙ 1 e − i E 1 t / ℏ + E 1 a 1 e − i E 1 t / ℏ ) | 1 ⟩ + ( i ℏ a ˙ 2 e − i E 2 t / ℏ + E 2 a 2 e − i E 2 t / ℏ ) | 2 ⟩ = a 1 e − i E 1 t / ℏ ( E 1 | 1 ⟩ + γ e − i ω t | 2 ⟩ ) + a 2 e − i E 2 t / ℏ ( E 2 | 2 ⟩ + γ e i ω t | 1 ⟩ ) {\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 )}
i ℏ a ˙ 1 = a 2 e i ( E 1 − E 2 ) t / ℏ γ e i ω t = γ e i ( ω − ω 21 ) t a 2 {\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}}
i ℏ a ˙ 2 = a 1 e i ( E 1 − E 2 ) t / ℏ γ e − i ω t = γ e − i ( ω − ω 21 ) t a 1 {\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}}
i ℏ e − i ( ω − ω 21 ) t a ˙ 1 = γ a 2 {\displaystyle i\hbar e^{-i(\omega -\omega _{21})t}{\dot {a}}_{1}=\gamma a_{2}}
( ω − ω 21 ) ℏ e − i ( ω − ω 21 ) t a ˙ 1 + i ℏ e − i ( ω − ω 21 ) t a ¨ 1 = γ a ˙ 2 = γ 2 i ℏ e − i ( ω − ω 21 ) t a 1 {\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}}
− ℏ 2 a ¨ 1 + i ℏ 2 ( ω − ω 21 ) a ˙ 1 − γ 2 a 1 = 0 < m a t h >< m a t h > a ∼ e i λ t {\displaystyle -\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}}
ℏ 2 λ 2 − ℏ 2 λ ( ω − ω 21 ) − γ 2 = 0 {\displaystyle \hbar ^{2}\lambda ^{2}-\hbar ^{2}\lambda (\omega -\omega _{21})-\gamma ^{2}=0}
λ ± = 1 2 ( ω − ω 21 ) ± 1 2 ( ω − ω 21 ) 2 + 4 γ 2 ℏ 2 {\displaystyle \lambda _{\pm }={\frac {1}{2}}(\omega -\omega _{21})\pm {\frac {1}{2}}{\sqrt {(\omega -\omega _{21})^{2}+4{\frac {\gamma ^{2}}{\hbar ^{2}}}}}}
a 1 ( t ) = C 1 e i λ + t + B 1 e i λ − t {\displaystyle a_{1}(t)=C_{1}e^{i\lambda _{+}t}+B_{1}e^{i\lambda _{-}t}}
a 2 ( t ) = C 2 e i λ + t + B 2 e i λ − t {\displaystyle a_{2}(t)=C_{2}e^{i\lambda _{+}t}+B_{2}e^{i\lambda _{-}t}}
a 2 ( 0 ) = 0 {\displaystyle a_{2}(0)=0}
C 2 = − B 2 {\displaystyle C_{2}=-B_{2}}
a 2 ( t ) = 2 i C 2 e i ( ω − ω 21 ) t / 2 sin [ t ( ω − ω 21 ) 2 4 + γ 2 ℏ 2 ] {\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}}}}}]}
a 1 ( t ) = i ℏ γ e i ( ω − ω 21 ) t a ˙ 2 = − 2 ℏ γ C 2 { i ω − ω 21 2 sin [ t ( ω − ω 21 ) 2 4 + γ 2 ℏ 2 ] + ( ω − ω 21 ) 2 4 + γ 2 ℏ 2 cos [ t ( ω − ω 21 ) 2 4 + γ 2 ℏ 2 ] } e i 3 2 ( ω − ω 21 ) t {\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}}
a 1 ( 0 ) = 1 {\displaystyle a_{1}(0)=1}
C 2 = − γ 2 ℏ ( ω − ω 21 ) 2 4 + γ 2 ℏ 2 {\displaystyle C_{2}=-{\frac {\gamma }{2\hbar {\sqrt {{\frac {(\omega -\omega _{21})^{2}}{4}}+{\frac {\gamma ^{2}}{\hbar ^{2}}}}}}}}
| a 2 ( t ) | 2 = ( γ / ℏ ) 2 1 4 ( ω − ω 21 ) 2 + γ 2 ℏ 2 sin 2 [ t ( ω − ω 21 ) 2 4 + γ 2 ℏ 2 ] {\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}}}}}]}
| a 1 ( t ) | 2 = 1 1 4 ( ω − ω 21 ) 2 + γ 2 ℏ 2 { [ ( ω − ω 21 ) 2 4 + γ 2 ℏ 2 ] cos 2 [ t ( ω − ω 21 ) 2 4 + γ 2 ℏ 2 ] + ( ω − ω 21 ) 2 4 sin 2 [ t ( ω − ω 21 ) 2 4 + γ 2 ℏ 2 ] } = 1 − γ 2 / ℏ 2 ( ω − ω 21 ) 2 4 + γ 2 ℏ 2 sin 2 [ t ( ω − ω 21 ) 2 4 + γ 2 ℏ 2 ] = 1 − | a 2 ( t ) | 2 {\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}}
F ( ω ) = ( 2 γ / ℏ ) 2 ( ω − ω 21 ) 2 + ( 2 γ / ℏ ) 2 {\displaystyle F(\omega )={\frac {(2\gamma /\hbar )^{2}}{(\omega -\omega _{21})^{2}+(2\gamma /\hbar )^{2}}}}
ω = ω 21 = E 2 − E 1 ℏ {\displaystyle \omega =\omega _{21}={\frac {E_{2}-E_{1}}{\hbar }}}
4 γ / ℏ {\displaystyle 4\gamma /\hbar }
![{\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}}}}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b29aaf1ad95160db526fef499ac2e79ac7727a12)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bdbc62ca832ea4c168cf3e005cf6370f364a0dc)
![{\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}}}}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a3d3ac3448b475240fd86031d9d534474ba8ae7)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a90f4dd8ff0cab9362f8a8de52b456fb841724b)
