Time Dependent Perturbation Theory

Given an arbitary state,R2(410)

$$\tilde{\psi}(x,t) = \sum_j C_j \Phi_j(x) e^{-\frac{i}{\hbar}E_jt},$$

for the initially unperturbed system described by the Hamiltonian $\hat{H}$, for which $\hat{H} \hat{\Phi} _j= E_j \Phi_j$ and $i \hbar \frac{\partial \tilde{\psi}} {\partial t} = \hat{H} \tilde{\psi},$ let us obtain the solution of the time dependent Schrödinger equation:

$$i \hbar \frac{\partial \psi} {\partial t} = [\hat{H} + \lambda \hat{\omega} (t)] \psi,$$ (13)

assuming that such solution can be written as a rapidly convergent expansion in powers of $\lambda$,

$$\psi_{\lambda}(x,t)=\sum_j\sum_{l=0}^{\infty}C_{jl}(t)\lambda^l \Phi_j(x) e^{-\frac{i}{\hbar}E_j t}.$$ (14)

Substituting Eq. (14) into Eq. (13) we obtain,

$$i\hbar \sum_{l=0}^{\infty} \left(\dot{C}_{kl}(t)\lambda^l + C_{kl... ...+ \lambda <\Phi_k\vert\hat{\omega}\vert\Phi_j>\right) e^{-\frac{i}{\hbar}E_jt}.$$

Terms with $\lambda^0$: (Zero-order time dependent perturbation theory)

$$+i\hbar[\dot{C}_{k_0}(t)e^{-\frac{i}{\hbar}E_kt} + C_{k_0}(t)(-\f... ...lta_{kj} E_j e^{-\frac{i}{\hbar}E_jt} = C_{k_0}(t)E_k e^{-\frac{i}{\hbar}E_kt}.$$

Since,

$$\dot{C}_{k_0}(t)=0, \qquad \Rightarrow \qquad C_{k_0}(t) =C_{k_0}(0).$$

Therefore, the unperturbed wave function is correct to zeroth order in $\lambda$.

Terms with $\lambda$: (First-order time dependent perturbation theory)

$$i\hbar[\dot{C}_{k_1}(t)e^{-\frac{i}{\hbar}E_kt} + C_{k_1}(t)(-\fr... ..._jt} + C_{j_0}(t)<\Phi_k\vert\hat{\omega}\vert\Phi_j> e^{-\frac{i}{\hbar}E_jt},$$

$$\dot{C}_{k_1}(t)= - \frac{i}{\hbar}\sum_j \left(C_{j_0}(0)<\Phi_k\vert\hat{\omega}\vert\Phi_j> e^{-\frac{i}{\hbar}(E_j-E_k)t}\right).$$

Therefore,

$$\dot{C}_{k_1}(t)= - \frac{i}{\hbar}\sum_j C_{j_0}(0)<\Phi_k\vert ... ...^{\frac{i}{\hbar}\hat{H}t}\hat{\omega}e^{-\frac{i}{\hbar}\hat{H}t}\vert\Phi_j>,$$ (15)

Eq. (15) was obtained by making the substitution $e^{-\frac{i}{\hbar}\hat{H}t}\vert\Phi_j> = e^{-\frac{i}{\hbar}E_jt}\vert\Phi_j>$, which is justified in the note that follows this derivation. Integrating Eq. (15) we obtain,

$$C_{k_1}(t)= - \frac{i}{\hbar}\int_{-\infty}^{t} dt' \sum _j C_{j_... ...hat{H}t'}\hat{\omega}e^{-\frac{i}{\hbar}\hat{H}t'}\vert\Phi_j>. \hspace{1.0cm}$$

which can also be written as follows:

$$C_{k_1}(t)= - \frac{i}{\hbar}\int_{-\infty}^{t} dt' <\Phi_k \vert... ...{\hbar}\hat{H}t'}\hat{\omega}e^{-\frac{i}{\hbar}\hat{H}t'}\vert\tilde{\psi}_0>.$$

This expression gives the correction of the expansion coefficients to first order in $\lambda$.

Note: The substitution made in Eq. (15) can be justified as follows. The exponential function is defined in powers series as follows,

$$e^A = \sum_{n=0}^{\infty} \frac{A^n}{n!} = 1+ A + \frac{1}{2!}A A + ...., \qquad \qquad \bf R4(169) \normalfont$$

In particular, when $A= -i \hat{H} t/\hbar$,

$$e^{-\frac{i}{\hbar}\hat{H}t} = 1 + (-\frac{i}{\hbar}\hat{H}t) + \frac{1}{2!}(-\frac{i}{\hbar}t)^2\hat{H}\hat{H} + ....$$

Furthermore, since

$$\hat{H}\vert\Phi_j> = E_j \vert \Phi_j>,$$

and,

$$\hat{H}\hat{H}\vert\Phi_j> =E_j\hat{H}\vert\Phi_j> = E_j^2\vert\phi_j>,$$

we obtain,

$$e^{-\frac{i}{\hbar}\hat{H}t}\vert\Phi_j> = [1 + (-\frac{i}{\hbar}... ...{i}{\hbar}t)^2E_j^2 + ...]\vert\Phi_j> = e^{-\frac{i}{\hbar}E_jt}\vert\Phi_j>,$$

which is the substitution implemented in Eq. (15).

Terms with $\lambda^2$: (Second-order time dependent perturbation theory)

$$i\hbar[\dot{C}_{k_2}(t) + C_{k_2}(t)(-\frac{i}{\hbar}E_k)]e^{-\fr... ..._j + C_{j_1}(t)<\Phi_k\vert\hat{\omega}\vert\Phi_j> ] e^{-\frac{i}{\hbar}E_jt},$$

$$\dot{C}_{k_2}(t)= - \frac{i}{\hbar}\sum_j <\Phi_k \vert e^{\frac{... ...ar}\hat{H}t}\hat{\omega} e^{-\frac{i}{\hbar} \hat{H}t} \vert\Phi_j> C_{j_1}(t),$$

$$C_{k_2}(t)= \left(- \frac{i}{\hbar} \right) \int_{-\infty}^{t} dt... ...ar}\hat{H}t'}\hat{\omega}e^{-\frac{i}{\hbar}\hat{H}t'}\vert\Phi_j> C_{j_1}(t'),$$

$$C_{k_2}(t)= \left(- \frac{i}{\hbar} \right)^2 \sum_j \int_{-\inft... ...hbar}\hat{H}t''}\hat{\omega}e^{-\frac{i}{\hbar}\hat{H}t''}\vert\tilde{\psi}_0>.$$

Since $1=\sum_j \vert\Phi_j><\Phi_j\vert$,

$$C_{k_2}(t)= \left(- \frac{i}{\hbar} \right)^2 \int_{-\infty}^{t} ... ...H}(t'-t'')} \hat{\omega} e^{-\frac{i}{\hbar} \hat{H}t''} \vert \tilde{\psi}_0>.$$

This expression gives the correction of the expansion coefficients to second order in $\lambda$.

Limiting Cases

(1) Impulsive Perturbation:

The perturbation is abruptly "switched on":R2(412)

According to the equations for first order time dependent perturbation theory,

$$C_{k_1}(t)= - \frac{i}{\hbar}\sum_j <\Phi_k\vert\bar{\omega}\vert\Phi_j> C_{j_0}(0) \int_{0}^{t} dt' e^{-\frac{i}{\hbar}(E_j-E_k)t'},$$

therefore,

$$C_{k_1}(t)= (- \frac{i}{\hbar})\sum_j \frac {C_{j_0}(0)<\Phi_k\ve... ...{i}{\hbar}(E_j-E_k) \right)} \left[ e^{-\frac{i}{\hbar}(E_j-E_k)t} - 1 \right].$$

Assuming that initially: $C_j = \delta_{lj}, \qquad \Rightarrow \qquad C_{j0} = \delta_{lj}.$ Therefore,

$$C_{k_1}(t)=-\frac {<\Phi_k\vert\bar{\omega}\vert\Phi_l>}{(E_l-E_k)}[1-e^{-\frac{i}{\hbar}(E_l-E_k)t}],$$

when $k \neq l$. Note that $C_{l_1}(t)$ must be obtained from the normalization of the wave function expanded to first order in $\lambda$.

Exercise 10: Compare this expression of the first order correction to the expansion coefficients, due to an impulsive perturbation, with the expression obtained according to the time-independent perturbation theory.

(2) Adiabatic limit:

The perturbation is "switched-on" very slowly ( $\frac{d\omega_t}{dt}<<\epsilon$, with $\epsilon$ arbitrarily small):R2(448)

$$C_{k_1}(t)= (- \frac{i}{\hbar})\int_{-\infty}^t dt' <\Phi_k\vert\omega(t')\vert\Phi_l> e^{-\frac{i}{\hbar}(E_l-E_k)t'}.$$

Integrating by parts we obtain,

$$C_{k_1}(t)= (- \frac{i}{\hbar})\left[\frac { e^{-\frac{i}{\hbar}(... ...) (E_l-E_k)} <\Phi_k \vert \frac{\partial w}{\partial t'} \vert\Phi_l> \right],$$

and, since $<\Phi_k \vert w(-\infty)\vert\Phi_l> = 0$,

$$C_{k_1}(t)= \frac {<\Phi_k\vert\omega(t)\vert\Phi_l>}{(E_l-E_k)} e^{-\frac{i}{\hbar}(E_l-E_k)t},$$

when $k \neq l$. Note that $C_{l_1}(t)$ must be obtained from the normalization of the wave function expanded to first order in $\lambda$.

Exercise 11: Compare this expression for the first order correction to the expansion coefficients, due to an adiabatic perturbation, with the expression obtained according to the time-independent perturbation theory.

(3) Sinusoidal Perturbation:

The sinusoidal perturbation is defined as follows, $\hat{\omega}(t,x)= \bar{\omega}(x)$   Sin$(\Omega t)$ when $t \geq 0$ and $\hat{\omega}(t,x)=0$, otherwise.

It is, however, more conveniently defined in terms of exponentials,

$$\hat{\omega}= \frac{\bar{\omega}(x)}{2i} [e^{i \Omega t}-e^{-i\Omega t}].$$

Therefore,

$$C_{k_1}(t)= -\frac{i}{\hbar} \int_0^t dt'<\Phi_k\vert e^{\frac{i}... ...{H}t'} \hat{\omega} (t') e^{-\frac{i} {\hbar} \hat{H}t'} \vert \tilde{\psi}_0>,$$ (16)

with $\vert\tilde{\psi}_0 > = \sum_jC_j\vert\Phi_j>$, and $\hat{H} \Phi_j =E_j \Phi_j$. Substituting these expressions into Eq. (16) we obtain,

$$C_{k_1}(t)=-\frac{1}{2\hbar} \sum_j C_j <\Phi_k\vert\bar{\omega}\... ...k-E_j)+ \hbar \Omega]t} -e^{\frac{i}{\hbar}[(E_k-E_j)- \hbar \Omega]t} \right),$$

and therefore,

$$C_{k_1}(t)=\frac{1}{i2\hbar} \sum_j C_j \bar{\omega}_{kj} \left[ ... ...c{i}{\hbar}[(E_k-E_j)-\hbar \Omega]t}}{\frac{E_k -E_j}{\hbar}-\Omega} \right ].$$

Without lost of generality, let us assume that $C_j=\delta_{nj}$ (i.e., initially only state $n$ is occupied). For $k \ge n$ we obtain,

$$\vert C_{k_1}(t)\vert^2=\frac{\vert\bar{\omega}_{kn}\vert^2}{k\hb... ...E_k-E_n)}{\hbar}- \Omega]t}}{\frac{(E_k -E_n)}{\hbar} - \Omega} \right \vert^2.$$

Factor $\vert\bar{\omega}\vert _{kn}$ determines the intensity of the transition (e.g., the selection rules). The first term (called anti-resonant) is responsible for emission. The second term is called resonant and is responsible for absorption.

For $k \neq n$, $P_{k_1}(t)= \lambda^2\vert C_{k_1}(t)\vert^2$ is the probability of finding the system in state $k$ at time $t$ (to first order in $\lambda$).

It is important to note that $P_{k1} << 1$ indicates that the system has been slightly perturbed. Such condition is satisfied only when $t << \frac{2 \hbar}{\vert\bar{\omega}_{kn}\vert\lambda}$. Therefore, the theory is useful only at sufficiently short times.