Time-dependent perturbation theory (David Miller’s book, chapter 7)

Time-dependent perturbation theory

• Method very similar to time-independent pertubation theory

Steps to reach to the solutions:

1. \(\,\,\hat{H} = \hat{H}_0 + \gamma\hat{H}_p\) with the perturbation in time small
2. Expand wave function \(\,\,\Psi(x,t) = \sum_n a_n(t) e^{-i\omega_n t} \psi_n(x)\)
3. Then expand coefficients \(\,a_n(t)\) into power series in \(\gamma\)
4. Calculate \(\,a_n(t)\) up to some order to find \(\,\Psi\)

Perturbing Hamiltonian

• Solve the time-dependent Schrodinger equation
• The time-dependent part is a (small) perturbation

\[ \hat{H} = \hat{H}_0 + \gamma\hat{H}_p(t), \qquad i\hbar \frac{\partial}{\partial t} | \Psi\rangle = \hat{H} | \Psi \rangle \]

• Unperturbed \(\,\hat{H}_0\) does not depend on time
• \(\,\hat{H}_0\) has known eigenvalues and eigenstates

\[ \hat{H}_0 |\psi_n\rangle = E_n |\psi_n\rangle \]

The time-dependent wave function can be expanded in \(\,|\psi_n\rangle\) with extra time factors

\[ |\Psi\rangle = \sum_n \, a_n(t) e^{-iE_nt/\hbar} \,| \psi_n \rangle \]

Some notations

Notation: We will most of the time stop writing the time argument to simplify:

\[ a_n(t) = a^{(0)}_n + \gamma \, a^{(1)}_n + \gamma^2 \, a^{(2)}_n + \gamma^3 \, a^{(3)}_n + \dots \]

Notation: We write a little dot on top of a function to indicate a derivative to time:

\[ \frac{\partial a^{(j)}_q(t)}{\partial t} \equiv \dot{a}^{(j)}_q(t) \equiv \dot{a}^{(j)}_q \]

Notation: prime derivative notation for spacial derivatives (to \(x\)):

\[ f'(x) \equiv \frac{\partial f(x)}{\partial x} \qquad f''(x) \equiv \frac{\partial^2 f(x)}{\partial x^2} \]

Notation: shorter partial derivative notation (I will try to avoid to use it):

\[ \partial_x f(x) \equiv \frac{\partial f(x)}{\partial x} \qquad \partial_{x}^2 f(x) = \partial_{xx} f(x) \equiv \frac{\partial^2 f(x)}{\partial x \partial y} \qquad \partial_{xy} f(x) \equiv \frac{\partial^2 f(x)}{\partial x \partial y} \]

Equation in the coefficients

We know everything except of \(\,a_n(t)\), how to find them?

\[ \hat{H} = \hat{H}_0 + \gamma\hat{H}_p(t), \qquad i\hbar \frac{\partial}{\partial t} |\Psi\rangle = \hat{H} |\Psi \rangle \]

• Fill in the wave function and our approximate Hamiltonian

\[ i\hbar \frac{\partial}{\partial t} \sum_n \, a_n(t) e^{-iE_nt/\hbar} \,| \psi_n \rangle = \left(\hat{H}_0 + \gamma\hat{H}_p(t)\right) \sum_n \, a_n(t) e^{-iE_nt/\hbar} \,| \psi_n \rangle \]

\[ \begin{aligned} \Rightarrow \quad \sum_n \, \left( i\hbar\, \dot{a}_n(t) e^{-iE_nt/\hbar} + a_n(t) E_n e^{-iE_nt/\hbar}\right) \,| \psi_n \rangle &= \\ \qquad\qquad\qquad\qquad\left(\hat{H}_0 + \gamma\hat{H}_p(t)\right) &\sum_n \, a_n(t) e^{-iE_nt/\hbar} \,| \psi_n \rangle\\ \end{aligned} \]

Equation in the coefficients CTU’d

\[ \begin{aligned} \sum_n \, \left( i\hbar\, \dot{a}_n(t) e^{-iE_nt/\hbar} + a_n(t) E_n e^{-iE_nt/\hbar}\right) \,| \psi_n \rangle = \qquad \qquad \qquad\qquad\qquad\\ \qquad\qquad\qquad\qquad\left(\hat{H}_0 + \gamma\hat{H}_p(t)\right) \sum_n \, a_n(t) e^{-iE_nt/\hbar} \,| \psi_n \rangle\\ \\ \Longrightarrow \quad \sum_n \, \left( i\hbar \dot{a}_n + a_n E_n \right) e^{-iE_nt/\hbar} \,| \psi_n \rangle = \sum_n \, a_n \left(E_n + \gamma\hat{H}_p(t)\right) e^{-iE_nt/\hbar} \,| \psi_n \rangle\\ \end{aligned} \]

• Time-independent terms in energy \(E_n\): they cancel out

\[ \Longrightarrow \quad \sum_n \, i\hbar \dot{a}_n e^{-iE_nt/\hbar} \,| \psi_n \rangle = \sum_n \, a_n \gamma\hat{H}_p(t) e^{-iE_nt/\hbar} \,| \psi_n \rangle\\ \]

Equation in the coefficients CTU’d

\[ \sum_n \, i\hbar \dot{a}_n e^{-iE_nt/\hbar} \,| \psi_n \rangle = \sum_n \, a_n \gamma\hat{H}_p(t) e^{-iE_nt/\hbar} \,| \psi_n \rangle\\ \]

• Then left-multiply with \(\langle \psi_q |\)

\[ \Longrightarrow \quad \sum_n \, i\hbar \dot{a}_n e^{-iE_nt/\hbar} \, \langle \psi_q | \psi_n \rangle = \sum_n \, a_n \gamma e^{-iE_nt/\hbar} \langle \psi_q |\hat{H}_p(t) \,| \psi_n \rangle\\ \]

\[ \Longrightarrow \quad i\hbar \dot{a}_q e^{-iE_qt/\hbar} = \sum_n \, a_n \gamma e^{-iE_nt/\hbar} \langle \psi_q |\hat{H}_p(t) \,| \psi_n \rangle\\ \]

\[ \Longrightarrow \quad \dot{a}_q = \frac{1}{i\hbar}\sum_n \, a_n \gamma e^{-i(E_n-E_q)t/\hbar} \langle \psi_q |\hat{H}_p(t) \,| \psi_n \rangle\\ \]

Now we will make the actual approximation using \(\gamma\) (perturbation)

Perturbation expansion coefficients

Power series of the expansion coefficients \(a_n(t)\) in \(\gamma\)

\[ a_n(t) = a^{(0)}_n(t) + \gamma \, a^{(1)}_n(t) + \gamma^2 \, a^{(2)}_n(t) + \gamma^3 \, a^{(3)}_n(t) + \dots \]

Notation: We will stop writing the time argument to simplify:

\[ a_n(t) = a^{(0)}_n + \gamma \, a^{(1)}_n + \gamma^2 \, a^{(2)}_n + \gamma^3 \, a^{(3)}_n + \dots \]

Perturbation expansion coefficients

\[ \dot{a}_q = \frac{1}{i\hbar}\sum_n \, a_n \gamma e^{-i(E_n-E_q)t/\hbar} \langle \psi_q |\hat{H}_p(t) \,| \psi_n \rangle\\ \]

Power series of the expansion coefficients \(a_n(t)\) in \(\gamma\)

\[ a_n(t) = a^{(0)}_n + \gamma \, a^{(1)}_n + \gamma^2 \, a^{(2)}_n + \gamma^3 \, a^{(3)}_n + \dots \]

Derivative to time \(\quad\longrightarrow\quad\) time-derivatives of the power series coefficients

\[ \dot{a}_n(t) = \dot{a}^{(0)}_n + \gamma \, \dot{a}^{(1)}_n + \gamma^2 \, \dot{a}^{(2)}_n + \gamma^3 \, \dot{a}^{(3)}_n + \dots \]

Perturbation expansion coeffs CTU’d

\[ \dot{a}^{(0)}_q + \gamma \, \dot{a}^{(1)}_q + \gamma^2 \, \dot{a}^{(2)}_q + \dots = \qquad \qquad \qquad\qquad\qquad \qquad \qquad \qquad \qquad \\ \qquad\frac{1}{i\hbar}\sum_n \, \left(a^{(0)}_n + \gamma \, a^{(1)}_n + \gamma^2 \, a^{(2)}_n + \dots\right) \gamma e^{-i(E_n-E_q)t/\hbar} \langle \psi_q |\hat{H}_p(t) \,| \psi_n \rangle\\ \]

First order perturbation

\[ \dot{a}^{(0)}_q + \gamma \, \dot{a}^{(1)}_q + \gamma^2 \, \dot{a}^{(2)}_q + \dots = \qquad \qquad \qquad\qquad\qquad \qquad \qquad \qquad \qquad \\ \qquad\frac{1}{i\hbar}\sum_n \, \left(a^{(0)}_n + \gamma \, a^{(1)}_n + \gamma^2 \, a^{(2)}_n + \dots\right) \gamma e^{-i(E_n-E_q)t/\hbar} \langle \psi_q |\hat{H}_p(t) \,| \psi_n \rangle\\ \]

• First order perturbed coefficients

\[ \, \dot{a}^{(1)}_q = \frac{1}{i\hbar}\sum_n \, a^{(0)}_n \gamma e^{i\omega_{qn}t} \langle \psi_q |\hat{H}_p(t) \,| \psi_n \rangle\\ \]

• with \(\omega_{qn} = (E_q - E_n)/\hbar\)
• \(a_n^{(0)}\) all constant in time

Higher orders

• Higher order perturbed coefficients
• Suppose you calculated up to order \(p\)
• Next order \(a_n^{(p+1)}\) computed from previous orders

\[ \, \dot{a}^{(p+1)}_q = \frac{1}{i\hbar}\sum_n \, a^{(p)}_n \gamma e^{i\omega_{qn}t} \langle \psi_q |\hat{H}_p(t) \,| \psi_n \rangle\\ \]

• with \(\omega_{qn} = (E_q - E_n)/\hbar\)

For examples: Chapter 7 of David Miller’s book.