Time-Dependent Perturbation Theory and Selection Rules#

Setup: Stationary States and a Time-Dependent Perturbation#

We consider a Hamiltonian split into an exactly solvable part and a weak, time-dependent perturbation:

\[ \hat{H}(t) = \hat{H}_0 + \hat{V}(t) \]

The unperturbed Hamiltonian \(\hat{H}_0\) has known eigenstates and eigenvalues:

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

We expand the full time-dependent state in the stationary basis:

\[ |\Psi(t)\rangle = \sum_n c_n(t), e^{-iE_n t/\hbar} |n\rangle \]

The time-dependent coefficients \(c_n(t)\) encode transitions between stationary states caused by \(\hat{V}(t)\).

Deriving the First-Order Transition Amplitude#

Insert the expansion into the time-dependent Schrödinger equation

\[ i\hbar \frac{\partial}{\partial t}|\Psi(t)\rangle = \hat{H}(t)|\Psi(t)\rangle \]

After some algebra and using orthonormality \(\langle m|n\rangle = \delta_{mn}\), you obtain the equation of motion for the coefficients:

\[ i\hbar, \dot{c}*m(t) = \sum_n c_n(t) V*{mn}(t), e^{i\omega_{mn} t} \]

where

\[ V_{mn}(t) = \langle m|\hat V(t)|n\rangle, \qquad \omega_{mn} = \frac{E_m - E_n}{\hbar} \]

Assume the system starts in state \(|i\rangle\) at \(t=0\):

\[ c_n(0) = \delta_{ni}, \quad c_i(0) = 1, \quad c_{f\neq i}(0) = 0 \]

In first-order perturbation theory, we approximate \(c_n(t) \approx \delta_{ni}\) on the right-hand side (the perturbation is weak, so population remains mostly in the initial state). Then the equation for \(c_f(t)\) becomes

\[ \dot{c}*f^{(1)}(t) = -\frac{i}{\hbar} V*{fi}(t), e^{i\omega_{fi} t} \]

Integrating from \(0\) to \(t\):

\[ c_f^{(1)}(t) = -\frac{i}{\hbar} \int_0^t V_{fi}(t'), e^{i\omega_{fi} t'} dt' \]

This is the first-order transition amplitude from state \(|i\rangle\) to \(|f\rangle\).

Coupling to Light: Dipole Approximation#

For an atom or molecule in a classical electromagnetic field, in the electric dipole approximation the perturbation is

\[ \hat{V}(t) = -\hat{\boldsymbol{\mu}}\cdot \mathbf{E}(t) \]

For a monochromatic linearly polarized field

\[ \mathbf{E}(t) = \mathbf{E}_0 \cos(\omega t) \]

the matrix element is

\[ V_{fi}(t) = \langle f|\hat{V}(t)|i\rangle = -\langle f|\hat{\boldsymbol{\mu}}\cdot \mathbf{E}_0|i\rangle \cos(\omega t) \]

Define the (possibly complex) dipole matrix element

\[ \mu_{fi} = \langle f|\hat{\boldsymbol{\mu}}\cdot \hat{\mathbf{e}}|i\rangle \]

where \(\hat{\mathbf{e}}\) is the polarization direction of the field, and let \(E_0 = |\mathbf{E}_0|\). Then

\[ V_{fi}(t) = -\mu_{fi} E_0 \cos(\omega t) \]

Plugging into the transition amplitude:

\[ c_f^{(1)}(t) = \frac{i E_0}{\hbar} \mu_{fi} \int_0^t \cos(\omega t'), e^{i\omega_{fi} t'} dt' \]

Resonance Condition and Energy Conservation#

Use the exponential form of the cosine:

\[ \cos(\omega t') = \frac{1}{2}\left(e^{i\omega t'} + e^{-i\omega t'}\right) \]

Then the integral in \(c_f^{(1)}(t)\) contains terms of the form

\[ \int_0^t e^{i(\omega_{fi} \pm \omega)t'} dt' \]

If \(\omega_{fi} \pm \omega\) is large, the exponential oscillates rapidly and the integral averages out to a small value. A large transition amplitude occurs when the exponent is nearly stationary:

\[ \omega_{fi} - \omega \approx 0 \quad \Rightarrow \quad \omega_{fi} \approx \omega \]

This gives the familiar energy conservation condition for absorption:

\[ E_f - E_i = \hbar\omega \]

Similarly, for stimulated emission one finds \(E_f - E_i = -\hbar\omega\).

From Transition Amplitudes to Selection Rules#

Selection Rules

The transition probability (to first order) is

\[ P_{i\to f}(t) \approx |c_f^{(1)}(t)|^2 \propto |\mu_{fi}|^2 \]

  • A transition \(|i\rangle \to |f\rangle\) is only allowed (electric-dipole allowed) if the dipole matrix element

\[ \mu_{fi} = \langle f|\hat{\boldsymbol{\mu}}\cdot \hat{\mathbf{e}}|i\rangle \neq 0 \]

If the matrix element is exactly zero due to symmetry, the transition is dipole-forbidden (in first order).

Selection rules therefore come from:

  • the symmetry properties (angular, parity, etc.) of the wavefunctions \(|i\rangle\) and \(|f\rangle\)

  • the transformation properties of the operator \(\hat{\boldsymbol{\mu}} \sim \mathbf{r}\) (a vector operator)

Electric Dipole Selection Rules (Hydrogen-like Orbitals)#

For an electron in a central potential (e.g., hydrogenic atom), stationary states are labeled by \(|n, l, m_l\rangle\).

The position operator \(\mathbf{r}\) transforms like an angular momentum 1 object (similar to the spherical harmonics \(Y_{1m}\)). From angular momentum coupling rules one obtains:

Orbital angular momentum selection rule

\[ \Delta l = l_f - l_i = \pm 1 \]

Magnetic quantum number selection rule

\[ \Delta m_l = m_{l,f} - m_{l,i} = 0, \pm 1 \]

Summary

  • Start from the time-dependent Schrödinger equation with \(\hat{H}(t) = \hat{H}_0 + \hat{V}(t)\).

  • Expand the state in the eigenbasis of \(\hat{H}_0\) to obtain equations for \(c_n(t)\).

  • First-order perturbation theory gives a transition amplitude $\( c_f^{(1)}(t) = -\frac{i}{\hbar} \int_0^t \langle f|\hat{V}(t')|i\rangle e^{i\omega_{fi} t'} dt' \)$

  • For an oscillating field, the integral is large only when \(E_f - E_i = \pm \hbar\omega\) (energy conservation).

  • The transition probability is proportional to \(|\langle f|\hat{\boldsymbol{\mu}}|i\rangle|^2\).

  • If the dipole matrix element is zero by symmetry, the transition is forbidden.

  • For electric dipole transitions in atoms: \(\Delta l = \pm 1\), \(\Delta m_l = 0, \pm 1\), and parity changes.

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