Time-Dependent Perturbation Theory and Selection Rules
Transitions between states: When a system is exposed to a time-dependent perturbation, such as an oscillating electromagnetic field, it can make transitions between the stationary states of the unperturbed Hamiltonian.
First-order transition amplitude: Expanding the time-dependent state in the unperturbed eigenbasis and keeping the leading term gives a simple integral for the amplitude of going from an initial state ∣ i ⟩ |i\rangle ∣ i ⟩ ∣ f ⟩ |f\rangle ∣ f ⟩
Resonance and energy conservation: For an oscillating field, the transition is large only when the photon energy matches the level spacing, E f − E i = ± ℏ ω E_f - E_i = \pm\hbar\omega E f − E i = ± ℏ ω
Selection rules: The transition probability is proportional to ∣ μ f i ∣ 2 |\mu_{fi}|^2 ∣ μ f i ∣ 2 Δ l = ± 1 \Delta l = \pm 1 Δ l = ± 1 Δ m l = 0 , ± 1 \Delta m_l = 0, \pm 1 Δ m l = 0 , ± 1
Setup: stationary states and a time-dependent perturbation ¶ We consider a Hamiltonian split into an exactly solvable part and a weak, time-dependent perturbation:
H ^ ( t ) = H ^ 0 + V ^ ( t ) \hat{H}(t) = \hat{H}_0 + \hat{V}(t) H ^ ( t ) = H ^ 0 + V ^ ( t ) The unperturbed Hamiltonian H ^ 0 \hat{H}_0 H ^ 0
H ^ 0 ∣ n ⟩ = E n ∣ n ⟩ \hat{H}_0 |n\rangle = E_n |n\rangle H ^ 0 ∣ n ⟩ = E n ∣ n ⟩ We expand the full time-dependent state in the stationary basis:
∣ Ψ ( t ) ⟩ = ∑ n c n ( t ) e − i E n t / ℏ ∣ n ⟩ |\Psi(t)\rangle = \sum_n c_n(t)\, e^{-iE_n t/\hbar} |n\rangle ∣Ψ ( t )⟩ = n ∑ c n ( t ) e − i E n t /ℏ ∣ n ⟩ The time-dependent coefficients c n ( t ) c_n(t) c n ( t ) V ^ ( t ) \hat{V}(t) V ^ ( t )
Deriving the first-order transition amplitude ¶ Insert the expansion into the time-dependent Schrodinger equation:
i ℏ ∂ ∂ t ∣ Ψ ( t ) ⟩ = H ^ ( t ) ∣ Ψ ( t ) ⟩ i\hbar \frac{\partial}{\partial t}|\Psi(t)\rangle = \hat{H}(t)|\Psi(t)\rangle i ℏ ∂ t ∂ ∣Ψ ( t )⟩ = H ^ ( t ) ∣Ψ ( t )⟩ After some algebra, using orthonormality ⟨ m ∣ n ⟩ = δ m n \langle m|n\rangle = \delta_{mn} ⟨ m ∣ n ⟩ = δ mn
i ℏ c ˙ m ( t ) = ∑ n c n ( t ) V m n ( t ) e i ω m n t i\hbar\, \dot{c}_m(t) = \sum_n c_n(t)\, V_{mn}(t)\, e^{i\omega_{mn} t} i ℏ c ˙ m ( t ) = n ∑ c n ( t ) V mn ( t ) e i ω mn t where
V m n ( t ) = ⟨ m ∣ V ^ ( t ) ∣ n ⟩ , ω m n = E m − E n ℏ V_{mn}(t) = \langle m|\hat V(t)|n\rangle, \qquad \omega_{mn} = \frac{E_m - E_n}{\hbar} V mn ( t ) = ⟨ m ∣ V ^ ( t ) ∣ n ⟩ , ω mn = ℏ E m − E n Assume the system starts in state ∣ i ⟩ |i\rangle ∣ i ⟩ t = 0 t=0 t = 0
c n ( 0 ) = δ n i , c i ( 0 ) = 1 , c f ≠ i ( 0 ) = 0 c_n(0) = \delta_{ni}, \quad c_i(0) = 1, \quad c_{f\neq i}(0) = 0 c n ( 0 ) = δ ni , c i ( 0 ) = 1 , c f = i ( 0 ) = 0 In first-order perturbation theory we approximate c n ( t ) ≈ δ n i c_n(t) \approx \delta_{ni} c n ( t ) ≈ δ ni c f ( t ) c_f(t) c f ( t )
c ˙ f ( 1 ) ( t ) = − i ℏ V f i ( t ) e i ω f i t \dot{c}_f^{(1)}(t) = -\frac{i}{\hbar} V_{fi}(t)\, e^{i\omega_{fi} t} c ˙ f ( 1 ) ( t ) = − ℏ i V f i ( t ) e i ω f i t Integrating from 0 to t t t
c f ( 1 ) ( t ) = − i ℏ ∫ 0 t V f i ( t ′ ) e i ω f i t ′ d t ′ c_f^{(1)}(t) = -\frac{i}{\hbar} \int_0^t V_{fi}(t')\, e^{i\omega_{fi} t'} dt' c f ( 1 ) ( t ) = − ℏ i ∫ 0 t V f i ( t ′ ) e i ω f i t ′ d t ′ This is the first-order transition amplitude from state ∣ i ⟩ |i\rangle ∣ i ⟩ ∣ f ⟩ |f\rangle ∣ f ⟩
Coupling to light: the dipole approximation ¶ For an atom or molecule in a classical electromagnetic field, the electric dipole approximation gives the perturbation
V ^ ( t ) = − μ ^ ⋅ E ( t ) \hat{V}(t) = -\hat{\boldsymbol{\mu}}\cdot \mathbf{E}(t) V ^ ( t ) = − μ ^ ⋅ E ( t ) For a monochromatic, linearly polarized field
E ( t ) = E 0 cos ( ω t ) \mathbf{E}(t) = \mathbf{E}_0 \cos(\omega t) E ( t ) = E 0 cos ( ω t ) the matrix element is
V f i ( t ) = ⟨ f ∣ V ^ ( t ) ∣ i ⟩ = − ⟨ f ∣ μ ^ ⋅ E 0 ∣ i ⟩ cos ( ω t ) V_{fi}(t) = \langle f|\hat{V}(t)|i\rangle = -\langle f|\hat{\boldsymbol{\mu}}\cdot \mathbf{E}_0|i\rangle \cos(\omega t) V f i ( t ) = ⟨ f ∣ V ^ ( t ) ∣ i ⟩ = − ⟨ f ∣ μ ^ ⋅ E 0 ∣ i ⟩ cos ( ω t ) Define the (possibly complex) dipole matrix element
μ f i = ⟨ f ∣ μ ^ ⋅ e ^ ∣ i ⟩ \mu_{fi} = \langle f|\hat{\boldsymbol{\mu}}\cdot \hat{\mathbf{e}}|i\rangle μ f i = ⟨ f ∣ μ ^ ⋅ e ^ ∣ i ⟩ where e ^ \hat{\mathbf{e}} e ^ E 0 = ∣ E 0 ∣ E_0 = |\mathbf{E}_0| E 0 = ∣ E 0 ∣
V f i ( t ) = − μ f i E 0 cos ( ω t ) V_{fi}(t) = -\mu_{fi} E_0 \cos(\omega t) V f i ( t ) = − μ f i E 0 cos ( ω t ) Plugging into the transition amplitude:
c f ( 1 ) ( t ) = i E 0 ℏ μ f i ∫ 0 t cos ( ω t ′ ) e i ω f i t ′ d t ′ c_f^{(1)}(t) = \frac{i E_0}{\hbar} \mu_{fi} \int_0^t \cos(\omega t')\, e^{i\omega_{fi} t'} dt' c f ( 1 ) ( t ) = ℏ i E 0 μ f i ∫ 0 t cos ( ω t ′ ) e i ω f i t ′ d t ′ Resonance condition and energy conservation ¶ Use the exponential form of the cosine:
cos ( ω t ′ ) = 1 2 ( e i ω t ′ + e − i ω t ′ ) \cos(\omega t') = \frac{1}{2}\left(e^{i\omega t'} + e^{-i\omega t'}\right) cos ( ω t ′ ) = 2 1 ( e iω t ′ + e − iω t ′ ) Then the integral in c f ( 1 ) ( t ) c_f^{(1)}(t) c f ( 1 ) ( t )
∫ 0 t e i ( ω f i ± ω ) t ′ d t ′ \int_0^t e^{i(\omega_{fi} \pm \omega)t'} dt' ∫ 0 t e i ( ω f i ± ω ) t ′ d t ′ If ω f i ± ω \omega_{fi} \pm \omega ω f i ± ω large transition amplitude occurs when the exponent is nearly stationary:
ω f i − ω ≈ 0 ⇒ ω f i ≈ ω \omega_{fi} - \omega \approx 0 \quad \Rightarrow \quad \omega_{fi} \approx \omega ω f i − ω ≈ 0 ⇒ ω f i ≈ ω This gives the familiar energy-conservation condition for absorption:
E f − E i = ℏ ω E_f - E_i = \hbar\omega E f − E i = ℏ ω Similarly, for stimulated emission one finds E f − E i = − ℏ ω E_f - E_i = -\hbar\omega E f − E i = − ℏ ω
From transition amplitudes to selection rules ¶ 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, and so on) of the wavefunctions ∣ i ⟩ |i\rangle ∣ i ⟩ ∣ f ⟩ |f\rangle ∣ f ⟩
the transformation properties of the operator μ ^ ∼ r \hat{\boldsymbol{\mu}} \sim \mathbf{r} μ ^ ∼ r
Electric dipole selection rules for hydrogen-like orbitals ¶ For an electron in a central potential (for example, a hydrogenic atom), the stationary states are labeled by ∣ n , l , m l ⟩ |n, l, m_l\rangle ∣ n , l , m l ⟩
The position operator r \mathbf{r} r Y 1 m Y_{1m} Y 1 m
Orbital angular momentum selection rule
Δ l = l f − l i = ± 1 \Delta l = l_f - l_i = \pm 1 Δ l = l f − l i = ± 1 Magnetic quantum number selection rule
Δ m l = m l , f − m l , i = 0 , ± 1 \Delta m_l = m_{l,f} - m_{l,i} = 0, \pm 1 Δ m l = m l , f − m l , i = 0 , ± 1 Start from the time-dependent Schrodinger equation with H ^ ( t ) = H ^ 0 + V ^ ( t ) \hat{H}(t) = \hat{H}_0 + \hat{V}(t) H ^ ( t ) = H ^ 0 + V ^ ( t )
Expand the state in the eigenbasis of H ^ 0 \hat{H}_0 H ^ 0 c n ( t ) c_n(t) c n ( t )
First-order perturbation theory gives a transition amplitude
c f ( 1 ) ( t ) = − i ℏ ∫ 0 t ⟨ f ∣ V ^ ( t ′ ) ∣ i ⟩ e i ω f i t ′ d t ′ c_f^{(1)}(t) = -\frac{i}{\hbar} \int_0^t \langle f|\hat{V}(t')|i\rangle\, e^{i\omega_{fi} t'} dt' c f ( 1 ) ( t ) = − ℏ i ∫ 0 t ⟨ f ∣ V ^ ( t ′ ) ∣ i ⟩ e i ω f i t ′ d t ′ For an oscillating field, the integral is large only when E f − E i = ± ℏ ω E_f - E_i = \pm \hbar\omega E f − E i = ± ℏ ω
The transition probability is proportional to ∣ ⟨ f ∣ μ ^ ∣ i ⟩ ∣ 2 |\langle f|\hat{\boldsymbol{\mu}}|i\rangle|^2 ∣ ⟨ f ∣ μ ^ ∣ i ⟩ ∣ 2
If the dipole matrix element is zero by symmetry, the transition is forbidden.
For electric-dipole transitions in atoms: Δ l = ± 1 \Delta l = \pm 1 Δ l = ± 1 Δ m l = 0 , ± 1 \Delta m_l = 0, \pm 1 Δ m l = 0 , ± 1