5.2 Perturbation theory: time-independent and time-dependent

Perturbation theory helps solve complex quantum systems by tweaking simpler ones. It's like figuring out how a small change affects a well-understood system, giving us insights into trickier problems without solving them directly.

This method comes in two flavors: time-independent for static systems and time-dependent for dynamic ones. Both are crucial for understanding how particles behave when nudged by external forces or fields in quantum chemistry.

Perturbation Theory Fundamentals

Perturbation Hamiltonian and Zeroth-Order Approximation

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• Perturbation theory is a method for finding approximate solutions to the Schrödinger equation when the Hamiltonian can be split into two parts: $H = H^{(0)} + \lambda V$
  • $H^{(0)}$ is the unperturbed Hamiltonian, which has known eigenstates and eigenvalues
  • $\lambda V$ is the perturbation, where $\lambda$ is a small parameter and $V$ is the perturbation operator
• The zeroth-order approximation assumes that the eigenstates and eigenvalues of the perturbed system are the same as those of the unperturbed system
  • Eigenstates: $\psi_n^{(0)} = \phi_n$, where $\phi_n$ are the eigenstates of $H^{(0)}$
  • Eigenvalues: $E_n^{(0)} = \varepsilon_n$, where $\varepsilon_n$ are the eigenvalues of $H^{(0)}$
• Examples of unperturbed Hamiltonians include the particle in a box and the harmonic oscillator

First-Order and Second-Order Corrections

• The first-order correction to the energy is given by $E_n^{(1)} = \langle \phi_n | V | \phi_n \rangle$
  • This is the expectation value of the perturbation operator in the unperturbed state
  • The first-order corrected energy is $E_n \approx E_n^{(0)} + E_n^{(1)}$
• The first-order correction to the wavefunction is given by $\psi_n^{(1)} = \sum_{m \neq n} \frac{\langle \phi_m | V | \phi_n \rangle}{E_n^{(0)} - E_m^{(0)}} \phi_m$
  • This is a sum over all unperturbed states except $\phi_n$, weighted by the matrix elements of the perturbation and the energy differences
• The second-order correction to the energy is given by $E_n^{(2)} = \sum_{m \neq n} \frac{|\langle \phi_m | V | \phi_n \rangle|^2}{E_n^{(0)} - E_m^{(0)}}$
  • This involves a sum over all unperturbed states except $\phi_n$, with each term being the square of the perturbation matrix element divided by the energy difference
• Examples of perturbations include an electric field applied to a hydrogen atom (Stark effect) or a magnetic field applied to a spin system (Zeeman effect)

Advanced Perturbation Techniques

Degenerate Perturbation Theory

• Degenerate perturbation theory is used when the unperturbed system has degenerate energy levels (multiple states with the same energy)
• In this case, the perturbation can lift the degeneracy, and the first-order correction to the energy is found by diagonalizing the perturbation matrix within the degenerate subspace
• The eigenstates of the perturbed system are linear combinations of the degenerate unperturbed states
• Examples of degenerate systems include the $2p$ orbitals of hydrogen or the $d$ orbitals in a cubic crystal field

Time-Dependent Perturbation Theory

• Time-dependent perturbation theory is used when the perturbation is time-dependent, such as an oscillating electric field
• The time-dependent Schrödinger equation is $i\hbar \frac{\partial \psi(t)}{\partial t} = [H^{(0)} + V(t)] \psi(t)$
• The wavefunction can be expanded in the basis of unperturbed states: $\psi(t) = \sum_n c_n(t) \phi_n e^{-i E_n^{(0)} t / \hbar}$
  • The coefficients $c_n(t)$ are time-dependent and satisfy a set of coupled differential equations
• Examples of time-dependent perturbations include the interaction of an atom with a laser field or the absorption of a photon by a molecule

Time-Dependent Perturbation Theory

Interaction Picture and Fermi's Golden Rule

• The interaction picture is a useful formalism for time-dependent perturbation theory
  • The wavefunction is transformed as $\psi_I(t) = e^{i H^{(0)} t / \hbar} \psi(t)$
  • The Schrödinger equation in the interaction picture is $i\hbar \frac{\partial \psi_I(t)}{\partial t} = V_I(t) \psi_I(t)$, where $V_I(t) = e^{i H^{(0)} t / \hbar} V(t) e^{-i H^{(0)} t / \hbar}$
• Fermi's golden rule gives the transition rate from an initial state $i$ to a final state $f$ under a perturbation $V$: $\Gamma_{i \to f} = \frac{2\pi}{\hbar} |\langle f | V | i \rangle|^2 \rho(E_f)$
  • $\rho(E_f)$ is the density of states at the final energy $E_f$
  • This rule is valid for weak perturbations and long times (first-order time-dependent perturbation theory)
• Examples of processes described by Fermi's golden rule include spontaneous emission, photoelectric effect, and Raman scattering