15.3 Perturbation Theory and Approximation Methods

Perturbation theory helps solve complex quantum systems by splitting them into solvable parts and small disturbances. It calculates energy levels and states as corrections to the undisturbed system, using power series expansions based on the perturbation's strength.

Time-independent perturbation theory handles constant disturbances, while time-dependent theory deals with changing ones. Both methods find applications in real-world scenarios, like the Stark effect in hydrogen atoms or transitions between energy states in atoms and molecules.

Perturbation Theory

Concept of perturbation theory

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• Perturbation theory approximates solutions to complex quantum systems by splitting the Hamiltonian into an exactly solvable part $H_0$ and a small perturbation $H'$
  • The total Hamiltonian is expressed as $H = H_0 + H'$
  • The perturbation $H'$ is assumed to be much smaller in magnitude compared to the exactly solvable part $H_0$
• Perturbation theory calculates approximate energy levels and eigenstates as corrections to the unperturbed system
  • The corrections are expressed as power series expansions in terms of the perturbation strength $\lambda$
    • Example: $E_n = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + \ldots$
• Two main types of perturbation theory are used depending on the nature of the perturbation:
  • Time-independent perturbation theory handles time-independent perturbations $H'$
  • Time-dependent perturbation theory deals with time-dependent perturbations $H'(t)$

Time-independent perturbation theory applications

• Time-independent perturbation theory is applied when the perturbation $H'$ does not depend on time
• First-order energy corrections are calculated using the expectation value of the perturbation in the unperturbed eigenstates:
  • $E_n^{(1)} = \langle n^{(0)} | H' | n^{(0)} \rangle$, where $|n^{(0)}\rangle$ represents the unperturbed eigenstate
• First-order corrections to the eigenstates are obtained by summing over the contributions from all other unperturbed states:
  • $|n^{(1)}\rangle = \sum_{m \neq n} \frac{\langle m^{(0)} | H' | n^{(0)} \rangle}{E_n^{(0)} - E_m^{(0)}} |m^{(0)}\rangle$
• Higher-order corrections are calculated using recursive formulas that involve lower-order corrections
• Non-degenerate perturbation theory is used when the unperturbed energy levels are non-degenerate
  • Example: Stark effect in hydrogen atom, where an external electric field splits the degenerate energy levels
• Degenerate perturbation theory is employed when the unperturbed energy levels are degenerate
  • It requires diagonalizing the perturbation matrix within the degenerate subspace to lift the degeneracy
    • Example: Zeeman effect, where an external magnetic field splits the degenerate energy levels of an atom

Time-Dependent Perturbation Theory and Variational Methods

Time-dependent perturbation theory calculations

• Time-dependent perturbation theory is used when the perturbation $H'(t)$ varies with time
• The transition probability from an initial state $|i\rangle$ to a final state $|f\rangle$ is given by:
  • $P_{i \to f} = \left| \langle f | U(t) | i \rangle \right|^2$, where $U(t)$ is the time-evolution operator expanded in a perturbation series
• Fermi's Golden Rule gives the first-order transition probability:
  • $P_{i \to f}^{(1)} = \frac{2\pi}{\hbar} \left| \langle f | H' | i \rangle \right|^2 \delta(E_f - E_i - \hbar\omega)$
    • $\hbar\omega$ is the energy of the perturbation photon
    • The delta function ensures energy conservation during the transition
• Higher-order transition probabilities are calculated using more advanced techniques like the Dyson series

Variational methods for ground states

• Variational methods estimate the ground state energy and wavefunction by providing an upper bound on the true ground state energy $E_0$
  • The variational principle states that the energy expectation value of any trial wavefunction is greater than or equal to the true ground state energy: $E[\psi] \geq E_0$
• The trial wavefunction $|\psi_\text{trial}\rangle$ depends on a set of variational parameters $\{\alpha_i\}$:
  • $|\psi_\text{trial}\rangle = |\psi(\alpha_1, \alpha_2, \ldots)\rangle$
• The variational energy is calculated as the expectation value of the Hamiltonian in the trial state:
  • $E[\psi_\text{trial}] = \frac{\langle \psi_\text{trial} | H | \psi_\text{trial} \rangle}{\langle \psi_\text{trial} | \psi_\text{trial} \rangle}$
• Minimizing the variational energy with respect to the variational parameters yields an estimate of the ground state energy and wavefunction
  • The minimization condition is given by $\frac{\partial E[\psi_\text{trial}]}{\partial \alpha_i} = 0$ for all variational parameters $\alpha_i$
• Common variational methods include:
  • Rayleigh-Ritz method expands the trial wavefunction as a linear combination of basis functions
    • Example: Using hydrogen-like orbitals as basis functions for the helium atom
  • Gaussian variational method employs Gaussian functions as the trial wavefunction
    • Example: Modeling the ground state of a harmonic oscillator with a Gaussian wavefunction