4.5 Perturbation Theory and Variational Principle

Quantum mechanics can get tricky, but perturbation theory and the variational principle are here to help. These methods let us tackle complex problems by starting with simpler, solvable systems and making clever approximations.

We'll learn how to use perturbation theory to find energy corrections and improved wave functions. We'll also explore the variational principle for estimating ground state energies. These tools are super useful for real-world quantum systems.

Perturbation Theory for Energy Corrections

Perturbation Theory Basics

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• Perturbation theory finds approximate solutions to quantum mechanical problems by treating the system as a small perturbation from a known, exactly solvable system
• The Hamiltonian of the perturbed system is the sum of the unperturbed Hamiltonian (H₀) and the perturbation (H'): H = H₀ + λH', where λ is a small parameter
• The energy and wave function of the perturbed system are expanded in a power series of λ:
  • E = E⁽⁰⁾ + λE⁽¹⁾ + λ²E⁽²⁾ + ...
  • ψ = ψ⁽⁰⁾ + λψ⁽¹⁾ + λ²ψ⁽²⁾ + ...
• The zeroth-order terms (E⁽⁰⁾ and ψ⁽⁰⁾) correspond to the unperturbed system, while higher-order terms represent corrections due to the perturbation

Energy Corrections and Wave Function Corrections

• The first-order energy correction is given by E⁽¹⁾ = ⟨ψ⁽⁰⁾|H'|ψ⁽⁰⁾⟩
  • This correction represents the expectation value of the perturbation Hamiltonian with respect to the unperturbed wave function
• The second-order energy correction is E⁽²⁾ = Σ_{n≠m} |⟨ψ_n⁽⁰⁾|H'|ψ_m⁽⁰⁾⟩|² / (E_m⁽⁰⁾ - E_n⁽⁰⁾), where m and n are the unperturbed states
  • This correction accounts for the mixing of unperturbed states due to the perturbation
  • The summation is over all unperturbed states except the state of interest (n≠m)
• The first-order correction to the wave function is ψ⁽¹⁾ = Σ_{n≠m} (⟨ψ_n⁽⁰⁾|H'|ψ_m⁽⁰⁾⟩ / (E_m⁽⁰⁾ - E_n⁽⁰⁾)) ψ_n⁽⁰⁾
  • This correction represents the admixture of other unperturbed states into the state of interest due to the perturbation
  • The summation is over all unperturbed states except the state of interest (n≠m)

Time-Independent vs Time-Dependent Perturbation Theory

Time-Independent Perturbation Theory

• Time-independent perturbation theory is used when the perturbation is constant in time
• The goal is to find the energy corrections and perturbed wave functions for stationary states
• Examples of time-independent perturbations include:
  • Electric fields in the Stark effect
  • Magnetic fields in the Zeeman effect

Time-Dependent Perturbation Theory

• Time-dependent perturbation theory is used when the perturbation varies with time
• It studies the transition probabilities between states and the time evolution of the system under the influence of a time-varying perturbation
• The interaction picture is often used, where the time dependence of the perturbation is separated from the time evolution of the unperturbed system
• Fermi's Golden Rule, derived from time-dependent perturbation theory, gives the transition rate between states due to a perturbation: Γ_{i→f} = (2π/ħ) |⟨ψ_f|H'|ψ_i⟩|² ρ(E_f), where ρ(E_f) is the density of final states
• Examples of time-dependent perturbations include:
  • Electromagnetic radiation in spectroscopy
  • Oscillating electric or magnetic fields

Ground State Energy Estimation with the Variational Principle

The Variational Principle

• The variational principle states that the expectation value of the Hamiltonian calculated using any trial wave function will always be greater than or equal to the true ground state energy
• This principle provides a way to estimate the ground state energy of a system by minimizing the expectation value of the Hamiltonian with respect to adjustable parameters in a trial wave function
• The expectation value of the Hamiltonian is given by ⟨H⟩ = ⟨ψ_trial|H|ψ_trial⟩ / ⟨ψ_trial|ψ_trial⟩

Variational Method

• The variational method involves choosing a trial wave function ψ_trial with adjustable parameters and minimizing the expectation value of the Hamiltonian with respect to those parameters
• The minimization of ⟨H⟩ is typically done by setting the derivative of ⟨H⟩ with respect to each adjustable parameter equal to zero and solving the resulting equations
• The accuracy of the variational method depends on the choice of the trial wave function
  • A well-chosen trial wave function that captures the essential physics of the system will yield a better estimate of the ground state energy
• Examples of systems where the variational method is applied include:
  • Helium atom
  • Hydrogen molecule

Trial Wave Functions and Variational Optimization

Constructing Trial Wave Functions

• Trial wave functions are approximate wave functions with adjustable parameters used in the variational method to estimate the ground state energy and wave function of a system
• The choice of the trial wave function depends on the system being studied and should incorporate the known physical properties and symmetries of the system
• Common types of trial wave functions include:
  • Linear combinations of basis functions, such as Gaussian or Slater-type orbitals
  • Correlated wave functions, such as Jastrow factors

Optimizing Trial Wave Functions

• The adjustable parameters in the trial wave function can be optimized by minimizing the expectation value of the Hamiltonian
• Optimization techniques include:
  • The Ritz method
  • Stochastic optimization methods like variational Monte Carlo
• The quality of the optimized trial wave function can be assessed by comparing the calculated properties, such as the energy and electron density, with experimental data or more accurate computational methods
• The optimized trial wave function can be used as a starting point for more sophisticated methods to obtain more accurate results
  • Examples of such methods include diffusion Monte Carlo or coupled cluster theory