4.5 Perturbation Theory and Variational Principle
4 min read•august 14, 2024
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
Top images from around the web for Perturbation Theory Basics
Frontiers | Perturbation Theory for Time-Dependent Quantum Systems Involving Complex Potentials View original
Is this image relevant?
Hamiltonian Simulation by Qubitization – Quantum View original
Is this image relevant?
Amplification of quadratic Hamiltonians – Quantum View original
Is this image relevant?
Frontiers | Perturbation Theory for Time-Dependent Quantum Systems Involving Complex Potentials View original
Is this image relevant?
Hamiltonian Simulation by Qubitization – Quantum View original
Is this image relevant?
1 of 3
Top images from around the web for Perturbation Theory Basics
Frontiers | Perturbation Theory for Time-Dependent Quantum Systems Involving Complex Potentials View original
Is this image relevant?
Hamiltonian Simulation by Qubitization – Quantum View original
Is this image relevant?
Amplification of quadratic Hamiltonians – Quantum View original
Is this image relevant?
Frontiers | Perturbation Theory for Time-Dependent Quantum Systems Involving Complex Potentials View original
Is this image relevant?
Hamiltonian Simulation by Qubitization – Quantum View original
Is this image relevant?
1 of 3
- 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 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 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
Key Terms to Review (18)
Degenerate perturbation theory: Degenerate perturbation theory is a method in quantum mechanics used to analyze systems with multiple states having the same energy, known as degenerate states, when subjected to a small perturbation. This approach focuses on how these degenerate states interact with each other under the influence of a slight change in the Hamiltonian, allowing for the calculation of energy shifts and new eigenstates as the system evolves.
Eigenvalue problem: The eigenvalue problem is a mathematical formulation that involves finding eigenvalues and eigenvectors of an operator or a matrix. In the context of quantum mechanics and physical chemistry, this problem is fundamental for understanding the behavior of quantum systems, particularly when applying methods like perturbation theory and the variational principle to approximate solutions to complex systems.
Energy eigenstate: An energy eigenstate is a specific quantum state of a system that corresponds to a definite energy value, described mathematically by an eigenfunction of the Hamiltonian operator. In quantum mechanics, these states are crucial because they represent the possible states that a particle can occupy, and each eigenstate is associated with a particular energy level that the system can have when it is not influenced by external forces.
Energy expectation value: The energy expectation value is a statistical measure that represents the average energy of a quantum system in a given state, calculated using the wave function of that state. This value is crucial in quantum mechanics as it provides insights into the energy distribution of particles and helps predict the behavior of systems under perturbations. Understanding the energy expectation value is particularly important when applying perturbation theory and the variational principle, as it allows for the comparison of different states and the assessment of system stability.
First-order perturbation: First-order perturbation refers to a mathematical approach in quantum mechanics used to approximate the effects of a small perturbation on a system's energy levels and wave functions. This method is particularly useful when the changes in the system are minor, allowing for the calculation of corrections to the energies and states of a quantum system based on its unperturbed state.
Non-degenerate perturbation theory: Non-degenerate perturbation theory is a mathematical approach used to approximate the eigenvalues and eigenstates of a quantum mechanical system when a small perturbation is applied to a Hamiltonian that has distinct energy levels. This theory assumes that the unperturbed states do not share the same energy, allowing for a straightforward calculation of corrections to both energy levels and wavefunctions due to the perturbation.
Paul Dirac: Paul Dirac was a pioneering physicist known for his contributions to quantum mechanics and quantum field theory, particularly through the formulation of the Dirac equation. His work laid the groundwork for understanding the behavior of fermions and the principles of particle physics, connecting deeply with concepts like perturbation theory and variational principles in modern physics.
Perturbative vs. Non-Perturbative Methods: Perturbative methods are techniques used to find an approximate solution to a problem by starting from a known solution and adding small corrections, while non-perturbative methods seek solutions that cannot be achieved by small perturbations, often dealing with more complex systems. In the context of quantum mechanics and physical chemistry, perturbative methods are often easier to implement and can provide quick insights, while non-perturbative methods address scenarios where perturbation theories break down, such as strong coupling situations.
Perturbed energy levels: Perturbed energy levels refer to the adjusted energy states of a quantum system when it is subjected to small external influences or perturbations. These changes in energy levels are essential in understanding how systems behave under non-ideal conditions, linking them to various quantum mechanical principles and calculations.
Perturbing hamiltonian: A perturbing hamiltonian is a modification of the original Hamiltonian operator that accounts for small disturbances or changes in a quantum system. This concept is essential in perturbation theory, where it allows us to analyze how these small changes affect the energy levels and wavefunctions of the system, helping to approximate solutions for systems that cannot be solved exactly.
Rayleigh-Ritz Variational Principle: The Rayleigh-Ritz Variational Principle is a method used in quantum mechanics and physical chemistry to estimate the ground state energy of a quantum system. It involves approximating the wave function of the system by a linear combination of basis functions, allowing one to compute an upper bound for the energy, which can be particularly useful in perturbation theory and variational methods.
Second-order perturbation: Second-order perturbation refers to a method in quantum mechanics used to approximate the effects of a small perturbation on a quantum system's energy levels and wave functions. This approach allows for corrections to the first-order approximation, providing a more accurate representation of how a system behaves when it is subjected to small changes in potential energy, especially in the context of more complex systems or interactions.
Small parameter approximation: The small parameter approximation is a mathematical technique used in perturbation theory where a small parameter is introduced to simplify complex problems by expanding a solution in terms of that parameter. This approach is particularly useful when dealing with systems where the effects of small perturbations can be treated incrementally, leading to approximate solutions that are easier to handle. In the context of physical systems, this method allows for the analysis of the impact of slight changes on energy levels and wave functions.
Stability criteria: Stability criteria refer to the conditions that determine whether a system, particularly in quantum mechanics and physical chemistry, will remain in a stable state or return to equilibrium after a disturbance. These criteria help in understanding the behavior of systems under perturbations, ensuring that perturbation methods and variational principles yield reliable predictions about energy states and their properties.
Trial wave function: A trial wave function is an assumed mathematical representation of a quantum state used in variational methods to approximate the true wave function of a system. It serves as a starting point to evaluate the energy of a quantum system and helps in finding the ground state energy by minimizing the expectation value of the Hamiltonian with respect to this function. This approach is key in perturbation theory and the variational principle, as it allows for approximations when dealing with complex systems.
Variational Method: The variational method is a mathematical technique used to approximate the ground state energy of quantum systems by optimizing a trial wave function. It relies on the variational principle, which states that for any trial wave function, the expectation value of the Hamiltonian will always be greater than or equal to the true ground state energy. This method is crucial for solving complex quantum problems where exact solutions are not feasible.
Variational vs. Exact Methods: Variational methods are mathematical techniques that provide approximate solutions to complex problems by optimizing a functional, while exact methods yield precise and often closed-form solutions. These approaches are crucial in understanding how systems behave under different conditions, with variational methods being particularly useful in quantum mechanics and perturbation theory, as they help estimate ground state energies and wave functions.
Walter Heitler: Walter Heitler was a prominent theoretical physicist known for his significant contributions to quantum mechanics and quantum chemistry. He is particularly recognized for developing the first rigorous treatment of electron-electron interactions in quantum chemistry, laying foundational work for perturbation theory and the variational principle.