7.1 Variational principle and its applications

The variational principle is a powerful tool in quantum mechanics for estimating ground state energies. It states that any trial wave function will yield an energy expectation value higher than or equal to the true ground state energy, providing a systematic way to approximate solutions.

This method involves choosing a trial wave function with adjustable parameters and minimizing the energy expectation value. While it's computationally efficient and can be systematically improved, its accuracy depends on the choice of trial function and it's limited to ground state calculations.

Variational Principle in Quantum Mechanics

Basic Concept and Formulation

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  Variational Quantum Singular Value Decomposition – Quantum View original

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• The variational principle states that the expectation value of the Hamiltonian operator, calculated using any trial wave function, is always greater than or equal to the true ground state energy of the system
• The variational method involves choosing a trial wave function with adjustable parameters and minimizing the expectation value of the Hamiltonian with respect to these parameters to obtain an upper bound on the ground state energy
• The trial wave function should satisfy the boundary conditions and symmetry properties of the quantum system under consideration
• The variational principle is based on the Rayleigh-Ritz method, which states that the expectation value of the Hamiltonian is stationary with respect to small variations in the trial wave function near the true ground state
• The variational principle can be mathematically expressed as $⟨ψ|H|ψ⟩ ≥ E₀$, where $|ψ⟩$ is any normalized trial wave function, $H$ is the Hamiltonian operator, and $E₀$ is the true ground state energy

Ground State Energy Estimation

Variational Method Application

• The variational method involves choosing a suitable trial wave function with adjustable parameters, such as a linear combination of basis functions or a function with variational parameters
• The expectation value of the Hamiltonian is calculated using the trial wave function, resulting in an expression that depends on the adjustable parameters
• The optimal values of the parameters are determined by minimizing the expectation value of the Hamiltonian, which can be done using analytical or numerical methods, such as setting the derivatives of the expectation value with respect to the parameters equal to zero
• The minimized expectation value of the Hamiltonian provides an upper bound on the ground state energy of the quantum system

Examples and Accuracy

• Examples of quantum systems where the variational principle can be applied include the harmonic oscillator, the hydrogen atom, and the helium atom
• The accuracy of the estimated ground state energy depends on the choice of the trial wave function and the number of adjustable parameters used
  • A well-chosen trial wave function that closely resembles the true ground state wave function will yield a more accurate estimate of the ground state energy
  • Increasing the number of adjustable parameters in the trial wave function can improve the accuracy of the variational method, as it allows for a more flexible representation of the wave function
• The variational method can be systematically improved by using more sophisticated forms of the trial wave function, such as correlated wave functions or Jastrow factors, which account for the correlation effects between particles

Variational Method Advantages and Limitations

Advantages

• The variational method is computationally less demanding compared to other methods, such as the perturbation theory or the exact diagonalization of the Hamiltonian matrix, making it suitable for systems with a large number of particles or degrees of freedom
• The variational method can be systematically improved by increasing the number of adjustable parameters in the trial wave function or by using more sophisticated forms of the trial wave function, such as correlated wave functions or Jastrow factors
• The variational method provides a rigorous upper bound on the ground state energy, which can be used as a benchmark for other approximate methods or as a starting point for more accurate calculations

Limitations

• The variational method provides an upper bound on the ground state energy, but it does not guarantee the exact ground state energy unless the trial wave function coincides with the true ground state wave function
• The accuracy of the variational method depends on the choice of the trial wave function, and a poorly chosen trial wave function may lead to a significant error in the estimated ground state energy
  • If the trial wave function does not capture the essential features of the true ground state wave function, such as the correct symmetry or the presence of nodes, the variational method may yield a poor estimate of the ground state energy
• The variational method is not suitable for estimating excited state energies, as it only provides an upper bound on the ground state energy. Other methods, such as the linear variational method or the diffusion Monte Carlo method, can be used to estimate excited state energies

Trial Wave Functions and Optimization

Constructing Trial Wave Functions

• The choice of the trial wave function depends on the physical properties and symmetries of the quantum system under consideration, such as the number of particles, the potential energy, and the boundary conditions
• For simple systems, such as the infinite square well or the harmonic oscillator, the trial wave function can be chosen as a linear combination of the eigenfunctions of the unperturbed Hamiltonian, with the coefficients serving as variational parameters
• For more complex systems, such as atoms or molecules, the trial wave function can be constructed using Slater determinants, which ensure the antisymmetry of the wave function for fermions, and Jastrow factors, which account for the correlation effects between particles
• The trial wave function should satisfy the boundary conditions and the normalization condition, which can be imposed by using appropriate basis functions or by including normalization factors in the trial wave function

Optimization and Quality Assessment

• The variational parameters in the trial wave function are optimized by minimizing the expectation value of the Hamiltonian, which can be done using analytical methods, such as solving the Euler-Lagrange equations, or numerical methods, such as the steepest descent or the conjugate gradient method
  • Analytical methods involve deriving the equations that the variational parameters must satisfy to minimize the expectation value of the Hamiltonian and solving these equations to obtain the optimal values of the parameters
  • Numerical methods involve iteratively updating the values of the variational parameters based on the gradient of the expectation value of the Hamiltonian with respect to these parameters until convergence is achieved
• The quality of the optimized trial wave function can be assessed by comparing the estimated ground state energy with the exact value (if known) or with the results obtained using other methods, such as the perturbation theory or the quantum Monte Carlo method
  • If the estimated ground state energy is close to the exact value or agrees well with the results obtained using other reliable methods, it indicates that the optimized trial wave function is a good approximation to the true ground state wave function
  • If there is a significant discrepancy between the estimated ground state energy and the reference values, it suggests that the trial wave function may need to be improved by using a more flexible parametrization or by including additional physical effects