Glossary

11.3 Variational method and Hellmann-Feynman theorem

2 min read•july 25, 2024

The variational method is a powerful tool in quantum mechanics for approximating ground state energies. It's based on the principle that any 's energy is always higher than the true , providing an upper bound.

This method shines when dealing with complex systems where exact solutions are hard to find. By optimizing trial wavefunctions and using techniques like the , we can tackle challenging problems in molecular systems and materials science.

Variational Method

Variational principle for ground states

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  • fundamentals underpin approximation of ground state energies stating expectation value of Hamiltonian for any trial wavefunction always greater than or equal to true ground state energy
  • Mathematically expressed as ⟨ψtrial∣H∣ψtrial⟩≥E0\langle \psi_{\text{trial}}|H|\psi_{\text{trial}}\rangle \geq E_0 providing upper bound for true ground state energy
  • Energy expectation value calculated using E[ψtrial]=⟨ψtrial∣H∣ψtrial⟩⟨ψtrial∣ψtrial⟩E[\psi_{\text{trial}}] = \frac{\langle \psi_{\text{trial}}|H|\psi_{\text{trial}}\rangle}{\langle \psi_{\text{trial}}|\psi_{\text{trial}}\rangle}
  • Crucial in quantum mechanics for systems where exact solutions challenging (many-body problems, complex molecules)

Trial wavefunction optimization

  • Trial wavefunction construction involves choosing functional form with adjustable parameters satisfying boundary conditions and symmetry requirements
  • Optimization process minimizes energy expectation value with respect to variational parameters using analytical or numerical methods (gradient descent, Newton-Raphson)
  • Common trial wavefunction types include linear combination of basis functions (Gaussian orbitals), product of single-particle wavefunctions (Hartree-Fock), and Jastrow factors for electron correlation
  • Iterative improvement systematically refines trial wavefunction based on optimization results enhancing accuracy of ground state energy approximation

Hellmann-Feynman theorem applications

  • Theorem states dEλdλ=⟨ψλ∣∂Hλ∂λ∣ψλ⟩\frac{d E_\lambda}{d \lambda} = \left\langle \psi_\lambda \left| \frac{\partial H_\lambda}{\partial \lambda} \right| \psi_\lambda \right\rangle enabling calculation of expectation values and forces in quantum systems
  • Applications include:
    1. Calculation of forces in molecular systems (bond lengths, angles)
    2. Evaluation of electric dipole moments (polarizability)
  • Connects to perturbation theory providing method for computing first-order energy corrections
  • Implementation steps:
    1. Identify parameter of interest (λ\lambda)
    2. Determine Hamiltonian dependence on λ\lambda
    3. Calculate expectation value of ∂Hλ∂λ\frac{\partial H_\lambda}{\partial \lambda}

Variational method: pros and cons

  • Advantages include providing upper bounds for ground state energies, applicability to complex systems (proteins, nanostructures), and systematic improvement capability
  • Limitations involve inaccurate excited state energies, sensitivity to trial wavefunction choice, and slow convergence for strongly correlated systems (transition metals)
  • Compares favorably to perturbation theory as non-perturbative method and more scalable than exact diagonalization for large systems
  • Complementary techniques involve combining with Monte Carlo sampling (variational Monte Carlo) and using results as starting points for other methods (coupled cluster)

Key Terms to Review (16)

Approximation of wave functions: The approximation of wave functions involves creating simplified representations of quantum states that can be used to solve complex problems in quantum mechanics. This concept is crucial in the variational method and the Hellmann-Feynman theorem, as it allows physicists to find reasonable estimates for energy levels and other properties of quantum systems without needing exact solutions.
David Bohm: David Bohm was a theoretical physicist known for his significant contributions to quantum mechanics and his philosophical interpretations of the theory. His work, particularly on the causal interpretation of quantum mechanics, challenges traditional views and connects deeply with concepts like nonlocality and the role of the observer in quantum systems.
Eigenvalue Problem: The eigenvalue problem is a mathematical challenge where one seeks to find scalar values, known as eigenvalues, and corresponding vectors, known as eigenvectors, that satisfy the equation $$A\mathbf{v} = \lambda \mathbf{v}$$ for a linear operator or matrix A. This concept is fundamental in various areas such as quantum mechanics and linear algebra, helping to describe the behavior of physical systems and facilitating the process of solving differential equations.
Energy Minimization: Energy minimization refers to the process of finding the lowest possible energy configuration of a system, which is often associated with stability and equilibrium. This concept is central to various fields, including physics and chemistry, as systems tend to evolve towards states of lower energy. In quantum mechanics, this principle is key to understanding how physical systems behave and how approximations can lead to effective solutions for complex problems.
Expectation Value: The expectation value is a fundamental concept in quantum mechanics that represents the average outcome of a measurement for a given observable in a quantum state. It connects to the mathematical framework through linear operators and is central to understanding how measurements affect the state of a system over time, as well as how different states relate to energy and spin.
Functional Derivatives: Functional derivatives are a generalization of ordinary derivatives that apply to functionals, which are mappings from a space of functions to the real numbers. They help us understand how a functional changes when the function it depends on is varied slightly. This concept is crucial in many areas such as variational principles, mechanics, and quantum mechanics, allowing us to derive equations of motion and optimize functionals.
Ground state energy: Ground state energy is the lowest energy level that a quantum mechanical system can occupy, representing the most stable state of the system. In this state, all quantum numbers are at their minimum values, and it serves as a reference point for measuring the energies of excited states. Understanding ground state energy is crucial in applying variational methods and the Hellmann-Feynman theorem, which are tools used to calculate and approximate the properties of quantum systems.
Hellmann-Feynman Theorem: The Hellmann-Feynman Theorem states that the derivative of the energy eigenvalue of a quantum system with respect to a parameter is equal to the expectation value of the derivative of the Hamiltonian with respect to that parameter in the corresponding eigenstate. This theorem connects variational principles to quantum mechanics by providing a way to calculate energy changes when a system's parameters are varied.
Hermitian operator: A Hermitian operator is a linear operator on a Hilbert space that is equal to its own adjoint, meaning that it satisfies the condition \( A = A^\dagger \). This property ensures that the eigenvalues of the operator are real, making Hermitian operators vital in the context of observables in quantum mechanics, where they correspond to measurable physical quantities. Their spectral properties also play a crucial role in understanding the structure of quantum systems.
Lagrange multipliers: Lagrange multipliers are a mathematical tool used to find the local maxima and minima of a function subject to equality constraints. This technique helps in optimizing a function while adhering to specific conditions, making it essential in various fields such as physics, engineering, and economics. By introducing additional variables, known as multipliers, the method transforms a constrained problem into an unconstrained one, enabling the application of traditional optimization techniques.
Rayleigh-Ritz Variational Method: The Rayleigh-Ritz variational method is a mathematical technique used to approximate the eigenvalues and eigenfunctions of differential operators, particularly in quantum mechanics. By selecting a trial wave function that depends on adjustable parameters, this method minimizes the expectation value of the Hamiltonian to estimate energy levels, providing insights into the behavior of quantum systems.
Richard Feynman: Richard Feynman was a renowned American theoretical physicist known for his contributions to quantum mechanics and quantum electrodynamics. His innovative ideas, such as the path integral formulation, revolutionized our understanding of particle physics and made complex concepts more accessible through intuitive visuals like Feynman diagrams.
Self-adjoint operator: A self-adjoint operator is a linear operator that is equal to its own adjoint, meaning that it satisfies the property \( A = A^* \). This condition ensures that the operator has real eigenvalues and orthogonal eigenvectors, making it essential in quantum mechanics and various mathematical formulations. Self-adjoint operators also preserve the inner product structure of a space, which connects them to important concepts such as orthogonality and spectral theory.
Trial wavefunction: A trial wavefunction is an approximate solution to the Schrödinger equation used in quantum mechanics to estimate the ground state energy and properties of a quantum system. It is a key component in variational methods, where one can test different functional forms to find the best approximation, allowing for the calculation of energy expectations and insights into the system's behavior.
Variational parameter: A variational parameter is a quantity used in the variational method to minimize or maximize an energy functional in order to approximate the ground state energy and wave function of a quantum system. It serves as a variable that can be adjusted to find the best possible approximation to the true state of the system, effectively guiding the optimization process. This concept is closely linked to the principles of calculus of variations and has significant applications in both classical and quantum mechanics.
Variational Principle: The variational principle is a fundamental concept in physics that states the path taken by a system between two states is the one for which a certain quantity, typically the action, is minimized or made stationary. This principle underlies several formulations of mechanics and quantum mechanics, emphasizing that systems tend to evolve in a way that optimizes certain quantities. It connects to various concepts such as the dynamics of systems, energy conservation, and perturbation effects in quantum systems.
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