Comprehensive Study Notes

The Variation Method

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Variation Method: An approximation technique used to estimate the ground-state energy of a system without solving the Schrödinger equation 1. Variation Theorem: A theorem stating that for any normalized, well-behaved function that satisfies the boundary conditions of a system, the expectation value of the Hamiltonian is an upper bound to the ground-state energy1. Hamiltonian Operator (H n): The operator corresponding to the total energy of the system, including both kinetic and potential energies1. Ground-State Energy (E 1): The lowest energy eigenvalue of the Hamiltonian operator for a given system1. Trial Variation Function (f): A well-behaved function used in the variation method to approximate the ground-state energy1. Variational Integral: The integral of the product of the trial variation function, the Hamiltonian operator, and the trial variation function, divided by the integral of the square of the trial variation function1. Normalization Constant (N): A constant used to ensure that the trial variation function is normalized1. Eigenfunctions (c k_): The stationary-state wave functions that are solutions to the Schrödinger equation for a given Hamiltonian1. Eigenvalues (E k_): The energy values corresponding to the eigenfunctions of the Hamiltonian operator1. Orthonormal Set: A set of functions that are both orthogonal and normalized1. Kronecker Delta (d k j_): A function that is 1 if the indices are equal and 0 otherwise1. Parabolic Function: A function of the form f = x_1_l - _x_2 for a particle in a one-dimensional box1. Harmonic Oscillator: A system in which the potential energy is proportional to the square of the displacement from equilibrium1. Gaussian Elimination: A method for solving systems of linear equations by transforming the system's matrix into an upper triangular form1. Gauss–Jordan Elimination: An extension of Gaussian elimination that reduces the matrix to row echelon form1. Linear Variation Function: A linear combination of linearly independent functions used in the variation method1. Overlap Integral (S jk_): The integral of the product of two basis functions1. Secular Equation: An algebraic equation derived from the variation method that determines the approximate energies of the system1. Matrix Diagonalization: The process of finding the eigenvalues and eigenvectors of a matrix1. Hermitian Matrix: A matrix that is equal to its conjugate transpose1. Orthogonal Matrix: A matrix whose inverse is equal to its transpose1. Unitary Matrix: A matrix whose inverse is equal to its conjugate transpose1. Eigenvector: A non-zero vector that changes by only a scalar factor when a linear transformation is applied1. Characteristic Equation: An equation that determines the eigenvalues of a matrix1. Symmetric Matrix: A matrix that is equal to its transpose1. Diagonal Matrix: A matrix in which the entries outside the main diagonal are all zero1. Tridiagonal Matrix: A matrix that has non-zero elements only on the main diagonal and the diagonals immediately above and below it1. QR Method: An algorithm for finding the eigenvalues and eigenvectors of a matrix1. Cyclic Jacobi Method: An iterative method for diagonalizing a symmetric matrix1. Gaussian Variational Function: A trial function of the form e -cx 2 used in the variation method1. Block-Diagonal Form: A matrix form where the matrix is divided into smaller square matrices along the diagonal1. Normalization Condition: The condition that the integral of the square of the trial variation function is equal to 11. Expectation Value: The average value of a physical quantity in a given quantum state1. Schmidt Orthogonalization: A method for orthogonalizing a set of functions1. Symmetric Orthogonalization: A method for orthogonalizing a set of functions using the overlap matrix1. Rayleigh-Ritz Theorem: A theorem that provides an upper bound to the ground-state energy using the variation method1. Numerov Method: A numerical method for solving differential equations1. Particle-in-a-Box: A model system in quantum mechanics where a particle is confined to a one-dimensional box with infinite potential walls1. Quartic Oscillator: A system in which the potential energy is proportional to the fourth power of the displacement from equilibrium1. Double-Well Potential: A potential energy function with two minima separated by a barrier1. Harmonic Oscillator Basis Functions: The eigenfunctions of the harmonic oscillator Hamiltonian1. Particle-in-a-Box Basis Functions: The eigenfunctions of the particle-in-a-box Hamiltonian1. Radial Equation: The part of the Schrödinger equation that depends only on the radial coordinate in spherical coordinates1. Hydrogen Atom: A model system in quantum mechanics consisting of a single electron orbiting a proton1. Eigenvalue Problem: The problem of finding the eigenvalues and eigenvectors of a matrix or operator1. Matrix Algebra: The branch of mathematics that deals with matrices and their operations1. Linear Transformation: A transformation that preserves the operations of addition and scalar multiplication1. Unit Matrix: A square matrix with ones on the main diagonal and zeros elsewhere1. Inverse Matrix: A matrix that, when multiplied by the original matrix, yields the unit matrix1. Characteristic Polynomial: The polynomial obtained from the characteristic equation of a matrix1.