SchrödingerSolver Documentation
Welcome to the SchrödingerSolver documentation! This module provides utilities to solve the 1D Schrödinger equation using Gaussian basis functions.
Welcome to the SchrödingerSolver documentation! This module provides utilities to solve the 1D Schrödinger equation using Gaussian basis functions.
Settings
This document was generated with Documenter.jl version 1.7.0 on Wednesday 9 October 2024. Using Julia version 1.10.5.
Welcome to the SchrödingerSolver documentation! This module provides utilities to solve the 1D Schrödinger equation using Gaussian basis functions.
Settings
This document was generated with Documenter.jl version 1.7.0 on Friday 11 October 2024. Using Julia version 1.10.5.
This module handles interaction and plotting functions to display and visualize results.
Main.SchrödingerSolver.Interaction.display_menu — Methoddisplay_menu()Displays the main menu options to the user.
Main.SchrödingerSolver.Interaction.plot_wavefunctions — Methodplot_wavefunctions(
+Interaction · SchrödingerSolver Documentation Interaction Module
This module handles interaction and plotting functions to display and visualize results.
Main.SchrödingerSolver.Interaction.display_menu — Methoddisplay_menu()
Displays the main menu options to the user.
sourceMain.SchrödingerSolver.Interaction.plot_wavefunctions — Methodplot_wavefunctions(
x_vals::AbstractVector{Float64},
wavefunctions::Vector{Vector{Float64}},
num_levels::Int,
potential_name::String,
potential_params::Vector{Float64}
-)
Plots the wave functions.
Arguments
x_vals: Values of x.wavefunctions: Precomputed and normalized wave functions.num_levels: Number of energy levels to plot.potential_name: The name of the potential function.potential_params: Coefficients for the polynomial potential function.
sourceSettings
This document was generated with Documenter.jl version 1.7.0 on Wednesday 9 October 2024. Using Julia version 1.10.5.
Plots the wave functions.
Arguments
x_vals: Values of x.wavefunctions: Precomputed and normalized wave functions.num_levels: Number of energy levels to plot.potential_name: The name of the potential function.potential_params: Coefficients for the polynomial potential function.Settings
This document was generated with Documenter.jl version 1.7.0 on Friday 11 October 2024. Using Julia version 1.10.5.
This module provides methods for solving integrals and other calculations.
Main.SchrödingerSolver.Methods.build_matrices — Methodbuild_matrices(N, v, s, potential_gaussian_integral, potential_params)Builds the Hamiltonian matrix H and overlap matrix S for the variational method.
Arguments
N: Number of basis functions.v: Width parameters of the Gaussian basis functions.s: Centers of the Gaussian basis functions.potential_gaussian_integral: Function to compute the potential energy integral.potential_params: Additional parameters for the potential function.Returns
(H, S): The Hamiltonian and overlap matrices.Main.SchrödingerSolver.Methods.kinetic_integral — Methodkinetic_integral(v1, s1, v2, s2)Computes the kinetic energy integral between two Gaussian basis functions.
The kinetic energy integral is given by:
\[T_{ij} = \frac{v_1^{3/2} v_2^{3/2} \left( v_1 + v_2 - 2 v_1 v_2 (s_1 - s_2)^2 \right)}{\sqrt{\pi} (v_1 + v_2)^{5/2}} e^{- \frac{v_1 v_2 (s_1 - s_2)^2}{v_1 + v_2}}\]
Arguments
v1: Width parameter of the first Gaussian basis function.s1: Center of the first Gaussian basis function.v2: Width parameter of the second Gaussian basis function.s2: Center of the second Gaussian basis function.Returns
T_ij: The kinetic energy integral value.Main.SchrödingerSolver.Methods.overlap_integral — Methodoverlap_integral(v1, s1, v2, s2)Computes the overlap integral between two Gaussian basis functions with parameters (v1, s1) and (v2, s2).
The overlap integral is given by:
\[S_{ij} = \frac{\sqrt{v_1 v_2}}{\sqrt{\pi (v_1 + v_2)}} e^{- \frac{v_1 v_2 (s_1 - s_2)^2}{v_1 + v_2}}\]
Arguments
v1: Width parameter of the first Gaussian basis function.s1: Center of the first Gaussian basis function.v2: Width parameter of the second Gaussian basis function.s2: Center of the second Gaussian basis function.Returns
S_ij: The overlap integral value.Main.SchrödingerSolver.Methods.potential_integral_xn — Methodpotential_integral_xn(v1, s1, v2, s2, n)Computes the potential energy integral for ( V(x) = x^n ) between two Gaussian basis functions.
Arguments
v1, s1: Parameters of the first Gaussian basis function.v2, s2: Parameters of the second Gaussian basis function.n: The power of x in the potential function (integer from 0 to 4).Returns
V_ij: The potential energy integral value.Main.SchrödingerSolver.Methods.solve_schrodinger — Methodsolve_schrodinger(H, S, num_levels)Solves the generalized eigenvalue problem for the Hamiltonian H and overlap matrix S.
Arguments
H: Hamiltonian matrix.S: Overlap matrix.num_levels: Number of energy levels to compute.Returns
(energies, states): The lowest num_levels eigenvalues and eigenvectors.Settings
This document was generated with Documenter.jl version 1.7.0 on Wednesday 9 October 2024. Using Julia version 1.10.5.
This module provides methods for solving integrals and other calculations.
Main.SchrödingerSolver.Methods.build_matrices — Methodbuild_matrices(N, v, s, potential_gaussian_integral, potential_params)Builds the Hamiltonian matrix H and overlap matrix S for the variational method.
Arguments
N: Number of basis functions.v: Width parameters of the Gaussian basis functions.s: Centers of the Gaussian basis functions.potential_gaussian_integral: Function to compute the potential energy integral.potential_params: Additional parameters for the potential function.Returns
(H, S): The Hamiltonian and overlap matrices.Main.SchrödingerSolver.Methods.kinetic_integral — Methodkinetic_integral(v1, s1, v2, s2)Computes the kinetic energy integral between two Gaussian basis functions.
The kinetic energy integral is given by:
\[T_{ij} = \frac{v_1^{3/2} v_2^{3/2} \left( v_1 + v_2 - 2 v_1 v_2 (s_1 - s_2)^2 \right)}{\sqrt{\pi} (v_1 + v_2)^{5/2}} e^{- \frac{v_1 v_2 (s_1 - s_2)^2}{v_1 + v_2}}\]
Arguments
v1: Width parameter of the first Gaussian basis function.s1: Center of the first Gaussian basis function.v2: Width parameter of the second Gaussian basis function.s2: Center of the second Gaussian basis function.Returns
T_ij: The kinetic energy integral value.Main.SchrödingerSolver.Methods.overlap_integral — Methodoverlap_integral(v1, s1, v2, s2)Computes the overlap integral between two Gaussian basis functions with parameters (v1, s1) and (v2, s2).
The overlap integral is given by:
\[S_{ij} = \frac{\sqrt{v_1 v_2}}{\sqrt{\pi (v_1 + v_2)}} e^{- \frac{v_1 v_2 (s_1 - s_2)^2}{v_1 + v_2}}\]
Arguments
v1: Width parameter of the first Gaussian basis function.s1: Center of the first Gaussian basis function.v2: Width parameter of the second Gaussian basis function.s2: Center of the second Gaussian basis function.Returns
S_ij: The overlap integral value.Main.SchrödingerSolver.Methods.potential_integral_xn — Methodpotential_integral_xn(v1, s1, v2, s2, n)Computes the potential energy integral for ( V(x) = x^n ) between two Gaussian basis functions.
Arguments
v1, s1: Parameters of the first Gaussian basis function.v2, s2: Parameters of the second Gaussian basis function.n: The power of x in the potential function (integer from 0 to 4).Returns
V_ij: The potential energy integral value.Main.SchrödingerSolver.Methods.solve_schrodinger — Methodsolve_schrodinger(H, S, num_levels)Solves the generalized eigenvalue problem for the Hamiltonian H and overlap matrix S.
Arguments
H: Hamiltonian matrix.S: Overlap matrix.num_levels: Number of energy levels to compute.Returns
(energies, states): The lowest num_levels eigenvalues and eigenvectors.Settings
This document was generated with Documenter.jl version 1.7.0 on Friday 11 October 2024. Using Julia version 1.10.5.
The Utils module includes utility functions that assist with data processing, normalization, and parameter handling.
Main.SchrödingerSolver.Utils.check_positive_v — Methodcheck_positive_v(v)Checks if v is positive.
Arguments
v: Value to check.Throws
Error if v <= 0.Main.SchrödingerSolver.Utils.get_parameters — Methodget_parameters(choice::String)Retrieves parameters based on the user's selection.
Arguments
choice: User selection for the potential type.Returns
(N, v, s, potential_gaussian_integral, potential_params, potential_name, num_levels): Parameters for solving the Schrödinger equation.Main.SchrödingerSolver.Utils.get_potential_function — Methodget_potential_function(choice::String)Selects the potential function based on user's choice.
Arguments
choice: User selection for the potential type.Returns
[(potential_gaussian_integral, potential_name, potential_params)]: Potential function and parameters.Main.SchrödingerSolver.Utils.normalize_wavefunction — Methodnormalize_wavefunction(x_vals::AbstractVector{Float64}, ψ::AbstractVector{Float64})::Vector{Float64}Normalizes the wave function ψ based on the integration over x_vals.
Arguments
x_vals: Vector of x values.ψ: Wave function to normalize.Returns
Main.SchrödingerSolver.Utils.read_int — Methodread_int(prompt::String, default::Int)Reads an integer from the user with a prompt. If the user presses enter without input, returns the default value.
Arguments
prompt: The prompt message to display.default: The default value to return if no input is provided.Returns
Main.SchrödingerSolver.Utils.read_number — Methodread_number(prompt::String, default::Float64)Reads a floating-point number from the user with a prompt. If the user presses enter without input, returns the default value.
Arguments
prompt: The prompt message to display.default: The default value to return if no input is provided.Returns
Main.SchrödingerSolver.Utils.trapezoidal_integration — Methodtrapezoidal_integration(x, y)Performs trapezoidal integration of y with respect to x.
Arguments
x: Vector of x values.y: Vector of y values.Returns
integral: The result of the integration.Settings
This document was generated with Documenter.jl version 1.7.0 on Wednesday 9 October 2024. Using Julia version 1.10.5.
The Utils module includes utility functions that assist with data processing, normalization, and parameter handling.
Main.SchrödingerSolver.Utils.check_positive_v — Methodcheck_positive_v(v)Checks if v is positive.
Arguments
v: Value to check.Throws
Error if v <= 0.Main.SchrödingerSolver.Utils.get_parameters — Methodget_parameters(choice::String)Retrieves parameters based on the user's selection.
Arguments
choice: User selection for the potential type.Returns
(N, v, s, potential_gaussian_integral, potential_params, potential_name, num_levels): Parameters for solving the Schrödinger equation.Main.SchrödingerSolver.Utils.get_potential_function — Methodget_potential_function(choice::String)Selects the potential function based on user's choice.
Arguments
choice: User selection for the potential type.Returns
[(potential_gaussian_integral, potential_name, potential_params)]: Potential function and parameters.Main.SchrödingerSolver.Utils.normalize_wavefunction — Methodnormalize_wavefunction(x_vals::AbstractVector{Float64}, ψ::AbstractVector{Float64})::Vector{Float64}Normalizes the wave function ψ based on the integration over x_vals.
Arguments
x_vals: Vector of x values.ψ: Wave function to normalize.Returns
Main.SchrödingerSolver.Utils.read_int — Methodread_int(prompt::String, default::Int)Reads an integer from the user with a prompt. If the user presses enter without input, returns the default value.
Arguments
prompt: The prompt message to display.default: The default value to return if no input is provided.Returns
Main.SchrödingerSolver.Utils.read_number — Methodread_number(prompt::String, default::Float64)Reads a floating-point number from the user with a prompt. If the user presses enter without input, returns the default value.
Arguments
prompt: The prompt message to display.default: The default value to return if no input is provided.Returns
Main.SchrödingerSolver.Utils.trapezoidal_integration — Methodtrapezoidal_integration(x, y)Performs trapezoidal integration of y with respect to x.
Arguments
x: Vector of x values.y: Vector of y values.Returns
integral: The result of the integration.Settings
This document was generated with Documenter.jl version 1.7.0 on Friday 11 October 2024. Using Julia version 1.10.5.