Variational Method Eigenvalue Calculator
Variational Method Eigenvalue Calculator computes approximate eigenvalues and eigenfunctions for the 1D Schrödinger equation \\( -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V(x) \psi = E \psi \\) with boundary conditions \\( \psi(0) = \psi(L) = 0 \\), using the variational method with a trial wave function. Results are visualized with p5.js and steps are displayed with MathJax.
Variational Method Eigenvalue Calculator
This calculator approximates the ground state eigenvalue and eigenfunction for the 1D Schrödinger equation \\( -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V(x) \psi = E \psi \\) with boundary conditions \\( \psi(0) = \psi(L) = 0 \\), using the variational method. Input the domain length (\\( L \\)), potential function (\\( V(x) \\)), trial wave function (\\( \psi(x) \\)), Planck’s constant (\\( \hbar \\)), mass (\\( m \\)), and grid points (\\( M \\)). The eigenvalue is computed as \\( E = \frac{\langle \psi | H | \psi \rangle}{\langle \psi | \psi \rangle} \\), visualized with p5.js, and steps are shown with MathJax. Results can be shared or embedded.
Example 1: Particle in a Box
Parameters: \\( L = 1 \\), \\( V(x) = 0 \\), \\( \psi(x) = x(1-x) \\), \\( \hbar = 1 \\), \\( m = 1 \\), \\( M = 100 \\).
Step 1: Compute expectation value of Hamiltonian.
Step 2: Normalize trial wave function.
Result: Approximate eigenvalue close to \\( \frac{\pi^2}{2} \approx 4.9348 \\).
Example 2: Harmonic Oscillator
Parameters: \\( L = 2 \\), \\( V(x) = x^2 \\), \\( \psi(x) = e^{-x^2} \\), \\( \hbar = 1 \\), \\( m = 1 \\), \\( M = 100 \\).
Approximates the ground state energy of a harmonic oscillator.
Example 3: Linear Potential
Parameters: \\( L = 1 \\), \\( V(x) = x \\), \\( \psi(x) = x(1-x) \\), \\( \hbar = 1 \\), \\( m = 1 \\), \\( M = 100 \\).
Tests a linear potential with a simple trial function.
Example 4: Higher Mass
Parameters: \\( L = 1 \\), \\( V(x) = 0 \\), \\( \psi(x) = x(1-x) \\), \\( \hbar = 1 \\), \\( m = 2 \\), \\( M = 100 \\).
Higher mass reduces the kinetic energy contribution.
Example 5: Different Trial Function
Parameters: \\( L = 1 \\), \\( V(x) = 0 \\), \\( \psi(x) = \sin(\pi x) \\), \\( \hbar = 1 \\), \\( m = 1 \\), \\( M = 100 \\).
Exact eigenfunction yields exact eigenvalue \\( \frac{\pi^2}{2} \\).
Parameters: \\( L = 2 \\), \\( V(x) = x^2 \\), \\( \psi(x) = e^{-x^2} \\), \\( \hbar = 1 \\), \\( m = 1 \\), \\( M = 100 \\).
Approximates the ground state energy of a harmonic oscillator.
Parameters: \\( L = 1 \\), \\( V(x) = x \\), \\( \psi(x) = x(1-x) \\), \\( \hbar = 1 \\), \\( m = 1 \\), \\( M = 100 \\).
Tests a linear potential with a simple trial function.
Parameters: \\( L = 1 \\), \\( V(x) = 0 \\), \\( \psi(x) = x(1-x) \\), \\( \hbar = 1 \\), \\( m = 2 \\), \\( M = 100 \\).
Higher mass reduces the kinetic energy contribution.
Parameters: \\( L = 1 \\), \\( V(x) = 0 \\), \\( \psi(x) = \sin(\pi x) \\), \\( \hbar = 1 \\), \\( m = 1 \\), \\( M = 100 \\).
Exact eigenfunction yields exact eigenvalue \\( \frac{\pi^2}{2} \\).