Schrödinger Equation Solver

Solve the quantum wave equation for various potentials and visualize wave functions

Quantum System Parameters

Select potential and enter quantum system parameters

nm
eV
Energy Level (Eₙ)
0.094 eV
Energy of quantum state
Wavelength (λ)
1.23 nm
De Broglie wavelength
Zero-Point Energy
0.094 eV
Ground state energy
Quantum Number
1
State number (n)
Understanding the Schrödinger Equation

The Schrödinger equation describes how quantum systems evolve over time. The time-independent form is:

Ĥψ = Eψ

  • Ĥ is the Hamiltonian operator
  • ψ is the wave function
  • E is the energy eigenvalue
  • |ψ|² represents probability density
  • Wave functions are quantized
  • Energy levels are discrete
Quantum Systems Examples
System Potential Energy Levels Wave Function
Particle in a box Infinite square well Eₙ = n²h²/(8mL²) Sine waves
Harmonic oscillator V(x) = ½mω²x² Eₙ = ħω(n+½) Hermite polynomials
Hydrogen atom Coulomb potential Eₙ = -13.6/n² eV Laguerre polynomials
Finite square well Finite depth well Transcendental equation Exponential decay outside
Quantum tunneling Barrier potential Continuous spectrum Exponential decay in barrier

Quantum Mechanics Concepts

The Schrödinger equation is the fundamental equation of quantum mechanics that describes how quantum systems evolve over time.

1

Wavefunction (ψ): Mathematical description of quantum state

Contains all information about a quantum system

2

Probability Density: |ψ(x)|² gives probability of finding particle at position x

Wavefunction must be normalized: ∫|ψ|² dx = 1

3

Quantization: Energy levels are discrete in bound systems

Quantum number n determines energy state

4

Tunneling: Quantum particles can penetrate classically forbidden regions

Probability decreases exponentially with barrier width

Key Quantum Concepts

Wavefunction
ψ(x)
Quantum state description
Probability Density
|ψ(x)|²
Probability distribution
Energy Quantization
En
Discrete energy levels
Zero-Point Energy
E0 > 0
Minimum energy in QHO

Quantum Systems

Infinite Square Well

Particle confined in a box with infinite walls

Discrete energy levels En ∝ n²

Harmonic Oscillator

Particle in parabolic potential

Equally spaced energy levels En = ℏω(n+½)

Finite Square Well

Particle in a well with finite depth

Finite number of bound states

!!! IMPORTANT DISCLAIMER AND LIMITATIONS !!!

Please be fully aware and agree that this online Schrödinger equation solver is an educational demonstration tool based on specific numerical algorithms, with the primary purpose of assisting in the understanding of fundamental quantum mechanics concepts.

  • Model Simplifications: The solver deals exclusively with one-dimensional, non-relativistic, stationary problems. It does not account for complex factors such as electron spin, multi-particle interactions, or relativistic effects.
  • Numerical Approximations: The results are numerical approximations whose accuracy depends on factors such as grid discretization. There may be subtle differences compared to exact analytical solutions.
  • Research vs. Educational Tool: This tool cannot replace professional quantum chemistry software (such as Gaussian, ORCA, or VASP) for rigorous scientific research. For serious scientific investigations, these specialized software packages must be used.
  • Intended Use: Treat this tool as a "telescope" for exploring the quantum world - use it to observe fundamental phenomena and cross-verify with theoretical derivations.
By using this solver, you acknowledge and accept these limitations and agree to use this tool solely for educational purposes.

The wavefunction ψ(x) is a complex-valued function that contains all information about a quantum system:

  • Probability Amplitude: |ψ(x)|² gives the probability density of finding the particle at position x
  • Normalization: ∫|ψ(x)|² dx = 1 (total probability must be 1)
  • Phase: The complex phase of ψ(x) is important for interference phenomena
  • Evolution: The wavefunction evolves according to the Schrödinger equation

In the Copenhagen interpretation, the wavefunction "collapses" upon measurement to a definite state.

Energy quantization arises from the boundary conditions and wave nature of quantum particles:

  • Boundary Conditions: Wavefunctions must satisfy specific conditions at boundaries (e.g., ψ=0 at infinite walls)
  • Standing Waves: Only certain wavelengths "fit" in confined spaces, leading to discrete frequencies and energies
  • Mathematical Requirement: Solutions to the Schrödinger equation are only possible for specific energy values

Examples:

System Energy Quantization
Infinite Square Well En = n²π²ℏ²/(2mL²)
Harmonic Oscillator En = ℏω(n + ½)
Hydrogen Atom En = -13.6/n² eV

Quantum tunneling is a phenomenon where a particle passes through a potential barrier that it classically shouldn't be able to surmount:

  • Wave Nature: Quantum particles have wave-like properties that allow penetration into classically forbidden regions
  • Exponential Decay: Wavefunction decays exponentially inside barrier: ψ(x) ∝ e-κx
  • Transmission Probability: T ≈ e-2κL where κ = √(2m(V-E))/ℏ and L is barrier width

Applications:

  • Scanning Tunneling Microscopy (STM)
  • Nuclear fusion in stars
  • Flash memory devices
  • Quantum computing
  • Radioactive decay

Heisenberg's Uncertainty Principle states that certain pairs of physical properties cannot be simultaneously known to arbitrary precision:

Δx Δp ≥ ℏ/2

Where:

  • Δx = Uncertainty in position
  • Δp = Uncertainty in momentum
  • ℏ = Reduced Planck's constant (h/2π)

Key implications:

  • Position and momentum cannot both be precisely determined
  • Energy and time have a similar relationship: ΔE Δt ≥ ℏ/2
  • Fundamental limit, not due to measurement limitations
  • Reflects wave-particle duality

Example: In an infinite square well, the ground state has minimum uncertainty product.