Schrödinger Equation Calculator
Solve the 1D Schrödinger equation for a particle in an infinite potential box: energy levels, wave function, and probability density
Infinite Square Well Quantum Model:This solver models an ideal 1D infinite potential well (V=0 for 0 < x < L, V=∞ elsewhere). Bound states are strictly quantized with non-zero zero-point energy (ZPE) as mandated by the Heisenberg Uncertainty Principle.
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1D Particle in a Box Schrödinger Eigenstate & Wave Function Solver
Solve the time-independent Schrödinger equation for an idealized particle in an infinite 1D square potential well. Calculate quantized bound-state energy levels (E_n in Joules and eV), spatial wave functions ψ_n(x), Born probability density distributions |ψ_n(x)|², nodal structures, de Broglie wavelengths, and transition photon emissions for electrons, protons, and subatomic particles.
Governing Quantum Mechanics & Infinite Potential Well Equations
Imposing Dirichlet boundary conditions (ψ(0) = 0 and ψ(L) = 0) at impenetrable potential barriers produces discrete spatial standing waves with strictly quantized kinetic energy eigenvalues.
- Quantized Bound-State Energy Levels: E_n = (n^2 π^2 hbar^2) / (2 m L^2) = (n^2 h^2) / (8 m L^2). Discrete energy eigenvalues for quantum number n ∈ {1, 2, 3...}, where m is particle mass (kg) and L is well width (m).
- Normalized Spatial Wave Function: psi_n(x) = √((2) / (L)) sin((n π x) / (L)). Stationary state wave function satisfying the spatial normalization condition ∫ |ψ|² dx = 1.
- Born Probability Density: P_n(x) = |psi_n(x)|^2 = (2) / (L) sin^2((n π x) / (L)). Probability per unit length of observing the particle at position x inside the potential well.
- De Broglie Matter Wavelength: lambda_n = (h) / (p_n) = (2L) / (n). De Broglie wavelength of the confined particle; exactly n half-wavelengths fit within the boundary length L.
- Transition Photon Emission Wavelength: lambda_photon = (h c) / (Delta E) = frach cE_n_2 - E_n_1. Electromagnetic photon wavelength emitted or absorbed during a quantum transition between energy levels.
Worked Example: Electron Confined in a 1.0 nm Quantum Dot
- Confined Particle: Electron (m = 9.109 × 10⁻³¹ kg)
- Box Width (L): 1.00 nm (1.00 × 10⁻⁹ m)
- Quantum State (n): n = 1 (Ground state) and n = 2 (First excited state)
- Constants: h = 6.62607 × 10⁻³⁴ J·s, 1 eV = 1.60218 × 10⁻¹⁹ J
- Ground State Kinetic Energy (n=1): E₁ = (1² × h²) / (8 × m × L²) = (4.3905 × 10⁻⁶⁷) / (8 × 9.109 × 10⁻³¹ × 1.0 × 10⁻¹⁸) = 6.025 × 10⁻²⁰ Joules
- Conversion to Electron-Volts (eV): E₁ = (6.025 × 10⁻²⁰ J) / (1.6022 × 10⁻¹⁹ J/eV) = 0.3761 eV
- Excited States Quadratic Scaling: E₂ = 2² × E₁ = 4 × 0.3761 = 1.5044 eV; E₃ = 3² × E₁ = 9 × 0.3761 = 3.3849 eV
- De Broglie Wavelength (n=1): λ₁ = 2L / 1 = 2.00 nm (half-wavelength spans the 1 nm box)
- Zero-Point Energy (ZPE): E₁ = 0.376 eV > 0; the particle cannot have zero kinetic energy due to the Heisenberg Uncertainty Principle (Δx · Δp ≥ ℏ/2).
- Transition Energy (n=2 → n=1): ΔE = E₂ - E₁ = 1.5044 - 0.3761 = 1.1283 eV (1.808 × 10⁻¹⁹ J)
- Emitted Photon Wavelength: λ_photon = (h · c) / ΔE = (6.626e-34 × 2.998e8) / 1.808e-19 = 1.099 × 10⁻⁶ m = 1,099 nm (near-infrared).
The ground state energy is 0.376 eV with a 2.0 nm de Broglie wavelength. Transitioning from n=2 to n=1 emits an infrared photon at 1,099 nm, precisely matching quantum dot fluorescence models.
Particles Database
- Name: Electron | Mass: 9.1093837e-31 | Symbol: e⁻ | Typical Length: 1.0 nm | Description: Standard quantum model for semiconductor quantum dots and conjugated carbon chains.
- Name: Proton | Mass: 1.6726219e-27 | Symbol: p⁺ | Typical Length: 0.01 nm | Description: Nuclear-confined hadron showing microscopic level spacing due to 1836x electron mass.
- Name: Neutron | Mass: 1.6749275e-27 | Symbol: n⁰ | Typical Length: 0.01 nm | Description: Neutral nucleon subject to strong nuclear potential wells.
- Name: Muon | Mass: 1.8835316e-28 | Symbol: μ⁻ | Typical Length: 0.1 nm | Description: Second-generation lepton (~207x electron mass) creating tighter orbital confinements.
Formulas Used
Quantized Bound-State Energy Levels
E_n = (n^2 π^2 hbar^2) / (2 m L^2) = (n^2 h^2) / (8 m L^2)Discrete energy eigenvalues for quantum number n ∈ {1, 2, 3...}, where m is particle mass (kg) and L is well width (m).
Normalized Spatial Wave Function
psi_n(x) = √((2) / (L)) sin((n π x) / (L))Stationary state wave function satisfying the spatial normalization condition ∫ |ψ|² dx = 1.
Born Probability Density
P_n(x) = |psi_n(x)|^2 = (2) / (L) sin^2((n π x) / (L))Probability per unit length of observing the particle at position x inside the potential well.
De Broglie Matter Wavelength
lambda_n = (h) / (p_n) = (2L) / (n)De Broglie wavelength of the confined particle; exactly n half-wavelengths fit within the boundary length L.
Transition Photon Emission Wavelength
lambda_photon = (h c) / (Delta E) = frach cE_n_2 - E_n_1Electromagnetic photon wavelength emitted or absorbed during a quantum transition between energy levels.
Frequently Asked Questions
The Schrödinger equation is the foundational differential equation of quantum mechanics, analogous to Newton's F = ma in classical mechanics. It describes how the quantum state or wave function (ψ) of a physical system evolves over space and time, governing probability amplitudes for finding particles at particular coordinates.
The 1D particle in a box is the quintessential quantum mechanical paradigm. It models a single particle free to move in a 1D region between 0 and L where potential energy is zero (V = 0), bounded by infinitely high potential energy walls (V = ∞) from which it can never escape.
Because the wave function ψ(x) must vanish at the impenetrable boundaries (ψ(0) = 0 and ψ(L) = 0) to ensure continuity, only standing wave patterns with integer numbers of half-wavelengths (n · λ/2 = L) can physically exist. This mathematical boundary constraint restricts allowed kinetic energy to discrete, quantized levels proportional to n².
The wave function ψ(x) is a complex probability amplitude that can take positive, negative, or imaginary values and undergoes wave interference. In contrast, the Born probability density P(x) = |ψ(x)|² is always real and non-negative, representing the actual probability per unit length of detecting the particle at position x.
If n = 0, the wave function ψ(x) = 0 everywhere, meaning no particle exists. Furthermore, according to the Heisenberg Uncertainty Principle (Δx · Δp ≥ ℏ/2), because the particle is confined within finite space L (Δx ≤ L), its momentum uncertainty Δp must be non-zero, requiring an irreducible minimum kinetic energy called Zero-Point Energy (ZPE).
The n-th eigenstate ψ_n(x) possesses exactly n - 1 interior nodes (points where ψ(x) = 0 inside the box, excluding the fixed wall boundaries at x = 0 and x = L). For example, the ground state n=1 has 0 interior nodes, n=2 has 1 node at x = L/2, and n=3 has 2 nodes at x = L/3 and 2L/3.
Energy level spacing scales inversely with mass and the square of length (ΔE ∝ 1 / (m · L²)). For a macroscopic marble of mass 10 grams in a 10 cm box, the ground state energy is on the order of 10⁻³⁰ Joules, with energy level separations far too infinitesimal to measure, blurring discrete quantum mechanics into classical continuous physics (Bohr's Correspondence Principle).
The de Broglie wavelength (λ = 2L/n) describes the spatial standing matter wave of the confined particle itself (e.g. 2 nm for an electron in a 1 nm box). The emitted photon wavelength (λ = hc / ΔE) is the electromagnetic radiation released when the particle drops from an excited state n₂ to a lower state n₁ (e.g. 1,099 nm).
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Quick Tips
- 1Use consistent units throughout calculations
- 2Consider significant figures in results
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