Schrödinger Wave Function Calculator
Compute the wave function ψ(x) for a particle in an infinite square well or as a free particle. Use this when studying quantum mechanics, particle-in-a-box problems, or verifying probability amplitude distributions.
About this calculator
The Schrödinger wave function ψ(x) describes the quantum state of a particle and determines the probability of finding it at position x. For a particle in an infinite square well of width L, the normalized stationary-state solution is ψₙ(x) = √(2/L) · sin(nπx/L), where n is the principal quantum number (n = 1, 2, 3, …). The probability density is |ψ(x)|², which integrates to 1 over the well. Higher quantum numbers produce more nodes in the wave function, corresponding to higher energy levels Eₙ = n²π²ℏ²/(2mL²). For a free Gaussian wave packet, the spatial envelope is ψ(x) = (2πσ²)^(−1/4) · exp(−x²/2σ²), where σ is the positional spread. This calculator evaluates ψ(x) numerically at any specified position inside the well.
How to use
Suppose you want ψ(x) at x = 0.3 m inside an infinite square well of width L = 1 m for the second energy level (n = 2). Select 'Infinite Well', enter n = 2, L = 1 m, and x = 0.3 m. The formula gives ψ = √(2/1) · sin(2π · 0.3 / 1) = 1.4142 · sin(1.8850) = 1.4142 · 0.9511 ≈ 1.345 m^(−1/2). The probability density at that point is |ψ|² ≈ 1.81 m⁻¹, meaning the particle is relatively likely to be found near x = 0.3 m.
Frequently asked questions
What does the wave function value actually represent physically?
The wave function ψ(x) itself is not directly observable; it is a complex-valued probability amplitude. Its squared magnitude |ψ(x)|² gives the probability density of finding the particle at position x upon measurement. Integrating |ψ(x)|² over any interval gives the probability of the particle being located within that interval. Proper normalization ensures this integral equals 1 over all space.
How do quantum numbers affect the shape of the wave function in an infinite square well?
The principal quantum number n determines how many half-wavelengths fit inside the well, and therefore how many nodes (zero-crossings) appear. For n = 1 the wave function has no internal nodes and one central peak. For n = 2 there is one node at the centre and two lobes of opposite sign. Each successive n adds one more node and raises the energy by a factor of n². This is directly analogous to standing waves on a string fixed at both ends.
Why is the infinite square well used as a model in quantum mechanics courses?
The infinite square well is one of the few quantum systems with exact analytical solutions, making it ideal for building intuition about quantization, nodes, and probability densities. It models electrons confined in quantum dots, neutrons in a nucleus, or any particle trapped in a region with very high potential barriers. While real potentials are never truly infinite, the model gives surprisingly good approximations when the barrier height greatly exceeds the particle's energy. It is the standard starting point before tackling more complex potentials like the harmonic oscillator or hydrogen atom.