0.841471
0.540302
0.708073
—
m
—
Hz
100,000
m/s
1.055000e-24
kg·m/s
1.055000e-19
J
0.841471
0.540302
0.708073
—
m
—
Hz
100,000
m/s
1.055000e-24
kg·m/s
1.055000e-19
J
The Wave Function Calculator evaluates quantum mechanical plane wave functions and their associated physical properties. In quantum mechanics, a free particle with definite momentum and energy is described by a traveling plane wave: $$\psi(x,t) = A \sin(kx - \omega t) \quad \text{or} \quad \psi(x,t) = A e^{i(kx - \omega t)}$$
Here A is the amplitude, k = 2π/λ is the wave number, and ω = 2πf is the angular frequency. The de Broglie relations connect wave properties to particle properties: momentum p = ħk and energy E = ħω. This wave-particle duality is one of the central concepts of quantum physics.
The plane wave ψ = Ae^{i(kx−ωt)} represents a particle with perfectly defined momentum p = ħk, but by the Heisenberg uncertainty principle, its position is completely undefined — the probability density |ψ|² = A² is uniform over all space. Real particles are described by wave packets: superpositions of plane waves with a spread of k values, producing a localized probability distribution.
This calculator evaluates both the sine and cosine forms of the real wave function at any position x and time t. It computes the probability density |ψ|², wavelength, frequency, phase velocity, momentum, and energy. These calculations are fundamental for understanding matter waves, electron diffraction, neutron optics, and the behavior of quantum particles in free space.
The phase velocity vₚ = ω/k tells how fast the wave crests move. For a non-relativistic particle, the phase velocity equals half the particle velocity (vₚ = p/2m = v/2), while the group velocity (the speed of the wave packet envelope) equals the particle velocity. This distinction between phase and group velocity is crucial in wave mechanics and signal propagation.
Wave functions are the mathematical backbone of quantum mechanics. Every observable quantity is extracted from ψ: position and momentum expectation values, uncertainties, transition probabilities, and scattering cross sections. This calculator provides a practical tool for evaluating plane waves and understanding the wave-like behavior of quantum particles.
The calculator evaluates a traveling plane wave and extracts physical quantities:
Wave Function (sine form):
$$\psi(x,t) = A \sin(kx - \omega t)$$
Wave Function (cosine form):
$$\psi(x,t) = A \cos(kx - \omega t)$$
Probability Density:
$$|\psi|^2 = A^2 \sin^2(kx - \omega t)$$
Wavelength:
$$\lambda = \frac{2\pi}{k}$$
Frequency:
$$f = \frac{\omega}{2\pi}$$
Phase Velocity:
$$v_p = \frac{\omega}{k}$$
de Broglie Momentum:
$$p = \hbar k$$
Energy:
$$E = \hbar \omega$$
The wave function value at (x, t) gives the quantum amplitude at that point in space and time. The probability density |ψ|² oscillates between 0 and A² for the sine form (note: for a true complex plane wave e^{i(kx−ωt)}, |ψ|² = A² everywhere). The wavelength and wave number characterize spatial oscillation, while frequency and angular frequency characterize temporal oscillation. The momentum and energy link wave properties to particle properties through de Broglie relations. The phase velocity describes how fast wave crests propagate.
Inputs
Results
A thermal neutron with wavelength 0.175 nm and wave number 3.6×10¹⁰ m⁻¹ has momentum 3.8×10⁻²⁴ kg·m/s and energy about 9.4 meV, typical for neutron diffraction experiments.
Inputs
Results
A 100 eV electron has a de Broglie wavelength of 0.123 nm, suitable for probing atomic-scale structures in electron diffraction and transmission electron microscopy.
A wave function ψ(x,t) is a complex-valued function that completely describes the quantum state of a particle. It contains all information about the particle's position, momentum, energy, and other properties. The squared modulus |ψ|² gives the probability density for finding the particle at position x at time t. The wave function evolves according to the Schrödinger equation.
The de Broglie wavelength λ = h/p = 2π/k relates a particle's wavelength to its momentum. Louis de Broglie proposed in 1924 that all matter has wave-like properties, with the wavelength inversely proportional to momentum. This was confirmed by electron diffraction experiments (Davisson-Germer, 1927). At macroscopic scales, the wavelength is immeasurably small, but for electrons and neutrons, it is on the order of atomic spacings.
Phase velocity vₚ = ω/k is the speed of individual wave crests. Group velocity vₑ = dω/dk is the speed of the wave packet envelope and represents the actual particle velocity. For a free non-relativistic particle, vₚ = p/2m = v/2 (half the particle speed) while vₑ = p/m = v (the full particle speed). Energy and information travel at the group velocity.
No, a pure plane wave Ae^{i(kx−ωt)} extends over all space with constant amplitude, so ∫|ψ|²dx diverges. It represents a particle with exact momentum but completely uncertain position. Physical states are wave packets — superpositions of plane waves — that can be normalized. Plane waves are still useful as mathematical building blocks (Fourier components) for constructing normalizable wave packets.
For a normalized wave packet, A is chosen so that the total probability integrates to 1. For a plane wave, A is a constant that determines the probability density |ψ|² = A² (for the complex exponential form). In practice, plane waves are often used with A = 1 for simplicity, and normalization is applied to the wave packet formed from superposition.
The wave function determines the probability of every possible measurement outcome. Position measurement probabilities come from |ψ(x)|², momentum probabilities from |φ(p)|² (the Fourier transform of ψ), and energy probabilities from expansion coefficients in the energy eigenbasis. Upon measurement, the wave function collapses to the eigenstate corresponding to the observed value — a process with no classical analogue.
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