Glossary
Quantum mechanics terms, equations, and simulator links.
A
Amplitude
The magnitude of a wave function at a given point. In quantum mechanics, the square of the amplitude gives the probability density for finding a particle at that location.
A = |\\psi(\\mathbf{r}, t)|B
Barrier
A region of potential energy higher than the particle's kinetic energy. Classically impenetrable, but quantum particles can tunnel through barriers with nonzero probability.
V(x) > EBohm, David
American-Brazilian theoretical physicist (1917-1992) who developed the pilot-wave interpretation of quantum mechanics, providing a deterministic alternative to the Copenhagen interpretation.
\\frac{d\\mathbf{Q}}{dt} = \\frac{\\nabla S}{m}Bohmian Mechanics
A deterministic interpretation of quantum mechanics where particles have definite positions at all times, guided by a pilot wave described by the Schrodinger equation. Also called de Broglie-Bohm theory.
\\mathbf{v} = \\frac{\\hbar}{m} \\mathrm{Im}\\!\\left(\\frac{\\nabla\\psi}{\\psi}\\right)Born Rule
The fundamental postulate that the probability of finding a particle at position r is proportional to the square modulus of the wave function. In Bohmian mechanics, this emerges from equivariance rather than being postulated.
P(\\mathbf{r}) = |\\psi(\\mathbf{r}, t)|^2C
Collapse
In the Copenhagen interpretation, the instantaneous reduction of the wave function upon measurement to a single eigenstate. In Bohmian mechanics, no collapse occurs -- the particle simply reveals which branch it was always in.
|\\psi\\rangle \\xrightarrow{\\text{measure}} |\\phi_k\\rangleCoherence
The property of a quantum system where different parts of the wave function maintain a fixed phase relationship, enabling interference. Loss of coherence (decoherence) destroys interference patterns.
\\langle \\psi_1 | \\psi_2 \\rangle \\neq 0D
de Broglie Wavelength
The wavelength associated with a massive particle, inversely proportional to its momentum. Every object has a de Broglie wavelength, but it is only significant at quantum scales.
\\lambda = \\frac{h}{p} = \\frac{h}{mv}Decoherence
The process by which a quantum system loses its coherence through interaction with its environment, causing the wave function branches to become effectively independent. This explains the quantum-to-classical transition.
\\rho \\to \\sum_k p_k |\\phi_k\\rangle\\langle\\phi_k|Detector
A measurement device that registers the arrival of a particle. In the simulator, detectors record particle positions to build up probability distributions over many runs.
N(x) \\propto |\\psi(x)|^2Double Slit
The canonical quantum experiment where particles pass through two narrow openings and create an interference pattern on a screen, demonstrating wave-particle duality.
\\psi = \\psi_1 + \\psi_2E
Eigenstate
A state of a quantum system that is unchanged (up to a scalar factor) when acted upon by an operator. Measurement outcomes correspond to eigenvalues of the measured observable.
\\hat{A} |a\\rangle = a |a\\rangleEntanglement
A quantum correlation between two or more particles where the state of the total system cannot be written as a product of individual states. Measurement of one particle instantly constrains what can be known about the other.
|\\Psi\\rangle \\neq |\\psi_A\\rangle \\otimes |\\psi_B\\rangleF
Fringe
A bright or dark band in an interference pattern. Bright fringes appear where wave components add constructively; dark fringes where they cancel destructively.
I = I_1 + I_2 + 2\\sqrt{I_1 I_2}\\cos(\\delta)G
Gaussian
A bell-shaped function commonly used to describe the initial wave packet of a free particle. Gaussian wave packets have the minimum uncertainty allowed by the Heisenberg principle.
\\psi(x) = \\left(\\frac{1}{2\\pi\\sigma^2}\\right)^{\\!1/4} e^{-(x-x_0)^2/4\\sigma^2} e^{ik_0 x}H
Hamiltonian
The operator corresponding to the total energy of the system (kinetic plus potential). It generates the time evolution of the wave function through the Schrodinger equation.
\\hat{H} = -\\frac{\\hbar^2}{2m}\\nabla^2 + V(\\mathbf{r}, t)Heisenberg Uncertainty
The fundamental limit on simultaneously knowing both the position and momentum of a particle. The more precisely one is known, the less precisely the other can be determined.
\\Delta x \\, \\Delta p \\geq \\frac{\\hbar}{2}I
Interference
The phenomenon where two or more waves combine to form a resultant wave of greater, lower, or equal amplitude. Quantum interference is responsible for the patterns seen in double-slit experiments.
|\\psi_1 + \\psi_2|^2 = |\\psi_1|^2 + |\\psi_2|^2 + 2\\,\\mathrm{Re}(\\psi_1^*\\psi_2)M
Momentum
In quantum mechanics, momentum is represented by a differential operator. The expectation value of momentum determines the group velocity of a wave packet.
\\hat{p} = -i\\hbar\\nablaN
Nodal Line
A line or surface where the wave function is exactly zero. Bohmian particles cannot cross nodal lines, which act as impenetrable barriers in the velocity field.
\\psi(\\mathbf{r}_\\text{node}, t) = 0O
Observer
In quantum mechanics, any apparatus or system that interacts with a quantum system to extract information. In Bohmian mechanics, the observer has no special role -- measurement is just another physical interaction.
\\hat{M} |\\psi\\rangle \\to \\sum_k c_k |\\phi_k\\rangle |d_k\\rangleP
Particle
In Bohmian mechanics, a point-like entity with a definite position at all times. Its trajectory is deterministic, guided by the wave function according to the guidance equation.
\\mathbf{Q}(t) \\in \\mathbb{R}^3Phase
The argument of the complex wave function. The gradient of the phase determines the velocity field that guides Bohmian particles. Phase singularities produce quantum vortices.
\\psi = R\\, e^{iS/\\hbar}, \\quad \\mathbf{v} = \\frac{\\nabla S}{m}Pilot Wave
The wave function regarded as a physically real field that guides particle motion. Proposed by Louis de Broglie in 1927 and developed by David Bohm in 1952.
\\frac{d\\mathbf{Q}}{dt} = \\frac{\\hbar}{m} \\mathrm{Im}\\!\\left(\\frac{\\nabla\\psi}{\\psi}\\right)\\!\\Bigg|_{\\mathbf{r}=\\mathbf{Q}(t)}Planck Constant
The fundamental constant that sets the scale of quantum effects. The reduced Planck constant (h-bar) appears throughout quantum mechanics as the quantum of action.
\\hbar = \\frac{h}{2\\pi} \\approx 1.055 \\times 10^{-34}\\;\\text{J}\\cdot\\text{s}Potential
A scalar field V(r, t) representing the external forces acting on a particle. In the simulator, potentials create barriers, slits, and other structures that shape the wave function.
V(\\mathbf{r}, t)Probability Density
The probability per unit volume of finding a particle at a given location. Given by the squared modulus of the wave function, it is the primary observable in quantum experiments.
\\rho(\\mathbf{r}, t) = |\\psi(\\mathbf{r}, t)|^2Q
Quantum Potential
A term in Bohmian mechanics that arises from the wave function and has no classical analogue. It produces nonlocal forces responsible for interference, tunneling, and other quantum effects.
Q = -\\frac{\\hbar^2}{2m} \\frac{\\nabla^2 R}{R}S
Schrodinger Equation
The fundamental equation governing the time evolution of the wave function. It is a linear partial differential equation that describes how quantum states change over time.
i\\hbar \\frac{\\partial \\psi}{\\partial t} = -\\frac{\\hbar^2}{2m}\\nabla^2\\psi + V\\psiSuperposition
The principle that if two states are valid solutions of the Schrodinger equation, their sum is also a valid solution. Superposition enables interference and is the basis of quantum parallelism.
|\\psi\\rangle = \\alpha|0\\rangle + \\beta|1\\rangle, \\quad |\\alpha|^2 + |\\beta|^2 = 1T
Tunneling
The quantum phenomenon where a particle passes through a potential barrier that it classically could not surmount. The probability decreases exponentially with barrier width and height.
T \\approx e^{-2\\kappa d}, \\quad \\kappa = \\sqrt{\\frac{2m(V_0 - E)}{\\hbar^2}}V
Vortex
A point or line where the phase of the wave function is undefined and the probability density is zero. Quantum vortices carry quantized circulation and cause Bohmian particles to orbit around them.
\\oint_{\\mathcal{C}} \\nabla S \\cdot d\\mathbf{l} = 2\\pi n\\hbarW
Wave Function
A complex-valued function that encodes the complete quantum state of a system. In the pilot-wave interpretation, it is a physically real field that exists in configuration space and guides particle motion.
\\psi(\\mathbf{r}, t) \\in \\mathbb{C}Wave Packet
A localized wave function formed by superposing plane waves with a range of momenta. Wave packets spread over time due to dispersion, with the rate determined by the mass and initial width.
\\psi(x, t) = \\int_{-\\infty}^{\\infty} \\hat{\\psi}(k) \\, e^{i(kx - \\omega(k)t)} \\, dk