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The wave function of an initially very localized free particle. In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters ψ and Ψ (lower-case and capital psi, respectively). Wave functions are complex ...
While the time-evolution process represented by the Schrödinger equation is continuous and deterministic, in that knowing the wave function at one instant is in principle sufficient to calculate it for all future times, wave functions can also change discontinuously and stochastically during a measurement. The wave function changes, according ...
The principle behind the condition is that, for example, if a wave is moving across a discrete spatial grid and we want to compute its amplitude at discrete time steps of equal duration, [2] then this duration must be less than the time for the wave to travel to adjacent grid points. As a corollary, when the grid point separation is reduced ...
The wave function cannot therefore be interpreted as a probability amplitude. The conserved quantity is instead interpreted as electric charge, and the norm squared of the wave function is interpreted as a charge density. The equation describes all spinless particles with positive, negative, and zero charge.
The condition m=0 is ruled out because = everywhere, meaning that the particle is not in the potential at all. Negative integers are also ruled out since they can easily be absorbed in the normalization condition. We then normalize the wave function, yielding a result where =. The normalized wave function is
A so-called eigenmode is a solution that oscillates in time with a well-defined constant angular frequency ω, so that the temporal part of the wave function takes the form e −iωt = cos(ωt) − i sin(ωt), and the amplitude is a function f(x) of the spatial variable x, giving a separation of variables for the wave function: (,) = ().
The variational theorem states that for a time-independent Hamiltonian operator, any trial wave function will have an energy expectation value that is greater than or equal to the true ground-state wave function corresponding to the given Hamiltonian. Because of this, the Hartree–Fock energy is an upper bound to the true ground-state energy ...
The Born rule is a postulate of quantum mechanics that gives the probability that a measurement of a quantum system will yield a given result. In one commonly used application, it states that the probability density for finding a particle at a given position is proportional to the square of the amplitude of the system's wavefunction at that position.