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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 time-independent Schrödinger equation for the wave function is ^ = [+ ()] = (), where Ĥ is the Hamiltonian, ħ is the reduced Planck constant, m is the mass, E the energy of the particle. The step potential is simply the product of V 0 , the height of the barrier, and the Heaviside step function : V ( x ) = { 0 , x < 0 V 0 , x ≥ 0 ...
In classical wave-physics, this effect is known as evanescent wave coupling. The likelihood that the particle will pass through the barrier is given by the transmission coefficient, whereas the likelihood that it is reflected is given by the reflection coefficient. Schrödinger's wave-equation allows these coefficients to be calculated.
2.1 Wave–particle duality and time evolution 2.1.1 Non-relativistic time-independent Schrödinger equation 2.1.2 Non-relativistic time-dependent Schrödinger equation
The Schrödinger equation describes the space- and time-dependence of the slow changing (non-relativistic) wave function of a quantum system. The solution of the Schrödinger equation for a bound system is discrete (a set of permitted states, each characterized by an energy level) which results in the concept of quanta.
The Schrödinger equation determines how wave functions evolve over time, and a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function", and gives rise to wave–particle duality.
In quantum mechanics, the Schrödinger equation describes how a system changes with time. It does this by relating changes in the state of the system to the energy in the system (given by an operator called the Hamiltonian). Therefore, once the Hamiltonian is known, the time dynamics are in principle known.
Time-dependent perturbation theory, initiated by Paul Dirac and further developed by John Archibald Wheeler, Richard Feynman, and Freeman Dyson, [12] studies the effect of a time-dependent perturbation V(t) applied to a time-independent Hamiltonian H 0. [13] It is an extremely valuable tool for calculating the properties of any physical system.