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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 ...
If the Hamiltonian has explicit time-dependence (for example, if the quantum system interacts with an applied external electric field that varies in time), it will usually be advantageous to include the explicitly time-dependent terms with ,, leaving , time-independent. We proceed assuming that this is the case.
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.
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.
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 ...
Since the time separation is infinitesimal and the cancelling oscillations become severe for large values of ẋ, the path integral has most weight for y close to x. In this case, to lowest order the potential energy is constant, and only the kinetic energy contribution is nontrivial.
The simplest approach is to focus on the description in terms of plane matter waves for a free particle, that is a wave function described by =, where is a position in real space, is the wave vector in units of inverse meters, ω is the angular frequency with units of inverse time and is time. (Here the physics definition for the wave vector is ...