Introduction
A stationary state is called stationary because a particle remains in the same state as time elapses, in every observable way. It has a constant probability distribution for its position, its velocity, its spin, etc. (This is true assuming the rest of the system is also static, i.e. the Hamiltonian is unchanging in time.) The wavefunction itself is not stationary: It continually changes its overall complex phase factor, so as to form a standing wave. The oscillation frequency of the standing wave, times Planck's constant, is the energy of the state according to the de Broglie relation.
Stationary states are quantum states that are solutions to the time-independent Schrödinger Equation:
- ,
where
- is a quantum state, which is a stationary state if it satisfies this equation;
- is the Hamiltonian operator;
- is a real number, and corresponds to the energy eigenvalue of the state .
This is an eigenvalue equation: is a linear operator on a vector space, is an eigenvector of, and is its eigenvalue.
If a stationary state is plugged into the time-dependent Schrödinger Equation, the result is:
Assuming that is time-independent (unchanging in time), this equation holds for any time t. Therefore this is a differential equation describing how varies in time. Its solution is:
Therefore a stationary state is a standing wave that oscillates with an overall complex phase factor, and its oscillation angular frequency is equal to its energy divided by .
Read more about this topic: Stationary State
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