In this theory, the state of a system is described by a wave function, i.e. by an element ψ of some complex Hilbert space. The wave function encodes the probabilities of measurement outcomes, and in general will depend on the position r and time t, so can be written ψ = ψ(r,t). The Schrödinger equation describes how ψ changes over time and is therefore of central importance in quantum mechanics. The general, time-dependent equation reads
where i is the imaginary unit, h equals Plancks constant h divided by 2π, and H is an operator on the Hilbert space, known as the Hamilton operator. The Hamilton operator corresponds to the total energy of the system and is therefore typically a sum of two operators, one corresponding to kinetic energy and the other to potential energy.
Many systems can be described by probability distributions which don't change over time. Examples are a confined electron or the hydrogen atom. The states ψ describing these systems have to be solutions of the time-independent Schrödinger equation
where E is the total energy of the system (which is constant). We see that φ is an eigenvector of the operator H. The time-independent Schrödinger equation follows from the more general time-dependent one if one uses the fact that two elements in H describe the same physical state of the system if and only if they differ only by a complex scalar factor of absolute value 1.
Solutions of the Schrödinger equation
Analytical solutions of the time-independent Schrödinger equation can be obtained for a variety of relatively simple conditions. These solutions provide insight into the nature of Quantum phenomena and sometimes provide a reasonable approximation of the behavior of more complex systems (eg. in Statistical Mechanics molecular vibrations are often approximated as harmonic oscillators). Several of the more common analytical solutions include:
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