In this theory, the instantanous state of a system is described by an element Ψ of some complex Hilbert space which encodes the probabilities of outcomes of all possible measurements applied to the system. The state of a system in general changes over time, Ψ = Ψ(t) is a function of time, and the Schrödinger equation describes this change quantitatively. The equation is therefore of central importance in quantum mechanics. The general, time-dependent equation reads
hd/dt Ψ = H Ψ
where i is the imaginary unit,
h equals Plancks constant h divided by 2π, and H is a self-adjoint linear operator on the Hilbert space, known as the Hamilton operator. The Hamilton operator describes the system under consideration and corresponds to the total energy of the system. It is therefore typically a sum of two operators, one corresponding to kinetic energy and the other to potential energy. In the special case of a system consisting of a single particle of mass m, the Hilbert space will consist of all square-integrable complex functions, Ψ = Ψ(r,t) will be a "wave function" depending on position r and time t, and the equation can be written as
hd/dt Ψ = - h2/2m ∇2 Ψ + V Ψ
where V=V(r) is the function describing the potential energy at position r and ∇2 is the Laplacian with respect to the space variables. Since the Laplacian involves squares of partial derivatives of Ψ, we are dealing with a non-linear partial differential equation, which is in general extremely difficult to solve explicitly.
Fortunately, many systems can be described by probability distributions which do not change over time. Examples are a confined electron or the hydrogen atom. These systems are described by the time-independent Schrödinger equation, which can be derived from the time-dependent one using the fact that two element of the Hilbert space encode the same probability distributions if and only if they differ only by a complex scalar factor of absolute value 1. The time-independent equation reads
- H φ = E φ
where H is again the Hamiltonian, E is the total energy of the system and is constant, and φ is an element of the Hilbert space (i.e. φ = φ(r) will be a square-integrable function depending only on space in the special case considered above). φ is related to the full time-dependent wave function Ψ by
- Ψ(t) = φ e-iE(t - τ) /
where τ is the phase of the wave. We see that the time-independent Schrödinger equation expresses E as an eigenvalue and φ as a corresponding eigenvector of the operator H.
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: