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 |
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 |
where i is the imaginary unit, |
where i is the imaginary unit, |
where V=V(r) is the function describing the potential energy at position r and ∇2 is the Laplacian?. |
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. |
Many systems can be described by probability distributions which don't 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 |
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 |
where the total energy of the system, E, is constant and φ depends only on space. φ is related to the full wave function Ψ by |
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(r,t) = φ(r) e-iE(t - τ) / |
:Ψ(t) = φ e-iE(t - τ) / |