Schrodinger wave equation

The Schrödinger equation, found by the Austrian physicist Erwin Schrödinger in 1925, describes the wave-like behavior of particles in quantum mechanics.

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

i h d/dt Ψ = H Ψ

where i is the imaginary unit, h equals Plancks constant h divided by 2π, and H is a linear 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. In the special case of a system consisting of a single particle of mass m, the equation can be written as

i h d/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?.

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

H φ = E φ

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

Ψ(r,t) = φ(r) e^{-iE(t - τ) / h}

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:

• The particle in a box
• The [quantum harmonic oscillator]?
• The [hydrogenic atom]?

---

/Talk