History of Schrodinger wave equation

Difference (from prior major revision) (author diff)

Changed: 3c3

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

Changed: 7c7

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

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

Changed: 11c11

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.

Changed: 13c13

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

Changed: 17c17

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

Changed: 19c19

:Ψ(r,t) = φ(r) e^{-iE(t - τ) / h}

:Ψ(t) = φ e^{-iE(t - τ) / h}

	Revision 14 . . October 21, 2001 8:38 am by AxelBoldt [incorporating (my own) comments from /Talk]
	Revision 13 . . October 17, 2001 11:40 pm by AxelBoldt
	Revision 12 . . (edit) October 17, 2001 9:06 am by Josh Grosse
	Revision 11 . . October 17, 2001 9:01 am by Josh Grosse [...ok, done]
	Revision 10 . . October 17, 2001 8:52 am by Josh Grosse [Making some changes, please hold...]
	Revision 9 . . (edit) October 17, 2001 8:42 am by (logged).5.185.xxx
	Revision 8 . . September 29, 2001 1:43 am by AxelBoldt [Time-independent equation]