Physical Chemistry Third Edition

15.2 The Schrödinger Equation 659

Exercise 15.2

Carry out the algebra to obtain Eq. (15.2-6) from Eq. (15.2-2).

The left-hand side of Eq. (15.2-6) is commonly abbreviated by defining

Ĥ−h ̄

2

2 m

d^2

dx^2

+V(x) (15.2-7)

so that we abbreviate the Schrödinger equation in the form:

Hψ̂ Eψ (15.2-8)

The symbolĤis amathematical operator, because it stands for the carrying out of

mathematical operations on the functionψ. It is called theHamiltonian operator.

The Time-Dependent Schrödinger Equation

The time-dependent Schrödinger equation is taken as one of thepostulates(fundamental

assumptions) of quantum theory:

ĤΨih ̄∂Ψ

∂t

(15.2-9)

whereiis the imaginary unit, defined to equal the square root of−1:

i

√

− 1 (15.2-10)

and where theĤoperator is the same as in the time-independent equation. The function

Ψis thetime-dependent wave functionand represents the displacement of a de Broglie

wave as a function of position and time. In this chapter and the next we will use a capital

psi (Ψ) for a time-dependent wave function and a lower-case psi (ψ) for a coordinate

factor (coordinate wave function).

There is no way to obtain the time-dependent Schrödinger equation from a classi-

cal wave equation. The classical wave equation of a vibrating string, Eq. (14.3-3), is

second order in time. It requires two initial conditions (an initial position and an initial

velocity) to make a general solution apply to a specific case. The uncertainty princi-

ple of quantum mechanics (to be discussed later) implies that positions and velocities

cannot be specified simultaneously with arbitrary accuracy. For this reason only one

initial condition is possible, which requires the Schrödinger equation to be first order

in time. The fact that the equation is first order in time also requires that the imaginary

unitimust occur in the equation in order for oscillatory solutions to exist.

The time-independent Schrödinger equation can be obtained from the time-dependent

equation by separation of variables. For motion in thexdirection, we assume the trial

function

Ψ(x,t)ψ(x)η(t) (15.2-11)