Physical Chemistry Third Edition

662 15 The Principles of Quantum Mechanics. I. De Broglie Waves and the Schrödinger Equation

If a system consists ofnparticles moving in three dimensions, its Hamiltonian

operator is

Ĥ−

∑n

j 1

h ̄^2

2 mj

∇j^2 +V(q) (15.2-27)

where∇j^2 is the Laplacian operator for the coordinates of particle numberj. The

potential energyVis a function of all of the coordinates, which are abbreviated by the

symbolq.

Once the Hamiltonian operator is obtained, the time-dependent Schrödinger equa-

tion is abbreviated as in Eq. (15.2-9):

ĤΨih ̄∂Ψ

∂t

(15.2-28)

whereΨdepends on all of the coordinates and ont. If we assume the trial solution

Ψψ(x 1 ,y 1 ,z 1 ,...,xn,yn,zn)η(t) (15.2-29)

then the time-independent Schrödinger equation can be extracted from the time-

dependent equation in exactly the same way as in Eqs. (15.2-11) through (15.2-15).

Eigenvalue Equations

The time-independent Schrödinger equation belongs to a class of equations called

eigenvalue equations. The word “eigenvalue” is a partial translation of the German

wordEigenwert. A full translation is “characteristic value.” An eigenvalue equation

has on one side an operator operating on a function, and on the other side a constant

called theeigenvaluemultiplying the same function, which is called theeigenfunction.

IfÂis a mathematical operator, its eigenvalue equation is

Af̂ nanfn (15.2-30)

wherefnis the eigenfunction andanis the eigenvalue. An eigenvalue equation gen-

erally has a set of solutions, so we have attached a subscriptnto the eigenfunction

and eigenvalue in Eq. (15.2-31) to specify a particular one of the solutions. Solving

an eigenvalue equation means finding not only the set of eigenfunctions that satisfy

the equation, but also the eigenvalue that belongs to each eigenfunction. Two common

cases occur. The first case is that the eigenvalue can take on any value within some

range of values (acontinuous spectrumof eigenvalues). The second case is that there

is a discrete set of eigenvalues with the values between the members of the set not

permitted (adiscrete spectrumof eigenvalues). The occurrence of a discrete spectrum

of eigenvalues corresponds to quantization.

The time-independent Schrödinger equation is the eigenvalue equation for the

Hamiltonian operator. The coordinate wave function is the eigenfunction of the

Hamiltonian operator, and is often called theenergy eigenfunction. The eigenvalue of

the Hamiltonian operator,E, is the value of the energy, and is called theenergy eigen-

value. There are other eigenvalue equations that are important in quantum mechanics,

and we will discuss some of them later.

In addition to satisfying the Schrödinger equation a wave function must satisfy other

conditions. Since it represents a wave, we assume that it has the following properties,