Physical Chemistry , 1st ed.

On the right,cancels, and the minus sign cancels i^2. Since the Hamiltonian

operator does not include time, the exponential on the left side can be re-

moved to outside the operator. Then we have:

eiEt/ Hˆ(x) E

(x) eiEt/

The exponentials on both sides cancel each other, and what is left is

Hˆ(x) E(x)

which is the time-independent Schrödinger equation.

The above example shows how the time-dependent Schrödinger equation

produces the time-independent Schrödinger equation, assuming a certain

form of(x,t) and a time-independent Hˆ. It is therefore more correct to say

that equation 10.29 is the fundamental equation of quantum mechanics, but

given the separability assumption, more attention in textbooks is devoted to

understanding the position-dependent part of the complete, time-dependent

Schrödinger equation. It is easy to show that wavefunctions of the form in

equation 10.31 are stationary states, because their probability distributions do

not depend on time. Some wavefunctions are not of the form in equation

10.31, so the time-dependent Schrödinger equation must be used.

10.15 Summary

Table 10.2 lists the postulates of quantum mechanics (even those not specifi-

cally discussed in this chapter). Different sources list different numbers of pos-

tulates, some broken into independent statements and some grouped together.

Hopefully, you can see how we applied these statements to the first ideal sys-

tem, the particle-in-a-box.

10.15 Summary 309

Table 10.2 The postulates of quantum mechanics
(Section 10.14) (If it is assumed that is separable into functions of
time and position, we find that this expression can be rewritten to get

the time-independent Schrödinger equation,HˆE.) (section 10.7)

Postulate V. The average value of an observable, O , is given by the ex-

pression

O 

all

*Oˆd

space

for normalized wavefunctions. (Section 10.9)

Postulate VI. The set of eigenfunctions for any quantum mechanical

operator is a complete mathematical set of functions.

Postulate VII. If, for a given system, the wavefunction is a linear

combination of nondegenerate wavefunctions nwhich have eigen-

values an:



n

cnn and Aˆnann

then the probability that anwill be the value of the corresponding

measurement is cn^2. The construction ofas the combination of

all possible n’s is called the superposition principle.

Postulate I. The state of a system of particles is given by a wavefunction
, which is a function of the coordinates of the particles and the time.
contains all information that can be determined about the state of
the system.must be single-valued, continuous, and bounded, and
^2 must be integrable. (Discussed in section 10.2)

Postulate II. For every physical observable or variable O, there exists a
corresponding Hermitian operator ˆO. Operators are constructed by
writing their classical expressions in terms of position and (linear) mo-
mentum, then replacing “xtimes” (that is,x ) for each xvariable and
i(/x) for each pxvariable in the expression. Similar substitutions
must be made for yand zcoordinates and momenta. (Section 10.3)

Postulate III. The only values of observables that can be obtained in a
single measurement are the eigenvalues of the eigenvalue equation con-
structed from the corresponding operator and the wavefunction :

OˆK


where Kis a constant. (Section 10.3)

Postulate IV. Wavefunctions must satisfy the time-dependent
Schrödinger equation:

Hˆi  (^) t