Page 281 - Essentials of physical chemistry

The Schrödinger Wave Equation 243

problem in precision measurements. Note we can sometimes use ‘‘Dirac notation’’ for the sandwich
integral ‘‘expectation value’’ expressed as a ‘‘bra-ket’’ where the complex part on the left is the
ð
*
‘‘bra’’ and the real part on the right is the ‘‘ket’’ for operator ‘‘O’’ in c Oc dt ¼hmjOjni:
m n
POSTULATES OF QUANTUM MECHANICS

Having solved only one problem in quantum mechanics we allow for the fact that there are only a
very few known solved problems and try to write down the postulated rules and then we will apply
the rules to another problem to reinforce the concepts.

Postulate I The state of a quantum-mechanical system is completely specified by a function C(r, t)
that depends on the coordinates of the particle and on time. This function, called the wave function or
the state function, has the important property that the product of C*(r, t) C(r, t) dxdydz is the
probability that the particle lies in the volume dxdydz located at r(x, y, z) at time t.

(Note that ‘‘completely specified’’ means that the wave function contains all the information that
can be obtained about the system using quantum mechanics! That provides tremendous motivation
to solve the Schrödinger equation and find the explicit wave function!)

Corollary In order for the wave function to be used in the Schrödinger equation, it must have
several mathematical properties:
1. It must be finite.
2. It must be continuous and single valued.
3. It must be defined for at least first and second derivatives.

Postulate II To every observable laboratory measurement in classical mechanics there
corresponds an operator in quantum mechanics.

Corollary Cartesian coordinates (x, y, z), spherical polar coordinates (r, u, f), or in general any set
of coordinates, q, merely become multiplicative operators, while the corresponding momentum

h q
operators, P q , become differential operators such as .
i qq

Postulate III In any measurement of the observable associated with the operator A, the only exact
values that will ever be observed are the eigenvalues a j which satisfy the eigenvalue equation

Ac ¼ a j c :
j
j
Corollary If the state function is not an eigenfunction of the operator A, then only an average
value can be obtained as from many measurements; see Postulate IV.

Postulate IV If a system is in a state described by a normalized wave function C, then the average
value of the observable corresponding to the operator A is given by

þ1
ð
C*ACdt:
<a> ¼
1