1. Physical Chemistry 2nd Edition
Chapter 14
The Quantum Mechanical Postulates
Physical Chemistry 2nd Edition
Thomas Engel, Philip Reid

2. Objectives
Introduce 5 postulates which relate to quantum mechanics.

3. Outline The Physical Meaning Associated with the Wave Function
Every Observable Has a Corresponding Operator
The Result of an Individual Measurement
The Expectation Value
The Evolution in Time of a Quantum Mechanical System

4. 14.1 The Physical Meaning Associated with the Wave Function
Postulate 1
The state of a quantum mechanical system is completely specified by a wave function
The probability that a particle will be found at time t0 in a spatial interval of width dx centered at x0 is given by

5. 14.1 The Physical Meaning Associated with the Wave Function
For sound wave, the wave function is associated with the pressure at a time t and position x.
For a water wave, is the height of the wave

6. 14.1 The Physical Meaning Associated with the Wave Function
The normalization condition for a particle confined in a 1-D space of infinite extent is
Ψ(x,t) must satisfy several mathematical conditions:
Wave function must be a single-valued function
The first derivative must be continuous function
Wave function cannot infinite amplitude over a finite interval

7. 14.2 Every Observable Has a Corresponding Operator
Postulate 2
For every measurable property of the system in classical mechanics such as position, momentum, and
energy, there exists a corresponding operator in
quantum mechanics. An experiment in the laboratory
to measure a value for such an observable is
simulated in the theory by operating on the wave
function of the system with the corresponding
operator.

8. 14.2 Every Observable Has a Corresponding Operator
All quantum mechanical operators belong to a mathematical class called Hermitian operators that have real eigenvalues.

9. 14.3 The Result of an Individual Measurement
Postulate 3
In any single measurement of the observable that corresponds to the operator , the only values that will ever be measured are the eigenvalues of that operator.

10. 14.3 The Result of an Individual Measurement
The measured energy values of an atom are the eigenvalues of the time-independent Schrödinger equation:

11. 14.4 The Expectation Value Postulate 4
If the system is in a state described by the wave function , and the value of the observable a is measured once each on many identically prepared systems, the average value (also called the expectation value) of all of these measurements is given by

12. 14.4 The Expectation Value
As eigenfunctions form an orthonormal set, it is normalized.
Thus

13. 14.5 The Evolution in Time of a Quantum Mechanical System
Postulate 5
The evolution in time of a quantum mechanical system is governed by the time-dependent Schrödinger equation:

14. 14.5 The Evolution in Time of a Quantum Mechanical System
We call this behavior deterministic in contrast to the probabilistic nature of Postulate 4.
When time at t0, Postulate 4 applies.
When t1 > t0, without carrying out a measurement in this time interval, Postulate 5 applies.
If at time t1, we carry out a measurement again, Postulate 4 will apply.