Physical Chemistry 2 nd Edition

1. 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.