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

Objectives Introduce 5 postulates which relate to quantum mechanics.

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

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

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

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

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.

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

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.

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

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

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

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:

14.5 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.