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Chapter 14 The Quantum Mechanical Postulates Physical Chemistry 2nd Edition Thomas Engel, Philip Reid Objectives • Introduce 5 postulates which relate to quantum mechanics. Chapter 14: The Quantum Mechanical Postulates Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd Outline 1. The Physical Meaning Associated with the Wave Function 2. Every Observable Has a Corresponding Operator 3. The Result of an Individual Measurement 4. The Expectation Value 5. The Evolution in Time of a Quantum Mechanical System Chapter 14: The Quantum Mechanical Postulates Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 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 x, t • The probability that a particle will be found at time t0 in a spatial interval of width dx centered at x0 is given by x0 , t0 x0 , t0 dx Chapter 14: The Quantum Mechanical Postulates Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 14.1 The Physical Meaning Associated with the Wave Function • • For sound wave, the wave function x, t is associated with the pressure at a time t and position x. For a water wave, x, t is the height of the wave Chapter 14: The Quantum Mechanical Postulates Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 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 * x, t x, t dx 1 • Ψ(x,t) must satisfy several mathematical conditions: 1. 2. 3. Wave function must be a single-valued function The first derivative must be continuous function Wave function cannot infinite amplitude over a finite interval Chapter 14: The Quantum Mechanical Postulates Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 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. Chapter 14: The Quantum Mechanical Postulates Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 14.2 Every Observable Has a Corresponding Operator • All quantum mechanical operators belong to a mathematical class called Hermitian operators that have real eigenvalues. Chapter 14: The Quantum Mechanical Postulates Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 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. Chapter 14: The Quantum Mechanical Postulates Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 14.3 The Result of an Individual Measurement • The measured energy values of an atom are the eigenvalues of the time-independent Schrödinger equation: Hˆ n x, t En n x, t Chapter 14: The Quantum Mechanical Postulates Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 14.4 The Expectation Value Postulate 4 If the system is in a state described by the wave function x, t , 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 a * x, t Aˆ x, t dx * x, t x, t dx Chapter 14: The Quantum Mechanical Postulates Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 14.4 The Expectation Value • As eigenfunctions  form an orthonormal set, it is normalized. • Thus 2 * a ambmbm bm am m 1 Chapter 14: The Quantum Mechanical Postulates Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd m 1 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: x, t Hψ x,t ih t Chapter 14: The Quantum Mechanical Postulates Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd 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. Chapter 14: The Quantum Mechanical Postulates Physical Chemistry 2nd Edition © 2010 Pearson Education South Asia Pte Ltd