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

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 14: The Quantum Mechanical Postulates Objectives Introduce 5 postulates which relate to quantum mechanics.

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 14: The Quantum Mechanical Postulates 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

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 14: The Quantum Mechanical Postulates 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 t 0 in a spatial interval of width dx centered at x 0 is given by

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 14: The Quantum Mechanical Postulates 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

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 14: The Quantum Mechanical Postulates 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: 1.Wave function must be a single-valued function 2.The first derivative must be continuous function 3.Wave function cannot infinite amplitude over a finite interval

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 14: The Quantum Mechanical Postulates 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.

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 14: The Quantum Mechanical Postulates 14.2 Every Observable Has a Corresponding Operator All quantum mechanical operators belong to a mathematical class called Hermitian operators that have real eigenvalues.

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 14: The Quantum Mechanical Postulates 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.

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 14: The Quantum Mechanical Postulates 14.3 The Result of an Individual Measurement The measured energy values of an atom are the eigenvalues of the time-independent Schrödinger equation:

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 14: The Quantum Mechanical Postulates 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

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 14: The Quantum Mechanical Postulates 14.4 The Expectation Value As eigenfunctions form an orthonormal set, it is normalized. Thus

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 14: The Quantum Mechanical Postulates 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:

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© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 14: The Quantum Mechanical Postulates 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 t 0, Postulate 4 applies. When t 1 > t 0, without carrying out a measurement in this time interval, Postulate 5 applies. If at time t 1, we carry out a measurement again, Postulate 4 will apply.

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