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Physical Chemistry 2Physical Chemistry 2ndnd Edition EditionThomas Engel, Philip Reid

Chapter 14 Chapter 14 The Quantum Mechanical PostulatesThe Quantum Mechanical Postulates

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Physical Chemistry 2nd EditionChapter 14: The Quantum Mechanical Postulates

ObjectivesObjectives

• Introduce 5 postulates which relate to quantum mechanics.

© 2010 Pearson Education South Asia Pte Ltd

Physical Chemistry 2nd EditionChapter 14: The Quantum Mechanical Postulates

OutlineOutline

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 Value5. The Evolution in Time of a Quantum

Mechanical System

© 2010 Pearson Education South Asia Pte Ltd

Physical Chemistry 2nd EditionChapter 14: The Quantum Mechanical Postulates

14.1 The Physical Meaning Associated with the 14.1 The Physical Meaning Associated with the Wave Wave Function 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

tx,

dxtxtx 0000 ,,

© 2010 Pearson Education South Asia Pte Ltd

Physical Chemistry 2nd EditionChapter 14: The Quantum Mechanical Postulates

14.1 The Physical Meaning Associated with the 14.1 The Physical Meaning Associated with the Wave Wave Function 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

tx,

tx,

© 2010 Pearson Education South Asia Pte Ltd

Physical Chemistry 2nd EditionChapter 14: The Quantum Mechanical Postulates

14.1 The Physical Meaning Associated with the 14.1 The Physical Meaning Associated with the Wave Wave Function 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 function2. The first derivative must be continuous function3. Wave function cannot infinite amplitude over a

finite interval

1,,*

dxtxtx

© 2010 Pearson Education South Asia Pte Ltd

Physical Chemistry 2nd EditionChapter 14: The Quantum Mechanical Postulates

14.2 Every Observable Has a Corresponding 14.2 Every Observable Has a Corresponding Operator Operator

Postulate 2For 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 laboratoryto 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.

© 2010 Pearson Education South Asia Pte Ltd

Physical Chemistry 2nd EditionChapter 14: The Quantum Mechanical Postulates

14.2 Every Observable Has a Corresponding 14.2 Every Observable Has a Corresponding Operator Operator

• All quantum mechanical operators belong to a mathematical class called Hermitian operators that have real eigenvalues.

© 2010 Pearson Education South Asia Pte Ltd

Physical Chemistry 2nd EditionChapter 14: The Quantum Mechanical Postulates

14.3 The Result of an Individual14.3 The Result of an Individual Measurement Measurement

Postulate 3In 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.

A

© 2010 Pearson Education South Asia Pte Ltd

Physical Chemistry 2nd EditionChapter 14: The Quantum Mechanical Postulates

14.3 The Result of an Individual14.3 The Result of an Individual Measurement Measurement

• The measured energy values of an atom are the eigenvalues of the time-independent Schrödinger equation:

txEtxH nnn ,,ˆ

© 2010 Pearson Education South Asia Pte Ltd

Physical Chemistry 2nd EditionChapter 14: The Quantum Mechanical Postulates

14.4 The Expectation Value14.4 The Expectation Value

Postulate 4If 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

tx,

dxtxtx

dxtxAtxa

,,*

,ˆ,*

© 2010 Pearson Education South Asia Pte Ltd

Physical Chemistry 2nd EditionChapter 14: The Quantum Mechanical Postulates

14.4 The Expectation Value14.4 The Expectation Value

• As eigenfunctions form an orthonormal set, it is normalized.

• Thusm

mmmm

mm abbbaa

1

2*

1

A

© 2010 Pearson Education South Asia Pte Ltd

Physical Chemistry 2nd EditionChapter 14: The Quantum Mechanical Postulates

14.5 The Evolution in Time of a Quantum 14.5 The Evolution in Time of a Quantum Mechanical System Mechanical System

Postulate 5The evolution in time of a quantum mechanical system is governed by the time-dependent Schrödinger equation:

t

txihx,tHψ

,

© 2010 Pearson Education South Asia Pte Ltd

Physical Chemistry 2nd EditionChapter 14: The Quantum Mechanical Postulates

14.5 The Evolution in Time of a Quantum 14.5 The Evolution in Time of a Quantum Mechanical System 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.