Physical Chemistry 2 nd Edition
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Physical Chemistry 2 Physical Chemistry 2 nd nd Edition Edition Thomas Engel, Philip Reid Chapter 14 Chapter 14 The Quantum Mechanical Postulates The Quantum Mechanical Postulates
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Chapter 14 The Quantum Mechanical Postulates. Physical Chemistry 2 nd 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 - PowerPoint PPT Presentation
Transcript of Physical Chemistry 2 nd Edition
Physical Chemistry 2Physical Chemistry 2ndnd Edition EditionThomas Engel, Philip Reid
Chapter 14 Chapter 14 The Quantum Mechanical PostulatesThe Quantum Mechanical Postulates
© 2010 Pearson Education South Asia Pte Ltd
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.
