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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