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Ch 3. The Quantum
Mechanical Postulates
- Summarized the rules for how information is obtained
from wave functions in a few postulates
- Comparison of postulated results from quantum
mechanics with those obtained from classical mechanics
MS310 Quantum Physical Chemistry
Basic concepts of Q.M : Postulates
1) What is postulates?
framework of Q.M
2) How many postulates in Q.M?
6 postulates
No exception of these postulates until now.
In this chapter, we explain the 5 postulates. In chapter 10, 6th
postulate is introduced.
MS310 Quantum Physical Chemistry
3.1 Physical meaning of wave function
Postulate 1 : The state of a quantum mechanical system is
completely specified by a wave function Ψ(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 Ψ*(x0,t0)Ψ(x0,t0)dx.
Meaning of wave function Ψ(x0,t0) in classical waves
sound wave : pressure at (x0,t0)
water wave : height of water at (x0,t0)
What is the meaning of Ψ(x0,t0) by Schrödinger equation?
→ probability of finding a particle(also has a wave character)
at position x0 , time t0 within an interval dx.
P ( x0 , t 0 )   * ( x0 , t 0 )( x0 , t 0 )dx | ( x0 , t 0 ) |2 dx
MS310 Quantum Physical Chemistry
Unlike the classical wave, amplitude of Ψ(x0,t0) has no physical
meaning in Q.M. Why?
→ probability P α square of the magnitude of Ψ(x0,t0)
Ψ(x0,t0) : complex function
→ can multiply -1 or change the phase by multiplying
the complex number eiθ (θ : phase angle)
However, take the square, Ψ*(x0,t0)Ψ(x0,t0), these effects are
cancelled.
(1)2  1, e i (e  i )  1
Therefore, all wave functions with a different phase angle
generate the same observable.
MS310 Quantum Physical Chemistry
Ψ*(x0,t0)Ψ(x0,t0) : probability
→ sum of probability over whole interval must be 1 :

normalization
*

 ( x, t )( x, t )dx  1

Therefore, Ψ*(x0,t0)Ψ(x0,t0) must satisfy the following conditions.
1) single-valued function(only one probability at each point)
2) 1st derivative exists and continuous(2nd derivative exist and
well-behaved)
3) no infinite amplitude over a finite interval(wave function
must be normalized)
MS310 Quantum Physical Chemistry
Example of doublevalued function and
single-valued function
Continuous and discontinuous function
MS310 Quantum Physical Chemistry
3.2 Every observable has a corresponding operator
Postulate 2 : For every measurable property of the system in
C.M such as position, momentum, and energy, there exists a
corresponding operator in Q.M. An experiment in the lab 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.
All Q.M operator : Hermitian operator(real eigenvalue)
Order of operation is important
Ex)  ( x )  sin x
pˆ x xˆ  ( x )   i(sin x  x cos x )
xˆ pˆ x ( x )   i cos x
MS310 Quantum Physical Chemistry
MS310 Quantum Physical Chemistry
3.3 The result of individual measurement
Postulate 3: In any single measurement of the observable that
corresponds to the operator Â, the only values that will ever be
measured are the eigenvalues of that operator.
Ex) Hydrogen atom
Measured energies in experiment : only eigenvalues of the
time-independent Schrödinger equation
ˆ  ( x, t )  E  ( x, t )
H
n
n n
This make senses because the energy levels of the hydrogen
atom is discrete and only those energies are allowed.
MS310 Quantum Physical Chemistry
3.4 The expectation value
Postulate 4 : If the system is in a state described by the wave
function Ψ(x,t), and the value of the observable a is measured
once each on many identically prepared systems, the average
value(also called expectation value) of all of those
measurement is given by

*

 ( x, t ) Aˆ ( x, t )dx
 a   
*

 ( x, t )( x, t )dx

If Ψ(x,t) is normalized, denominator is 1.
There are two cases
1) Ψ(x,t) is a normalized eigenfunction of Â
2) Ψ(x,t) is not a normalized eigenfunction of Â
MS310 Quantum Physical Chemistry
1) Ψ(x,t) is a normalized eigenfunction of Â, φj(x,t)
( x , t )   j ( x , t )
ˆ  ( x, t )  a  ( x, t )
A
j
j j




ˆ  ( x , t )dx  a  * ( x , t ) ( x , t )dx  a
 a    *j ( x , t )A
j
j  j
j
j
All measurements will give the same answer, aj
2) Ψ(x,t) is not a eigenfunction of Â
( x, t )   bnn ( x, t )
Ψ(x,t) is normalized → ∑bm* bm = ∑ | bm |2 = 1




 a    ( x , t )Aˆ  ( x , t )dx   [  b  ( x , t )][  a n bnn ( x , t )]dx
*
  m 1



* *
m m

  bm* bn a n  m* ( x , t )n ( x , t )dx
m 1 n 1

MS310 Quantum Physical Chemistry
n 1
Use eigenfunctions of  form an orthonormal set


 a   b b a
m 1 n 1
*
m n n



2

(
x
,
t
)

(
x
,
t
)
dx

b
b
a

|
b
|
am


n
m
m

*
m

m 1
*
m m
m 1
bm : expansion coefficient of the wave function
<a> : weighted average
| bm |2 : contribution of each eigenfunction to the wave
function Ψ(x,t)
No way of knowing the outcome of individual measurement
and <a> is only average value.
What happened in this case?
→ superposition state
MS310 Quantum Physical Chemistry
Superposition state : wave function has characters of ‘more
than 2 states’
In this case, electron has a 1s, 2s, 2p, 3s character all and
probability of each state after measurement is b12, b22, b32 and
b42. → probabilistic outcome
Meaning of measurement process : collapse
before the measurement : superposition state
after the measurement : one state, the measured eigenvalue
MS310 Quantum Physical Chemistry
φ1(x), φ2(x), φ3(x) : eigenfunctions of Â
each eigenvalue : a1, 4a1, 9a1
11
1
1
1 ( x )  2 ( x )  3 ( x )
4
4
2
1
1
11
 2 ( x )  1 ( x )  2 ( x ) 
3 ( x )
2
4
4
1
11
1
 1 ( x )  1 ( x ) 
2 ( x )  3 ( x )
2
4
4
 1( x) 
Result of individual result : regardless of state(only a1, 4a1,
9a1 in this case)
Probability of each eigenvalue : depends on state(related to
square of each coefficient)
MS310 Quantum Physical Chemistry
measurement process in quantum mechanics:
probabilistic ----> deterministic
The act of carrying out a quantum mechanical measurement
appears to convert the wave function of a system to the
eigen-function
of
the
operator
corresponding
measured quantity!!
MS310 Quantum Physical Chemistry
to
the
3.5 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 )
Hˆ ( x, t )  i
t
Meaning of this postulate : variation of wave function as
time.(can predict the time variation of wave function – state)
However, postulate 4 and 5 is not contradictory. Why?
If we measure the system at t0, and no more measurement
after t0, wave function follows the postulate 5 at t1> t0. However,
if we measure the system at t1 > t0, wave function follows the
postulate 4 after t1.
MS310 Quantum Physical Chemistry
If system is time-independent, wave function is given by
( x , t )   ( x )e  i ( E /  ) t
In this case, we can solve the eigenvalue equation for timeindependent operator Â.
ˆ ( x ) ( x , t )  a  ( x , t )
A
n
n n
ˆ ( x ) ( x , t )e  i ( E /  ) t  a  ( x , t )e  i ( E /  ) t
A
n
n n
ˆ ( x ) ( x , t )  a  ( x , t )
A
n
n
n
MS310 Quantum Physical Chemistry
Summary
- Quantum mechanics can be formulated in terms of six
postulates provided a convenient framework for summarizing
the basic concepts of quantum mechanics.
- The state of a quantum mechanical system is completely
specified by a wave function Ψ(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 Ψ*(x0,t0)Ψ(x0,t0)dx.
- For every measurable property of the system in C.M such as
position, momentum, and energy, there exists a
corresponding operator in Q.M. An experiment in the lab 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.
MS310 Quantum Physical Chemistry
Summary
- In any single measurement of the observable that
corresponds to the operator Â, the only values that will ever
be measured are the eigenvalues of that operator.
- If the system is in a state described by the wave function
Ψ(x,t), and the value of the observable a is measured once
each on many identically prepared systems, the average
value(also called expectation value) of all of those
measurement is given by

  ( x, t ) Aˆ ( x, t )dx
*
 a   
  ( x, t )( x, t )dx
*

- The evolution in time of a quantum mechanical system is
governed by the time-dependent Schrödinger equation :
( x, t )
Hˆ ( x, t )  i
t
MS310 Quantum Physical Chemistry