Physical Chemistry - The Postulates of Quantum Mechanics_lecture10-11

5.61 Fall 2007 Lecture #10 page 1



THE POSTULATES OF QUANTUM MECHANICS
(time-independent)

Postulate 1: The state of a system is completely described by a
wavefunction ψ r,t .( )
Postulate 2: All measurable quantities (observables) are described by
Hermitian linear operators.

Postulate 3: The only values that are obtained in a measurement of an
observable “A” are the eigenvalues “an” of the corresponding operator “ Â ”. The
measurement changes the state of the system to the eigenfunction of  with
eigenvalue an.

Postulate 4: If a system is described by a normalized wavefunction ψ,
then the average value of an observable corresponding to  is


a = ∫ ψ *Âψ dτ

Implications and elaborations on Postulates

2
#1] (a) The physically relevant quantity is ψ



( ) ( ) ( )
2
ψ * r,t ψ r,t = ψ r,t ≡ probability density at time t
and position r



(b) ( )
ψ r,t must be normalized


∫ ψ *ψ dτ = 1

(c) ( )
ψ r,t must be well behaved

, 5.61 Fall 2007 Lecture #10 page 2




(i) Single valued
(ii) ψ and ψ ′ continuous
(iii) Finite


#2] (a) Example: Particle in a box eigenfunctions of Ĥ

12
⎛ 2⎞ ⎛ nπ x ⎞
() ()
Ĥ x ψ n x = Enψ n x () ψn () x =⎜ ⎟
⎝ a⎠
sin ⎜
⎝ a ⎟⎠

But if ψ is not an eigenfunction of the operator, then the statement is not
true.


e.g. ()
ψ n x above with momentum operator


d ⎡⎛ 2 ⎞ ⎛ nπ x ⎞ ⎤
12
d
() dx
()
p̂nψ n x = −i! ψ n x = −i! ⎢⎜ ⎟ sin ⎜
dx ⎢⎝ a ⎠ ⎝ a ⎟⎠ ⎥
⎥
⎣ ⎦
⎡⎛ 2 ⎞ 1 2 ⎛ nπ x ⎞ ⎤
≠ pn ⎢⎜ ⎟ sin ⎜ ⎟⎠ ⎥
⎢⎣ ⎝ a ⎠ ⎝ a ⎥⎦

(b) In order to create a Q.M. operator from a classical observable, use
d
x̂ = x and p̂x = −i! and replace in classical expression.
dx
e.g.
1 2 1 !2 d 2
K.E. =
2m
p̂ =
2m
( )( )
p̂ p̂ = −
2m dx 2
(1D)

!2 ⎛ ∂2 ∂2 ∂2 ⎞
=− + + (3D)
2m ⎜⎝ ∂x 2 ∂y 2 ∂z 2 ⎟⎠


Another 3D example: Angular momentum L=r×p