Download Chapter 8 - Lecture 3

Peter Atkins • Julio de Paula
Atkins’ Physical Chemistry
Eighth Edition
Chapter 8
Quantum Theory:
Introduction and Principles
Copyright © 2006 by Peter Atkins and Julio de Paula
Born interpretation of the wavefunction
• The value of |Ψ|2
|Ψ|2
∝
(or Ψ*Ψ if complex)
probability of finding particle at that point
• For a 1-D system:
If the wavefunction of a particle has the value Ψ at
some point x, the probability of finding the particle
between x and x + dx is proportional to |Ψ|2 .
Fig 8.19 Probability of finding a particle in
some region of a 1-D system, Ψ(x)
Fig 8.20 Born Interpretation: Probability of finding
a particle in some volume of a 3-D system, Ψ(r)
Probability density = |Ψ|2
Probability = |Ψ|2 dτ
dτ = dx dy dz
Fig 8.21 Sign of wavefunction has no direct
physical significance
|Ψ|2 (or Ψ*Ψ if complex) > 0
However, the positive and
negative regions of Ψ1 can
constructively/destructively
interfere with the regions of
Ψ2.
Based on the Born
interpretation, an acceptable
wavefunction must be:
1) Continuous
2) Single-valued
3) Finite
To ensure that the particle
is in the system, the
wavefunction must be
normalized:
|Ψ|2 ∝ probability
|NΨ|2 = probability
Fig 8.24
|Ψ|2 ∝ probability
|NΨ|2 = probability
Normalization
Time-independent Schrodinger equation for particle
of mass m moving in one dimension, x:
 2 d2 Ψ

 V(x)  EΨ
2
2m dx
Sum of all probabilities must equal 1 or:

2
N

Ψ Ψdx  1

*
So:
N
1


 Ψ * Ψdx 


 



1/ 2
Normalized ψ in three dimensions:

Ψ* Ψdxdydz  1
Or:

Ψ* Ψdτ  1
For systems of spherical symmetry (atoms)
it is best to use spherical polar coordinates:
Fig 8.22 Spherical polar coordinates for systems
of spherical symmetry
Now:
Ψ(r, θ, φ)
x → r sin θ cos φ
y → r sin θ sin φ
z → r cos θ
Volume element becomes:
dτ = r2 sin θ dr dθ dφ
Fig 8.23 Spherical polar coordinates for systems
of spherical symmetry
r=0-∞
θ =0–π
φ = 0 - 2π
Consider a free particle of mass, m, moving in 1-D.
• Assume V = 0
 2 d2 

 E
2
2m dx
• From Schrodinger equation solutions are:
  A exp(ikx )  B exp(ikx )
k 2 2
E
2m
• Assume B = 0, then probability

2
 A
2
Particle may be found
anywhere!
Fig 8.25 Square of the modulus of a wavefunction
for a free particle of mass, m.
Assume B = 0, then probability:

2
 A
2
• Assume A = B, then probability

2
2
 4 A cos 2 kx
using:
eikx  cos kx  i sin kx
e -ikx  cos kx  i sin kx
• Now position is quantized!
Fig 8.25 Square of the modulus of a wavefunction
for a free particle of mass, m.
Assume A = B, then probability:

2
2
 4 A cos 2 kx
Operators, Eigenfunctions, and Eigenvalues
• Systematic method to extract info from wavefunction
• Operator for an observable is applied to wavefunction
to obtain the value of the observable
• (Operator)(function) = (constant)(same function)
• (Operator)(Eigenfunction) = (Eigenvalue)(Eigenfunction)

e.g., H   E
2 d2
where H  
 V(x)
2
2m dx

Operators, Eigenfunctions, and Eigenvalues

• e.g., x  x  is the position operator for one dimension

px 
 d
i dx
is the momentum operator
What is the linear momentum of a particle described by the
wavefunction: Ψ  A exp(ikx )

 d

d exp(ikx )
pΨ 
Ψ A
i dx
i
dx

d exp(ikx )
 Aik
i
dx
 kA exp(ikx )  kΨ
Operators, Eigenfunctions, and Eigenvalues

x  x

px 
is the position operator for one dimension
 d
i dx
is the momentum operator
Suppose we want operator for potential energy, V = ½ kx2:

V
1
2
kx

2
Likewise the operator for kinetic energy, EK = px2/2m:

EK 
    
1  d
2m i dx
 d
i dx
 2 d2
2m dx2
Fig 8.26 Kinetic energy of a particle with
a non-periodic wavefunction.
• 2nd derivative gives measure
of curvature of function
• The larger the 2nd derivative the
greater the curvature.
• The greater the curvature the
greater the EK.
Fig 8.27 Observed kinetic energy of a particle is
an average over the entire space
covered by the wavefunction.