Module CH2701:
Physical Chemistry II
Aims
Quantum Mechanics I
The aim of this course is to provide an introduction to
quantum mechanics in general and the quantum mechanics
of atoms.
Georg Haehner
Office 243, Purdie
Literature: e.g. Atkins’ Physical Chemistry, OUP;
‘Quantum Mechanics for Chemists’, David O Hayward, RSC Publishing
accessible online at
http://pubs.rsc.org/en/Content/eBook/978-0-85404-607-2#!divbookcontent
Objectives
1.
To understand the concept of quantization and how it
arises.
2.
To know the form of the time-independent Schrödinger
equation and the meaning of the quantities therein.
3.
To follow the treatment of 'free electron' systems and
understand how energy levels, quantum numbers,
degeneracy and the effects of symmetry result from
this.
Objectives
4.
To follow in broad outline the solution of the
Schrödinger equation for the hydrogen atom and to
know the general form of the energy levels and atomic
orbitals and how the shapes of the latter arise.
5.
To appreciate, in principle, how the electronic structure
of 'many-electron‘ atoms can be constructed and
understood.
Quantum mechanics animations can be found at
www.st-andrews.ac.uk/~qmanim/ and specifically for chemistry
students at http://www.st-andrews.ac.uk/~qmanim/chemistry/
1
1.2 Classical Mechanics
1.1 Why quantum mechanics?
m

r

p
According to classical mechanics an
object and its path in space are
characterized by its

position in space: r


and momentum: p  mv

(m: mass, v velocity )
Classical mechanics
1) predicts a precise trajectory for particles with precisely specified
positions and momenta at each instant
2) allows the translational, rotational and vibrational modes of
motion to be excited to any energy, simply by controlling the forces
that are applied.

The velocity , v, of a particle is
the rate of change of its position:

 dr
v
dt
The magnitude of the velocity is the speed, v.

The linear momentum, p, of a particle
of mass m is


p  mv
An object and its path in space are
characterized by:

position in space
r
and momentum
p
m
p
r
Conclusions are in agreement with everyday experience!
1.2.1 Linear Momentum
1.2.2 Newton‘s second law
The linear momentum of a
Newton’s second law of motion states that the rate of change
of momentum is equal to the force acting on the particle:
particle is a vector property
and points in the direction of
motion.
The linear momentum of
an object with mass m is
defined as


p  mv
v
d r 
r
dt
Worked example: linear momentum of a running car
dp 
F
dt
F 
dp
d
d  d r
m
vm
dt
dt
dt  dt
2


  m d 2r  m r

dt

(if the mass does not change)
The second derivative of the position with respect to time is the
accelaration of the particle
Worked examples: Free fall and accelerating car
2
1.3 The path of a particle
1.2.3 Potential (energy) and Force
The force acting on a particle
is determined by the slope of
the potential energy at each
point. The force points in the
direction of lower potential
energy.
The force, F, experienced
by a particle free to move in
any direction is related to its
potential energy, V, by
F 

r
V  V
(
  
, , )
x y z
Worked example:
climbing a slope
1.4 Energy
E   Fd r
The path of a particle (its trajectory) can be determined from
knowing its initial position and momentum and by integrating the
two equations:
 
dr p

dt m

dp
V
 
dt
r
m
The particle will have a definite
position and momentum
(trajectory) at each instant!
p
r
1.4.1 Potential Energy and Total energy
(Force opposing movement)
Potential energy
(depends often on the location of the object)

r
V (r )    F d r
Kinetic energy of an object:
 Fd r  
Classical Mechanics describes the behavior of objects in
terms of two equations.

r0
dp
d p dr
dr  
dt
dt
dt dt
F 

V  V
r
The total energy is:
Etotal  Ekin  E pot
Etotal 

1 2
mv  V (r )
2
Etotal 

p2
 V (r )
2m
2
p  1 p
1 2
  dp 
 mv
m
2 m 2
Worked examples: energy required to jack up a vehicle (car-jack)
and energy of a driving car
Worked example: free fall
3