CHM2S1-AIntroduction to Quantum Mechanics Dr R. L. Johnston

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Title: CHM2S1-AIntroduction to Quantum Mechanics Dr R. L. Johnston

1
CHM2S1-A Introduction to Quantum Mechanics Dr
R. L. Johnston

I Foundations of Quantum Mechanics
1. Classical Mechanics
1.1 Features of classical mechanics.
1.2 Some relevant equations in classical
mechanics.
1.3 Example The 1-Dimensional Harmonic
Oscillator
1.4 Experimental evidence for the breakdown of
classical mechanics.
1.5 The Bohr model of the atom.
2. Wave-Particle Duality
2.1 Waves behaving as particles.
2.2 Particles behaving as waves.
2.3 The De Broglie Relationship.

3. Wavefunctions
3.1 Definitions.
3.2 Interpretation of the wavefunction..
3.3 Normalization of the wavefunction.
3.4 Quantization of the wavefunction
3.5 Heisenbergs Uncertainty Principle.
4. Wave Mechanics
4.1 Operators and observables.
4.2 The Schrödinger equation.
4.3 Particle in a 1-dimensional box.
4.4 Further examples.

3
Learning Objectives

To appreciate the differences between Classical
(CM) and Quantum Mechanics (QM).
To know the failures in CM that led to the
development of QM.
To know how to interpret the wavefunction and how
to normalize it.
To appreciate the origins and implications of
quantization and the uncertainty principle.
To understand wave-particle duality and know the
relationships between momentum, frequency,
wavelength and energy for particles and
waves.
To be able to write down the Schrödinger equation
for particles in a 1-D box in 1- and 2-electron
atoms in 1- and 2-electron molecules.
To know the origins and allowed values of atomic
quantum numbers and how the energies and angular
momenta of hydrogen atomic orbitals depend on
them.
To be able to sketch the angular and radial nodal
properties of atomic orbitals.
To appreciate the origins of sheilding and its
effect on the ordering of orbital energies in
many-electron atoms.

To use the Aufbau Principle, the Pauli Principle
and Hunds Rule to predict the lowest energy
electron configuration for many-electron atoms.
To appreciate how the Born-Oppenheimer
approximation can be used to separate electronic
and nuclear motion in molecules.
To understand how molecular orbitals (MOs) can be
generated as linear combinations of atomic
orbitals and the difference between bonding and
antibonding orbitals.
To be able to sketch MOs and their corresponding
electron densities.
To construct MO diagrams for homonuclear and
heteronuclear diatomic molecules.
To predict the electron configurations for
diatomic molecules, calculate bond orders and
relate these to bond lengths, strengths and
vibrational frequencies.

5
References

Fundamentals
P. W. Atkins, J. de Paula, Atkins' Physical
Chemistry (7th edn.), OUP, Oxford, 2001.
D. O. Hayward, Quantum Mechanics for Chemists
(RSC Tutorial Chemistry Texts 14) Royal Society
of Chemistry, 2002.
W. G. Richards and P. R. Scott, Energy Levels in
Atoms and Molecules (Oxford Chemistry Primers 26)
OUP, Oxford, 1994.
More Advanced
P. W. Atkins and R. S. Friedman, Molecular
Quantum Mechanics (3rd edn.) OUP, Oxford, 1997.
P. A. Cox, Introduction to Quantum Theory and
Atomic Structure (Oxford Chemistry Primers 37)
OUP, Oxford, 1996.

6
1. Classical Mechanics

Do the electrons in atoms and molecules obey
Newtons classical laws of motion?
We shall see that the answer to this question is
No.
This has led to the development of Quantum
Mechanics we will contrast classical and
quantum mechanics.

1.1 Features of Classical Mechanics (CM)
1) CM predicts a precise trajectory for a
particle.
The exact position (r)and velocity (v) (and hence
the momentum p mv) of a particle (mass m) can
be known simultaneously at each point in time.
Note position (r),velocity (v) and momentum (p)
are vectors, having magnitude and direction ? v
(vx,vy,vz).

Any type of motion (translation, vibration,
rotation) can have any value of energy associated
with it
i.e. there is a continuum of energy states.
Particles and waves are distinguishable
phenomena, with different, characteristic
properties and behaviour.
Property Behaviour
mass momentum
Particles position ? collisions
velocity
Waves wavelength ? diffraction
frequency interference

1.2 Revision of Some Relevant Equations in CM
Total energy of particle
E Kinetic Energy (KE) Potential Energy (PE)
E ½mv2 V
? E p2/2m V (p mv)
Note strictly E, T, V (and r, v, p) are all
defined at a particular time (t) E(t) etc..

T - depends on v
V - depends on r
V depends on the system e.g. positional,
electrostatic PE
10

Consider a 1-dimensional system (straight line
translational motion of a particle under the
influence of a potential acting parallel to the
direction of motion)
Define position r x
velocity v dx/dt
momentum p mv m(dx/dt)
PE V
force F ?(dV/dx)
Newtons 2nd Law of Motion
F ma m(dv/dt) m(d2x/dt2)
Therefore, if we know the forces acting on a
particle we can solve a differential equation to
determine its trajectory x(t),p(t).

11
1.3 Example The 1-Dimensional Harmonic
Oscillator
NB assuming no friction or other forces act on
the particle (except F).

The particle experiences a restoring force (F)
proportional to its displacement (x) from its
equilibrium position (x0).
Hookes Law F ?kx
k is the stiffness of the spring (or stretching
force constant of the bond if considering
molecular vibrations)
Substituting F into Newtons 2nd Law we get
m(d2x/dt2) ?kx a (second order)
differential equation

k
12

Solution
position x(t) Asin(?t)
of particle
frequency ? ?/2?
(of oscillation)
Note frequency depends only on characteristics
of the system (m,k) not the amplitude (A)!

x
time period ? 1/ ?
?
A
t
?A
13

Assuming that the potential energy V 0 at x
0, it can be shown that the total energy of the
harmonic oscillator is given by
E ½kA2
As the amplitude (A) can take any value, this
means that the energy (E) can also take any value
i.e. energy is continuous.
At any time (t), the position x(t) and velocity
v(t) can be determined exactly i.e. the
particle trajectory can be specified precisely.
We shall see that these ideas of classical
mechanics fail when we go to the atomic regime
(where E and m are very small) then we need to
consider Quantum Mechanics.
CM also fails when velocity is very large (as v ?
c), due to relativistic effects.

14
1.4 Experimental Evidence for the Breakdown of
Classical Mechanics

By the early 20th century, there were a number of
experimental results and phenomena that could not
be explained by classical mechanics.
a) Black Body Radiation (Planck 1900)

l
/nm
15
Plancks Quantum Theory

Planck (1900) proposed that the light energy
emitted by the black body is quantized in units
of h? (? frequency of light).
?E nh? (n 1, 2, 3, )
High frequency light only emitted if thermal
energy kT ? h?.
h? a quantum of energy.
Plancks constant h 6.626?10?34 Js
If h ? 0 we regain classical mechanics.
Conclusions
Energy is quantized (not continuous).
Energy can only change by well defined amounts.

16
b) Heat Capacities (Einstein, Debye 1905-06)

Heat capacity relates rise in energy of a
material with its rise in temperature
CV (dU/dT)V
Classical physics ? CV,m 3R (for all T).
Experiment ? CV,m lt 3R (CV? as T?).
At low T, heat capacity of solids determined by
vibrations of solid.
Einstein and Debye adopted Plancks hypothesis.
Conclusion vibrational energy in solids is
quantized
vibrational frequencies of solids can
only have certain values (?)
vibrational energy can only change
by integer multiples of h?.

17
c) Photoelectric Effect (Einstein 1905)

Ideas of Planck applied to electromagnetic
radiation.
No electrons are ejected (regardless of light
intensity) unless n exceeds a threshold value
characteristic of the metal.
Ek independent of light intensity but linearly
dependent on n.
Even if light intensity is low, electrons are
ejected if n is above the threshold. (Number of
electrons ejected increases with light
intensity).
Conclusion Light consists of discrete packets
(quanta) of energy photons (Lewis, 1922).

18
d) Atomic and Molecular Spectroscopy

It was found that atoms and molecules absorb and
emit light only at specific discrete
frequencies ?? spectral lines (not
continuously!).
e.g. Hydrogen atom emission spectrum (Balmer
1885)
Empirical fit to spectral lines (Rydberg-Ritz)
n1, n2 (gt n1) integers.
Rydberg constant RH 109,737.3 cm-1 (but can
also be expressed in energy or frequency units).

n1 1 ? Lyman n1 2 ? Balmer n1 3 ?
Paschen n1 4 ? Brackett n1 5 ? Pfund
19
Revision Electromagnetic Radiation
A Amplitude l wavelength n -
frequency c n x l or n c / l
wavenumber n n / c 1 / l c (velocity of
light in vacuum) 2.9979 x 108 m s-1.
20
1.5 The Bohr Model of the Atom

The H-atom emission spectrum was rationalized by
Bohr (1913)
Energies of H atom are restricted to certain
discrete values
(i.e. electron is restricted to well-defined
circular orbits, labelled by quantum number n).
Energy (light) absorbed in discrete amounts
(quanta photons), corresponding to differences
between these restricted values
?E E2 ? E1 h?

Conclusion Spectroscopy provides direct
evidence for quantization of energies
(electronic, vibrational, rotational etc.) of
atoms and molecules.

Limitations of Bohr Model Rydberg-Ritz Equation
The model only works for hydrogen (and other one
electron ions) ignores e-e repulsion.
Does not explain fine structure of spectral
lines.
Note The Bohr model (assuming circular electron
orbits) is fundamentally incorrect.

22
2. Wave-Particle Duality

Remember Classically, particles and waves are
distinct
Particles characterised by position, mass,
velocity.
Waves characterised by wavelength, frequency.
By the 1920s, however, it was becoming apparent
that sometimes matter (classically particles) can
behave like waves and radiation (classically
waves) can behave like particles.

2.1 Waves Behaving as Particles
The Photoelectric Effect
Electromagnetic radiation of frequency ? can be
thought of as being made up of particles
(photons), each with energy E h ?.
This is the basis of Photoelectron Spectroscopy
(PES).
Spectroscopy
Discrete spectral lines of atoms and molecules
correspond to the absorption or emission of a
photon of energy h ?, causing the atom/molecule
to change between energy levels ?E h ?.
Many different types of spectroscopy are
possible.

The Compton Effect (1923)
Experiment A monochromatic beam of X-rays (?i)
incident on a graphite block.
Observation Some of the X-rays passing through
the block are found to have longer wavelengths
(?s).

Explanation The scattered X-rays undergo elastic
collisions with electrons in the graphite.
Momentum (and energy) transferred from X-rays to
electrons.
Conclusion Light (electromagnetic radiation)
possesses momentum.
Momentum of photon p h/?
Energy of photon E h? hc/ ?
Applying the laws of conservation
of energy and momentum we get

2.2 Particles Behaving as Waves
Electron Diffraction (Davisson and Germer, 1925)

Davisson and Germer showed that a beam of
electrons could be diffracted from the surface of
a nickel crystal. Diffraction is a wave property
arises due to interference between scattered
waves. This forms the basis of electron
diffraction an analytical technique for
determining the structures of molecules, solids
and surfaces (e.g. LEED).
NB Other particles (e.g. neutrons, protons,
He atoms) can also be diffracted by crystals.
27
2.3 The De Broglie Relationship (1924)

In 1924 (i.e. one year before Davisson and
Germers experiment), De Broglie predicted that
all matter has wave-like properties.
A particle, of mass m, travelling at velocity v,
has linear momentum p mv.
By analogy with photons, the associated
wavelength of the particle (?) is given by

28
3. Wavefunctions

A particle trajectory is a classical concept.
In Quantum Mechanics, a particle (e.g. an
electron) does not follow a definite trajectory
r(t),p(t), but rather it is best described as
being distributed through space like a wave.
3.1 Definitions
Wavefunction (?) a wave representing the
spatial distribution of a particle.
e.g. electrons in an atom are described by a
wavefunction centred on the nucleus.
? is a function of the coordinates defining the
position of the classical particle
1-D ?(x)
3-D ?(x,y,z) ?(r) ?(r,?,?) (e.g. atoms)
? may be time dependent e.g. ?(x,y,z,t)

The Importance of ?
? completely defines the system (e.g. electron in
an atom or molecule).
If ? is known, we can determine any observable
property (e.g. energy, vibrational frequencies,
) of the system.
QM provides the tools to determine ?
computationally, to interpret ? and to use ? to
determine properties of the system.

3.2 Interpretation of the Wavefunction
In QM, a particle is distributed in space like
a wave.
We cannot define a position for the particle.
Instead we define a probability of finding the
particle at any point in space.
The Born Interpretation (1926)
The square of the wavefunction at any point in
space is proportional to the probability of
finding the particle
at that point.
Note the wavefunction (?) itself has no physical
meaning.

1-D System
If the wavefunction at point x is ?(x), the
probability of finding the particle in the
infinitesimally small region (dx) between x and
xdx is
P(x) ? ??(x)?2 dx
??(x)? the magnitude of ? at point x.
Why write ???2 instead of ?2 ?
Because ? may be imaginary or complex ? ?2 would
be negative or complex.
BUT probability must be real and positive (0 ? P
? 1).
For the general case, where ? is complex (? a
ib) then
???2 ?? where ? is the complex conjugate
of ?.
(? a ib) (NB )

3-D System
If the wavefunction at r (x,y,z) is ?(r), the
probability of finding the particle in the
infinitesimal volume element d? ( dxdydz) is
P(r) ? ??(r)?2 d?
If ?(r) is the wavefunction describing
the spatial distribution of an electron
in an atom or molecule, then
??(r)?2 ?(r) the electron density at point r

3.3 Normalization of the Wavefunction
As mentioned above, probability P(r) ? ??(r)?2
d?
What is the proportionality constant?
If ? is such that the sum of ??(r)?2 at all
points in space 1, then
P(x) ??(x)?2 dx 1-D
P(r) ??(r)?2 d? 3-D
As summing over an infinite number of
infinitesimal steps integration, this means
i.e. the probability that the particle is
somewhere in space 1.
In this case, ? is said to be a normalized
wavefunction.

How to Normalize the Wavefunction
If ? is not normalized, then
A corresponding normalized wavefunction (?Norm)
can be defined
such that
The factor (1/?A) is known as the normalization
constant (sometimes represented by N).

3.4 Quantization of the Wavefunction
The Born interpretation of ? places restrictions
on the form of the wavefunction
(a) ? must be continuous (no breaks)
(b) The gradient of ? (d?/dx) must be continuous
(no kinks)
(c) ? must have a single value at any point in
space
(d) ? must be finite everywhere
(e) ? cannot be zero everywhere.

Other restrictions (boundary conditions) depend
on the exact system.
These restrictions on ? mean that only certain
wavefunctions and ? only
certain energies of the system are allowed.
? Quantization of ? ? Quantization of E

3.5 Heisenbergs Uncertainty Principle
It is impossible to specify simultaneously, with
precision, both the momentum
and the position of a particle
(if it is described by Quantum Mechanics)
Heisenberg (1927)
Dpx.Dx ? h / 4p (or ?/2, where ? h/2p).
Dx uncertainty in position
Dpx uncertainty in momentum (in the
x-direction)
If we know the position (x) exactly, we know
nothing about momentum (px).
If we know the momentum (px) exactly, we know
nothing about position (x).
i.e. there is no concept of a particle trajectory
x(t),px(t) in QM (which applies to small
particles).

How to Understand the Uncertainty Principle
To localize a wavefunction (?) in space (i.e. to
specify the particles position accurately, small
Dx) many waves of different wavelengths (?) must
be superimposed ? large Dpx (p h/?).
The Uncertainty Principle imposes a fundamental
(not experimental) limitation on how precisely we
can know (or determine) various observables.

Note to determine a particles position
accurately requires use of short radiation (high
momentum) radiation. Photons colliding with the
particle causes a change of momentum (Compton
effect) ? uncertainty in p.
? The observer perturbs the system.
Zero-Point Energy (vibrational energy Evib ? 0,
even at T 0 K) is also a consequence of the
Uncertainty Principle
If vibration ceases at T 0, then position and
momentum both 0 (violating the UP).
Note There is no restriction on the precision in
simultaneously knowing/measuring the position
along a given direction (x) and the momentum
along another, perpendicular direction (z)
Dpz.Dx 0

Similar uncertainty relationships apply to other
pairs of observables.
e.g. the energy (E) and lifetime (?) of a state
DE.D? ? ?
(a) This leads to lifetime broadening of
spectral lines
Short-lived excited states (? well defined, small
D?) possess large uncertainty in the energy
(large DE) of the state.
Broad peaks in the spectrum.
Shorter laser pulses (e.g. femtosecond,
attosecond lasers) have broader energy (therefore
wavelength) band widths.
(1 fs 10?15 s, 1 as 10?18 s)

40
4. Wave Mechanics

Recall the wavefunction (?) contains all the
information we need to know about any particular
system.
How do we determine ? and use it to deduce
properties of the system?
4.1 Operators and Observables
If ? is the wavefunction representing a system,
we can write
where Q observable property of system (e.g.
energy, momentum, dipole moment )
operator corresponding to observable Q.

This is an eigenvalue equation and can be
rewritten as
(Note ? cant be cancelled).
Examples d/dx (eax) a eax
d2/dx2 (sin ax) ?a2 sin ax

To find ? and calculate the properties
(observables) of a system
1. Construct relevant operator
2. Set up equation
3. Solve equation ? allowed values of ? and Q.
Quantum Mechanical Position and Momentum
Operators
1. Operator for position in the x-direction is
just multiplication by x
2. Operator for linear momentum in the
x-direction
?
(solve first order differential equation ? ? ,
px).

Constructing Kinetic and Potential Energy QM
Operators
1. Write down classical expression in terms of
position and momentum.
2. Introduce QM operators for position and
momentum.
Examples
1. Kinetic Energy Operator in 1-D
CM ? QM
2. KE Operator in 3-D
CM QM
3. Potential Energy Operator (a function
of position)
? PE operator corresponds to multiplication by
V(x), V(x,y,z) etc.

del-squared
44

4.2 The Schrödinger Equation (1926)
The central equation in Quantum Mechanics.
Observable total energy of system.
Schrödinger Equation
Hamiltonian Operator
E Total Energy
where and E T V.
SE can be set up for any physical system.
The form of depends on the system.
Solve SE ? ? and corresponding E.

Examples
1. Particle Moving in 1-D ?(x)
The form of V(x) depends on the physical
situation
Free particle V(x) 0 for all x.
Harmonic oscillator V(x) ½kx2
2. Particle Moving in 3-D ?(x,y,z)
SE ?
or

Note The SE is a second order differential
equation
46

4.3 Particle in a I-D Box
System
Particle of mass m in 1-D box of length L.
Position x 0?L.
Particle cannot escape from box as PE V(x) ? for
x 0, L (walls).
PE inside box V(x) 0 for 0lt x lt L.
1-D Schrödinger Eqn.
(V 0 inside box).

This is a second order differential equation
with general solutions of the form
? A sin kx B cos kx
SE ?
? (i.e. E depends on k).

Restrictions on ?
In principle Schrödinger Eqn. has an infinite
number of solutions.
So far we have general solutions
any value of A, B, k ? any value of ?,E.
BUT due to the Born interpretation of ?, only
certain values of ? are physically acceptable
outside box (xlt0, xgtL) V ? ? impossible for
particle to be outside the box
? ???2 0 ? ? 0 outside box.
? must be a continuous function
? Boundary Conditions ? 0 at x 0
? 0 at x L.

Effect of Boundary Conditions
x 0 ? A sin kx B cos kx B
? 0 ? B 0
? ? A sin kx for all x
x L ? A sin kL 0
sin kL 0 ? kL n? n 1, 2, 3,
(n ? 0, or ? 0 for all x)

Allowed Wavefunctions and Energies
k is restricted to a discrete set of values k
n?/L
Allowed wavefunctions ?n A sin(n?x/L)
Normalization A ?(2/L) ?
Allowed energies
?

Quantum Numbers
There is a discrete energy state (En),
corresponding to a discrete wavefunction (?n),
for each integer value of n.
Quantization occurs due to boundary conditions
and requirement for ? to be physically reasonable
(Born interpretation).
n is a Quantum Number labels each allowed state
(?n) of the system and determines its energy
(En).
Knowing n, we can calculate ?n and En.

Properties of the Wavefunction
Wavefunctions are standing waves
The first 5 normalized wavefunctions for the
particle in the 1-D box
Successive functions possess one more half-wave
(? they have a shorter wavelength).
Nodes in the wavefunction points at which ?n
0 (excluding the ends which are constrained to be
zero).
Number of nodes (n-1) ?1 ? 0 ?2 ? 1 ?3 ? 2

Curvature of the Wavefunction
If y f(x) dy/dx gradient of y (with respect
to x).
d2y/dx2 curvature of y.
In QM Kinetic Energy ? curvature of ?
Higher curvature ? (shorter ?) ? higher KE
For the particle in the 1-D box (V0)

Energies
En ? n2/L2 ? En? as n? (more nodes in ?n)
En? as L? (shorter box)
n? (or L?) ? curvature of ?n?
? KE? ? En?

En ? n2 ? energy levels get further apart as n?
Zero-Point Energy (ZPE) lowest energy of
particle in box
CM Emin 0
QM E 0 corresponds to ? 0 everywhere
(forbidden).

If V(x) V ? 0, everywhere in box, all energies
are shifted by V.

Density Distribution of the Particle in the 1-D
Box
The probability of finding the particle
between x and xdx (in the state
represented by ?n) is
Pn(x) ??n(x)?2 dx (?n(x))2 dx (?n is real)
?
Note probability is not uniform
?n2 0 at walls (x 0, L) for all ?n.
?n2 0 at nodes (where ?n 0).

4.4 Further Examples
(a) Particle in a 2-D Square or 3-D Cubic Box
Similar to 1-D case, but ? ? ?(x,y) or ?(x,y,z).
Solutions are now defined by 2 or 3 quantum
numbers
e.g. ?n,m, En,m ?n,m,l, En,m,l.
Wavefunctions can be represented as contour plots
in 2-D
(b) Harmonic Oscillator
Similar to particle in 1-D box, but PE V(x)
½kx2
(c) Electron in an Atom or Molecule
3-D KE operator
PE due to electrostatic interactions between
electron and all other electrons and nuclei.