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 Oscillator1.4 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.

THE UNIVERSITYOF BIRMINGHAM

3. Wavefunctions3.1 Definitions.3.2 Interpretation of the wavefunction..3.3 Normalization of the wavefunction.3.4 Quantization of the wavefunction 3.5 Heisenberg’s 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.

 

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 Hund’s 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.

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.

1. Classical Mechanics

• Do the electrons in atoms and molecules obey Newton’s 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).

position r = (x,y,z)

velocity v

2) 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.

3) 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 systeme.g. positional, electrostatic PE

• 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 = xvelocity v = dx/dtmomentum p = mv = m(dx/dt)

PE Vforce F = (dV/dx)

• Newton’s 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 it’s trajectory {x(t),p(t)}.

acceleration

1.3 Example – The 1-Dimensional Harmonic Oscillator

• The particle experiences a restoring force (F) proportional to its displacement (x) from its equilibrium position (x=0).

• Hooke’s Law F = kx

k is the stiffness of the spring (or stretching force constant of the bond if considering molecular vibrations)

• Substituting F into Newton’s 2nd Law we get:

m(d2x/dt2) = kx a (second order) differential equation

x

m

Fk

x = 0NB – assuming no friction or other forces act on the particle (except F).

k

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)!

mπ2

1 k

mω k

+A

A

x

t

time period = 1/

• 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.

• 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)

1.4 Experimental Evidence for the Breakdown of Classical Mechanics

0 2000 4000 6000

2000 K

1750 K

1250 K

Energy Radiated

Classical Mechanics(Rayleigh-Jeans)

“UV Catastrophe”

/nm

Planck’s 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.

• Planck’s constant h ~ 6.6261034 Js

• If h 0 we regain classical mechanics.

• Conclusions:• Energy is quantized (not continuous).• Energy can only change by well defined amounts.

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 < 3R (CV as T).• At low T, heat capacity of solids determined by

vibrations of solid.

• Einstein and Debye adopted Planck’s 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.

c) Photoelectric Effect (Einstein 1905)

• Ideas of Planck applied to electromagnetic radiation.• No electrons are ejected (regardless of light intensity) unless

exceeds a threshold value characteristic of the metal.• Ek independent of light intensity but linearly dependent on .• Even if light intensity is low, electrons are ejected if is above the

threshold. (Number of electrons ejected increases with light intensity).

• Conclusion: Light consists of discrete packets (quanta) of energy = photons (Lewis, 1922).

e P h o te lec tro n s-

h

Metal surface

work function =

ePhotoelectrons ejected with kinetic energy:

Ek = h -

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 (> n1) = integers.

• Rydberg constant RH = 109,737.3 cm-1 (but can also be expressed in energy or frequency units).

22

21

11

λ

1νν

nnR

c H

n1 = 1 Lyman

n1 = 2 Balmer

n1 = 3 Paschen

n1 = 4 Brackett

n1 = 5 Pfund

Revision: Electromagnetic Radiation

A – Amplitude – wavelength

- frequency c = x or = c /

wavenumber = c= 1 /

c (velocity of light in vacuum) = 2.9979 x 108 m s-1.

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

h

E1

E2

h

E1

E2

Absorption Emission

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

n2

n1

e

p+

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.

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

a) The Photoelectric EffectElectromagnetic 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).

b) SpectroscopyDiscrete 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.

c) 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).

i

s

Intensity

i 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:

i

s

e

p=h/s

p=mev

cos1λλΔλ is cm

h

e

2.2 Particles Behaving as Waves

Electron Diffraction (Davisson and Germer, 1925)

Davisson and Germer showed thata beam of electrons could be diffractedfrom the surface of a nickel crystal.

Diffraction is a wave property – arisesdue 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.

2.3 The De Broglie Relationship (1924)

• In 1924 (i.e. one year before Davisson and Germer’s 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:

mvpλ

hh

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.

3.1Definitions

• 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 x+dx 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 )

probability density

1i

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.

1dxxψD1P2

total

1dxdydzzy,x,ψdτψD3P22

total

r

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).

1dτψ A2

r

rr ψA

1ψ Norm

1dτψ2

Norm r

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 Heisenberg’s 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)

pxx h / 4(or /2, where = h/2).

x – uncertainty in position px – 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).

• NB – for macroscopic objects, x and px can be very small when compared with x and px so one can define a trajectory.

• Much of classical mechanics can be understood in the limit h 0.

How to Understand the Uncertainty Principle

• To localize a wavefunction () in space (i.e. to specify the particle’s position accurately, small x) many waves of

different wavelengths () must be superimposed large px

(p = h/).

• The Uncertainty Principle imposes a fundamental (not experimental) limitation on how precisely we can know (or determine) various observables.

2 ~ 1

• Note – to determine a particle’s 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):

pzx = 0

• Similar uncertainty relationships apply to other pairs of observables.

• e.g. the energy (E) and lifetime () of a state:

E.

(a)This leads to “lifetime broadening” of spectral lines:– Short-lived excited states ( well defined, small ) possess

large uncertainty in the energy (large E) of the state. Broad peaks in the spectrum.

(b)Shorter laser pulses (e.g. femtosecond, attosecond lasers) have broader energy (therefore wavelength) band widths.

(1 fs = 1015 s, 1 as = 1018 s)

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?

4.1Operators 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.

Qψψ Q

Q

• This is an eigenvalue equation and can be rewritten as:

(Note: can’t be cancelled).

Examples: d/dx (eax) = a eax

d2/dx2 (sin ax) = a2 sin ax

ψQψ Q

operator Q acting on function (eigenfunction)

function multiplied by a number Q (eigenvalue)

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).

Q

Qψψ Q

xψψ x

dx

d

i

ˆ xp

ψpψ xxp ψpdx

dψx

i

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.

xT

m2

pT

2x

x

2

222

dx

d

m2m2

xx

pT

ˆˆ

T

m2

ppp

m2

pT

2z

2y

2x

2

“del-squared”

22

2

2

2

2

2

222

m2zyxm2m2

pT

ˆˆ

partial derivatives operate on (x,y,z)

V

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.

Eψψ H H

VTH ˆˆˆ

H

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

Eψψψψ VTH ˆˆˆ EψψxVx

ψ

m2 2

22

Eψψzy,x,Vz

ψ

y

ψ

x

ψ

m2 2

2

2

2

2

22

Eψψzy,x,Vψm2

22

Note: The SE is a second order differential equation

4.3 Particle in a I-D Box

System– Particle of mass m in 1-D box of length L.– Position x = 0L.– Particle cannot escape from box as PE V(x)= for x = 0, L

(walls). – PE inside box: V(x)= 0 for 0< x < L.

1-D Schrödinger Eqn.

(V = 0 inside box).

0x0 L

PE (V)

Eψx

ψ

m2 2

22

• 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).

ψkkxcosBkxsinAkx

ψ 222

2

Eψψkm2x

ψ

m22

2

2

22

m2

kE

22

Eψx

ψ

m2 2

22

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 (x<0, x>L) 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

1. x = 0 = A sin kx + B cos kx = B

= 0 B = 0

= A sin kx for all x

2. x = L = A sin kL = 0

sin kL = 0 kL = n n = 1, 2, 3, …

(n 0, or = 0 for all x)

0 1

A=0 ? (or = 0 for all x)

sin kL = 0 ?

Allowed Wavefunctions and Energies

• k is restricted to a discrete set of values: k = n/L

• Allowed wavefunctions: n = A sin(nx/L)

• Normalization: A = (2/L)

• Allowed energies:

2

22222

n2mL

πn

m2

kE

2

22

n8mL

nE

h

L

xnsinψ

L2

n

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 …

L

xnsinψ

L2

n

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 (V=0):

.....

2

2

x

ψT

KE2

2

2n

2

nnL

n

x

ψT E

Energies

• En n2/L2 En as n (more nodes in n)

En as L (shorter box)

n (or L) curvature of n

KE En

2

22

n8mL

nE

h

L

1

2 nodeE

L1

E

L2

• 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).

En

1

2

3

02

2

18mL

Eh

2

2

28mL

4E

h

2

2

38mL

9E

h

2

2

8mLZPE

h

2

2

1min8mL

EEZPEh

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

V8mL

nE

2

22

n h

E1

E2

V = 0

VE1=E1+V

E2=E2+V

V 0

Density Distribution of the Particle in the 1-D Box

• The probability of finding the particle

between x and x+dx (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).

2

2

dxL

xnsinxP 2

L2

n

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. T

V