Quantum Mechanics Brighton College 3 rd March 2008 1 Everything you always wanted to know about...

1Quantum MechanicsBrighton College 3rd March 2008

Everything you always wanted to know about Quantum Mechanics

but were too embarrassed to ask

John F. C. Turner


1. The physical problem

2. The microscopic world

3. Microscopic vs macroscopic

4. Future possibilities

5. Summary

6. Questions

Talk Structure


Newtonian mechanics is a very successful system for describing macroscopic mechanical systems.

The physical problem

'We're living in a material world and I am a material girl...'

In the Newtonian world, waves and particles are distinct.

There are trajectories, positions and velocities

Energies, velocities, momenta are continuous


Shrinking the length scale of our observation using scientific instruments does reveal a distinct difference – when our observations of the system become measurable perturbations on the system – the Dirac definition of small*

In this regime,

• Energy is transferred in discrete units, the quanta • Positions and momenta become uncertain• Trajectories are no longer useful• The world becomes statistical

and this is the world in which the microscopic mechanisms of life, the technological transformations of matter and electronic age exist.

The microscopic world

* P. A. M. Dirac, Principles of Quantum MechanicsInternational Series of Monographs on Physics, 4 th Ed.


Despite the success of the Newtonian (Maxwellian) world, there is were major

failures:

Failures of the Newtonian world

This is impossible in the classical

world…..

Quantum MechanicsBrighton College 3rd March 2008

The spectrum of any body in the classical limit should diverge at small wavelengths

Failures of the Newtonian world

4

2

Tck

=λf B

The spectrum is given

by the Rayleigh-Jeans

Law:


Max Planck’s departure from the classical was an empirical adjustment – the energy

spectrum of all photons is not continuous but is discrete.

There was no basis for this, except that it worked…

This was still a classical theory however...

Planck’s departure

4

2

Tck

=λf B

1exp

125

2

Tkhc

hc=λf

B


Newtonian mechanics and classical electrodynamics cannot describe the observed

structure of the atom.

Classically, electrons ‘in orbit’ around the nucleus will lose energy through

radiation – the electron is accelerating as it travels around the orbit – and should

slowly descend and ‘land’ on the nucleus.

These findings, between ~ 1900 – 1930 stimulated the development of quantum

mechanics, which does account for the observed structure of the atom.

The Newtonian world and the classical atom


“Quantum world”

Objects are profoundly affected by the physics of observation.

No trajectories No positions No velocity

Calculations of properties are probabilistic, not certain.

Energy is transferred in discrete packets or quanta

Many other properties are no longer continuous but are also discrete

Applies to small objects only – those that are affected by the act of observation

The quantum world and the quantum atom


The atom is a quantum system.

Louis de Broglie formulated that a particle of

momentum p has an associated wavelength, λ, where

and p=mv

De Broglie relationship relates the motion of a

particle to wave-like properties and the electron in

an atom behaves as a wave. Standing waves have

fixed energies

violin strings

organ pipes

The quantum atom

h=pλ

Electrons in the atom


Atomic Structure

The first model of the atom was the Bohr model; it is incorrect but is still shown as

the model of the atom today.

It is the model in which electrons orbit the nucleus in a similar way that planets

orbit the sun.

The strong central and radial force is provided by the electric force between the

nucleus and the electron.

The energies of the electrons are empirically quantized so that the embarrassment of

the classical atom is avoided.

The energy of the atom in the absence of a field is given by

Where n is the ‘Principle’ quantum number

2n

B=E


Atomic Structure

It is the model in which electrons

orbit the nucleus in a similar way

that planets orbit the sun.


Atomic Structure

Erwin Schrödinger improved on the Bohr description and succeeded in explaining

the internal dynamics of the atom, which occur when atoms are examined

spectroscopically in the presence of a field.

The Schrödinger description is based on the wave properties of matter, detailed by

Louis de Broglie.


Atomic Structure

The solutions of the Schrödinger wave equations are called wavefunctions and have

discrete energies. The solutions are complicated – the Schrödinger equation is

Where

This equation is insoluble when written in terms of the Cartesian coordinates x, y, z.

When we allow the natural symmetry of the atom to dictate our approach, it

becomes soluble

Eψ=ψH

Eψψz)y,V(x,zyx

2

2

2

2

2

22

2m


Spherical Coordinates


The atom is spherical and there are

three important coordinates

Coordinates and waves

Each coordinate of the electron-wave has a label attached to it – the quantum

number.

r

The distance between the electron

and the nucleus - r

The 'longitude' - φ

The 'latitude' - θ

The electron-wave is defined by

all three coordinates.


Atomic Structure

The solutions of the Schrödinger wave

equations are called wavefunctions and have

discrete energies.

The solutions are products of three functions

The equations are only soluble for the hydrogen

atom and give the shapes of the orbitals – the

space in which the electron can be found as

well as the energies.

Each of these functions has an associated

quantum number.

φΦΘrR=ψ

r


Atomic Structure

By considering the electron in an atom as a wave, the energy of the electron

becomes quantized and gives the correct energy relation that Bohr described

empirically by

and we term n as the principle quantum number.

We associate n with the radial function

E=− B

n2

r


Atomic Structure

Classically a particle on a sphere can also move over the surface and this motion

is circular. In a similar way, the electron in an atom has properties that we can

associate with circular or angular motion.

Electrons in an atom have angular momentum – the momentum that is associated

with angular motion


Atomic Structure

Classically a particle on a sphere can also move over the surface and this motion

is circular. In a similar way, the electron in an atom has properties that we can

associate with circular or angular motion.

Electrons in an atom have angular momentum – the momentum that is associated

with angular motion

The angular motion is described by two

quantum numbers – l and ml termed the

angular quantum number and the

magnetic quantum number respectively.


Atomic Structure

The principle quantum number defines the energy of the electron.

The angular quantum numbers define the shape of the region of space in which

the electron is confined – these are termed the orbitals of the atom and they have

definite shapes:


Atomic Structure

n = 1, l = 0


Atomic Structure

n = 2, l = 0


Atomic Structure

n = 2, l = 1


Atomic Structure

n = 3, l = 0


Atomic Structure

n = 3, l = 1


Atomic Structure

n = 3, l = 2


Atomic Structure

n = 4, l = 0


Atomic Structure

n = 4, l = 1


Atomic Structure

n = 4, l = 2n = 4, l = 1


n = 4, l = 3

Atomic Structure


There are few if any divisions in the description of matter in our experiential world


Although there are differences in behaviour, due to mass and other factors, mostly, ice is ice.......


Important numbers in this regime are:

Avogadro's Number

Planck's Constant

These are enormously large and extremely small:

The population of the world today* is estimated to be

requiring Earths for a 'mole of people'


* http://www.census.gov/ipc/www/popclockworld.html

N A = 6.023 × 1023

h = 6.626 × 10−34 J s−1

6.654078261 × 109

~ 9.05 × 1013

1 mole of water occupies ~ 18 mL


Despite the statistical and quantum mechanical description of the microscopic world, we still require a macroscopic model that describes our sensible observations.

It cannot be 'mechanical' in the Newtonian sense and we require a way of describing spontaneous change simply because it is in these systems that the 'cleanest' display of principles is observed.

A spontaneous process is one that takes place without any influence external to the system.

The opposite of a spontaneous change is a non-spontaneous change – one where there must be an external influence to force the change.

For any observed, spontaneous change, the reverse process is non-spontaneous.

Microscopic vs macroscopic


If we use energy as the sole criterion of spontaneous change, we are using effectively a mechanical analogy – systems move to the local minimum in energy as the point of equilibrium.

In the example of a ball falling, we have one variable, position, in a field, the gravitational field of the earth, and there is no endothermic path.

Potential energy is converted into kinetic energy in flight and then into heat and sound (q and w) at impact.



In this case, minimization of the potential energy and conversion ultimately into heat defines the point of equilibrium and appears to be linked to the direction of spontaneous change. Early theories of chemical thermodynamics rested on the evolution of heat as the 'driving force' for a reaction, which fails.



It is certainly true that many reactions that are spontaneous are accompanied by the evolution of heat:

But some are not and are endothermic and yet are still spontaneous:

So, in the 'mechanical' view of chemical change, in the case of ammonium chloride, we have a system that spontaneously moves to a higher state of energy.

Physical changes such as melting and boiling are inherently endothermic and the same problem occurs.


NaOHs H2O

Naaq+ OHaq

- H = −44.51 kJ mol−1

NH4Cls H2O

NH4aq+ Claq

- H = 14.78.51 kJ mol−1


Both of these examples have an associated enthalpy change:


NaOHs H2O

Naaq+ OHaq

- H = −44.51 kJ mol−1

NH4Cls H2O

NH4aq+ Claq

- H = 14.78.51 kJ mol−1

q

T1 ≫ T2

T1 = T2

Some processes do not result in a net change of energy in the system and are still spontaneous:

In an insulated system that cannot exchange heat with the surroundings, heat will spontaneously move to equalize a temperature gradient between two bodies.


Similarly, there are no forces between a particles of a perfect gas and the internal energy of the perfect gas is independent of volume. Yet a perfect gas will always expand to fill the volume available, with no net change in energy.



So spontaneous changes can take place endothermically, exothermically or with no exchange of energy


q

T1 ≫ T2

T1 = T2

NaOHs H2O

Naaq+ OHaq

- H = −44.51 kJ mol−1

NH4Cls H2O

NH4aq+ Claq

- H = 14.78.51 kJ mol−1


Our descriptions of reactions and other chemical changes are on the basis of exothermicity or endothermicity

As a description of changes in heat content and work, these are adequate but they do not describe whether a process is spontaneous or not.

There are endothermic processes that are spontaneous – evaporation of water, the dissolution of ammonium chloride in water, the melting of ice and so on.

We need a thermodynamic description of spontaneous processes in order to fully describe a chemical system



The magnitude and sign of the change in enthalpy associated with a chemical or physical change does not reflect the spontaneity of the process. It is not a good measure.

However, any description of a molecular system such as a mole of a perfect gas is inherently statistical:

A mole contains particles and the number of ways that we can arrange a mole of particles is going to be of the order of

There are therefore many ways of describing a chemical system while conserving the observed macroscopic properties – internal energy, pressure, temperature etc.


6.023 × 1023

Number of ways = W ~ 101023


If we have a large number of ways of describing the inside of the system so that the outside stays the same, then we have a large number of ways of distributing the energy of the system amongst these different configurations.

There are many equivalent ways of distributing the thermal energy of a system given a certain macroscopic energy.

A change is spontaneous when the number of ways of distributing the energy in the universe increases.

The point of equilibrium occurs when this number of ways is maximized.



We name the property of the distribution of energy the entropy and we will refer to

• the entropy of the system – the bit of the universe that holds our local interest

• the entropy of the surroundings – all the rest......

Together, these sum to give the entropy of the Universe.


Ludwig Boltzmann quantified the entropy in terms of the number of ways of describing the system as

S = kBlnW



S = kBlnW


Any system above 0 K contains energy and can in principle do work with certain limitations. This energy is stored in modes inside the system; these modes can be translational, rotational, vibrational and electronic.

The type of modes, the number of them and the energy separations between individual modes depends critically on the system concerned and the state of matter.


ni

N~

exp{−Ei

kBT }∑

j

exp{−E j

kBT }

The availability of the modes depends on the temperature and is governed by the Boltzmann distribution. E

i is the energy of the particular mode and

ni is the population that is present in that mode. N is the

total number of particles in the system.



The population in each state is controlled by the energy spacing and the temperature

ni

N~

exp{−Ei

kBT }∑

j

exp{−E j

kBT }

Ei = 10 K

Ei = 100 K

Ei = 1000 K

Ei = 10000 K


Given the definition of the 'statistical' entropy and the distribution of particles over the energy states present in the system, with some mathematical elbow grease we can derive the thermodynamic entropy:

which allows us to determine the entropy on transfer of heat.


S = kBlnW S =qrevT


In order to predict the direction of spontaneous change, we need to consider the total entropy change in the universe.

We write this as

from our definition of entropy. We know that the heat change in the system is equivalent to the opposite of the heat change in the surroundings:

and we know, that for a system that can do work, qSystem

=H


SUniverse = SSurroundings SSystem

SUniverse =qSurroundings

T

qSystem

T

qSurroundings

T= −

qSystem

T


Now we can write the change in the universe solely in terms of changes in the system.

This is important because the system is the part of the universe that we know enough about for an accurate description in principle.

The entropy then becomes

We define

where G is the Gibbs function.


SUniverse = −HT SSystem

TSUniverse = −H TSSystem

−TSUniverse = H − TSSystem

G = H − TSSystem

qSurroundings

T= −

HT


The Gibbs function is a disguised form of entropy and has the units of energy; it is not an energy term in the First Law sense (H, U etc) but is a measure of the change in entropy of the universe.

For a spontaneous change,

i.e. the change in the Gibbs function must be zero or less than zero for a spontaneous change.

The Gibbs function allows us to define quantitatively the direction of spontaneous change in the universe – it allows us to determine what will take place amongst all the energetically possible changes allowed by the First Law


G 0


All conceivable chemical changes

The Gibbs function allows us to draw a 'map' of chemical change for the universe:

All possible chemical changes allowed by the First Law

All observed, spontaneous chemical changes allowed by the Second Law

You are here



Open questions:

There is no necessary requirement for the direction of change – can we find one?

How does entropy relate to the initial and final states of the Universe?

What happens when the size of the surroundings is not very much larger than the size of the system?

Future possibilities


The Universe is not mechanical

Spontaneous change concerns the change in the distribution of energy and nothing else

Entropy is straightforward and there is still much to learn

Conclusions

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