Physics AT3‐Part A The development of the quantum

Physics AT3‐Part A

The development of the quantum mechanical atomic theory – Bohr, Heisenberg and Pauli Research and Report: Secondary sources

Physics AT3 Part A‐ Research and Report: Secondary Sources

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Heisenburg/Pauli – Development of the Atomic Theory Contents Introduction ............................................................................................................................................ 2

Rutherford‐Bohr Model .......................................................................................................................... 3

A Brief History of the Atom ................................................................................................................. 3

Rutherford’s Model ............................................................................................................................. 4

The Bohr Model‐Structure, Postulates and Success ........................................................................... 6

The Rutherford‐Bohr Model‐Limitations .......................................................................................... 12

Werner Heisenberg ............................................................................................................................... 14

Brief History ...................................................................................................................................... 14

Contributions to the Atomic model .................................................................................................. 14

Impact of Heisenberg’s contribution to the atomic model .............................................................. 17

Impact on Society .............................................................................................................................. 18

Wolfgang Pauli ...................................................................................................................................... 20

Brief History ...................................................................................................................................... 20

Contributions to the Atomic Model .................................................................................................. 20

Impact of Pauli’s contribution on the atomic model ........................................................................ 22

Impact on Society .............................................................................................................................. 22

Future possible developments .............................................................................................................. 23

The Quark model/The Standard Model ............................................................................................ 23

Grand Unified Theories (GUTs) ......................................................................................................... 25

String Theory and Supersymmetry ................................................................................................... 25

Conclusion ............................................................................................................................................. 26

Bibliography .......................................................................................................................................... 27

Appendix‐Common Constants and Variables ....................................................................................... 28

Constants .......................................................................................................................................... 28

Variables............................................................................................................................................ 28


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Figure 1: Bohr, Heisenberg and Pauli in conversation (Source: http://history.aip.org)

Introduction The atomic model is perhaps one of the most intriguing concepts in modern physics, and a study of

its history is equally fascinating. From Democritus to Bohr, to Heisenberg and Pauli, and then the

Standard model. This report explores the earlier models of the atom, focusing on the development

of the Rutherford‐Bohr Model. A close study of the contributions of Werner Heisenberg and

Wolfgang Pauli will then be made, including the subsequent effects on society. Finally a summary of

the current understanding of the atom will be made, along with various avenues for future

development.


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Rutherford‐Bohr Model

A Brief History of the Atom The Rutherford‐Bohr model was a culmination of centuries of speculation on the structure of the

atom. Democritus (460‐370 BC) was the first to develop an atomic theory of atoms (Democritus, n.d.,

para 3):

1. All matter consists of indivisible particles called atoms

2. Atoms are indestructible

3. Atoms are solid but invisible

4. Atoms are homogenous

5. Atoms differ in size, shape, mass, position and arrangement.

John Dalton (1766‐1844) was the next to propose a significant atomic theory, with the following

postulates (Warren, 2008):

1. Matter is composed of small indivisible atoms

2. Elements contain only one type of atom; Different elements contain different atoms

3. Compounds contain more than one type of atom.

Up until this point, it was accepted that the atom is a fundamental particle. That is, they have no

measurable internal structure; they are not composed of other particles. This theory was dramatically

changed when Sir Joseph John Thomson determined the charge to mass ratio of a cathode ray particle.

The ‘electron’ as it was subsequently termed, had a charge to mass ratio of:

1.76 10 .

This ratio was around 1800 times larger than that of a hydrogen ion (a proton), and was the

same no matter what metal was used for the cathode, which suggested that the electron was a

fundamental particle which was contained in all atoms. J.J Thomson subsequently established his

‘plum‐pudding’ model, which consisted of electrons embedded in a mass of positive material with a

diameter of approximately 10‐10 m, to produce an overall negative charge, as seen in the figure below

(Bohr's Model of the Atom, n.d.):

Figure 2: Thomson's plum pudding model of the atom ((Source: http://www.quarkology.com)


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Rutherford’s Model Ernest Rutherford (1871‐1937) then established an atomic

model which contained a nucleus. This was a very high density region in

the centre of the atom, as opposed to Thomson’s low density positive

‘pudding’. Rutherford’s students, Geiger and Marsden conducted an

experiment in 1904 (Bohr's Model of the Atom, n.d.), which involved

bombarding a piece of thin gold foil with alpha particle (helium nuclei‐

two protons, two neutrons, charge of +2 C) from a radon source. The

particles could then be detected by a scintillation (fluorescent) screen.

(See diagram 1).

It was expected that the positive alpha particles would pass straight through the foil, with

some deflections due to interactions with the atoms in the foil. Multiple interactions produced various

deflections, even up to 90o although this was rare. In the actual experiment, it was observed that 1 in

every 8000 alpha particles was reflected (deflected by an angle greater than 90o. The probability of

multiple scatterings producing this reflection was so minute that Rutherford concluded that it had be

a result of an encounter with a single atom. Once this was established, Rutherford deduced that the

charge in that single atom must have been concentrated to a region that was 10 000 times smaller

than the radius of the atom. The alpha particles were known to have a velocity of approximately 1.6 x

107 ms‐1, and would penetrate the atom to within 3 x 10‐12 cm of its centre, before being turned back.

Rutherford then concluded that the charge must be situated in a very concentrated area in the centre

of the atom, known as the nucleus (Warren, 2008). The nucleus was predicted to be approximately

Figure 3: Ernest Rutherford (Source: http://www.nobelprize.org

Diagram 1: Rutherford's Experiment


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10‐15m1 across, and Einstein’s analysis of Brownian motion had predicted the diameter of an atom to

be 10‐10m across. This meant that the atom was mostly empty space, as seen in the diagram of

Rutherford/s model below. (Note that diagram 2 is not to scale. This would not be practical, as the

ratio of the diameter of the atom to the diameter of the nucleus is about 2.5 x 105.)

Diagram 2: Rutherford's Model

Rutherford’s model still contained several failings, such not being able to explain what the

nucleus consisted of or how the orbits of electrons are arranged around the nucleus. Another

significant limitation involved the energy of the orbiting electrons. When a charged particle

accelerates, or changes direction, it radiates energy, and an orbiting electron should theoretically

continuously lose energy and spiral into the nucleus. The spectral lines of atoms could also not be

explained.

1 Equal to one Femtometre, sometimes called a Fermi, written fm.


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The Bohr Model‐Structure, Postulates and Success Niels Bohr (1885‐1962) was a Danish Physicist who was the first

Physicist to introduce the new theory of quantum mechanics into his

atomic model, which was proposed in 1915. He first worked in at the

Cavendish laboratory with J.J. Thomson and then for Rutherford at the

University of Manchester. His model was a modification of the

Rutherford model, and is often called the Rutherford‐Bohr model. Unlike

the experimental physicists Thomson and Rutherford, Bohr was a

theoretical physicist.

Bohr recognised the problem of having electrons circle the

nucleus, as they would emit what was known as bremsstrahlung or

‘braking radiation’. As they accelerated in the electric field of the

nucleus, they would emit electromagnetic radiation, and their kinetic

energy would decrease. As their kinetic energy decreased, they would slow down and fall into a lower

orbit, spiralling into the nucleus. To address the limitations of the Rutherford model, Bohr announced

three postulates (Bosi, et al., 2010), (Andriessen, Pentland, McKay, Tacon, & Morante, 2003):

1. Electrons exist in stable orbits. An electron can exist in any of several special circular orbits

with no emission of radiation. These orbits are called stationary states, and any permanent

change in their motion must consist of a complete transition from one stationary state to

another.

2. Electrons absorb or emit specific quanta of energy when they transition between stationary

states (orbits). In contradiction to classical electromagnetic theory, a sudden transition

between two stationary states will produce an emission or absorption of quantised radiation

(a photon), described by the Planck‐Einstein relation:

h ∆

Where h=Planck’s constant, and E1 and E2 are values of the stationary states that form the

initial and final states of the atom.

3. Angular momentum1 of electrons is quantised. An electron in a stationary state (orbit) has a

quantised angular momentum that can only take values of where n is the principal

quantum number.

The frequent mention of quanta was part of the new avenue of physics being developed at the time,

quantum mechanics. The origination of quantum mechanics should be outlined briefly, as it will be

referred to and built on later.

Max Planck (1858‐1947) was the first to introduce the concept of quantised values. This was

originally a mathematical convenience to explain phenomena related to the intensity of radiation

emission. To create a black body radiation curve to match that found in experiments, a radiation

emitter was assumed by Planck to have the following properties (Bosi, et al., 2010):

1. A radiation emitter can only have energies E given by

1 The rotational equivalent of linear momentum, where the angular momentum L, of a point mass m, in a circular motion of radius r with velocity v, is given by

Figure 4: Niels Bohr (Source: http://www.nobelprize.org)


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where f is the emitted frequency in Hertz, h is the Planck’s constant (6.626 x 10‐34 J s) and n

is an integer. The energy E is measured in Joules.

2. An emitter can absorb or radiate energy in ‘jumps’ or quanta. Two consecutive energy states

differ by hf, as governed by n being an integer value.

Classical theory states that an object can have any energy, and these assumptions were

consequently so radical that Planck himself heatedly resisted the theory, which has become one of

the two pillars of modern physics, the other being relativity.

Einstein subsequently built on this theory, and suggested in 1905 that electromagnetic radiation

occurred in small packets, or photons. The properties of these photons are still however expressed

in wave form:

Where c is the speed of light (c=3.00 x 108 ms‐1), f is the frequency in Hertz and λ is the wavelength in

m. Einstein assumed that the energy E of each photon is related to the frequency f by the equation:

E being the energy in Joules, h being Planck’s constant (6.626 x 10‐34 J s), c being the speed of light

and λ the wavelength in m. The experimental evidence, such as blackbody radiation curves and the

photoelectric effect, confirmed Planck’s assumptions, and this period marked a new path away from

the classical Physics. Quantum mechanics had become the science of the very small, and classical

had become restricted to the science of the very large.

Bohr’s modifications of Rutherford’s model had made it into a hybrid model that spanned

both classical and quantum theories, and proved to be extremely successful in explaining many

experimental observations. The basic structure of the Rutherford‐Bohr atomic model can be seen in

diagram 3:

Diagram 3: Bohr's model of the atom


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Bohr’s model did not just consist of postulates and a graphical/diagrammatic model, it was also

backed up with significant mathematical framework. To display this success, emission spectra, the

Balmer series, and the revised Balmer equation must be outlined.

When white light is passed through a prism, it can be broken into its constituent colours, and is

known as a continuous spectrum, as it consist of a continuous spread of wavelengths/frequencies.

As seen in Figure 4, there are other types of spectra that can be produced. The radiation from a hot

gas can be analysed with a spectroscope, and bright lines will appear. If white light is passed through

a cold gas, dark lines will appear at the frequencies which have been absorbed. Each element has a

‘signature spectra’.

The emission spectra of hydrogen had been extensively studied before Bohr’s time, and in

1885, a Swiss schoolteacher named Johann Balmer (1825‐1898) derived an empirical formula1 which

gave the wavelengths of the emission lines for hydrogen:

2

Where b was a constant found empirically to be 364.56 nm and n is an integer related to a particular

spectral line, as shown in the table below:

1 A formula derived purely from experiments with no theoretical basis (Warren, 2008)

Figure 5: Different Spectra (Source: http://www.quarkology.com)

Figure 6: Emission Spectra of Hydrogen (Source: http://www.quarkology.com)


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Name of Line n λ (nm) Colour

Hα 3 656.2 Red

Hβ 4 486.1 Green

Hγ 5 434.0 Blue

Hδ 6 410.13 Violet Table 1: Balmer Series for Hydrogen (Warren, 2008)

Janne Rydberg modified Balmer’s equation to look like this:

1 1 1

Where RH is the Rydberg constant (RH=1.097 x 107 m‐1), λ is the wavelength of the emitted radiation,

nf is an integer equal to two, ni is an integer equal to 3, 4, 5, 6…

One of the greatest successes of the Bohr model, was its ability to theoretically produce this

equation, which had been previously only derived on an empirical basis. The mathematical workings

will now be demonstrated to show how the Balmer equation was theoretically replicated.

Firstly, the total ‘classical’ energy of the Rutherford hydrogen atom will be calculated. The

electrical potential energy of a proton‐electron system must be greater than the kinetic energy of

the electron, if the they are to remain as a bound system. Adding together these two quantities will

give the total energy of the system. (Mathematical progression as per (Warren, 2008)) Note: for ease

of progression, the values for common variables are given in the Appendix.

Kinetic energy of electron:

12

Electrical force on electron:

(qe is the charge on proton and electron (1.602 x 10‐19 C))

This electrical force provides the centripetal force of magnitude:

12

12

12

12


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Potential energy of electron is given by:

Total energy is the sum of potential and kinetic energies:

12

12

Remember: This is the total ‘classical’ energy of Rutherford’s Hydrogen atom.

Bohr’s ‘quantisation’ postulate however (as outlined), restricts the electron to stationary

states in which the angular momentum of the electron to integer multiples:

h2

In this equation, n is an integer known as the principle quantum number1. An expression of the

radius of the stationary states corresponding to each integer value n can be found:

h2

h2πm

4

From the earlier equation, , an expression for v2:

This can then be substituted:

h

4π mkqm r

h4π m kq

Where rn is the radius of the stationary state corresponding to the integer n.

The radius of the stationary state corresponding to n=1 will be:

h4π m kq

1 This will be discussed in later sections


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And the expressions for rn and r1 can be combined to give:

If we now return to the classical energy of the Rutherford Hydrogen atom ( ) and impose

the restriction that the only possible energies are those that correspond to values of radius

π, which then allows the calculation of the energy states:

12 h4π m kq

124

1 2h

It can again be seen the first energy state is equal to:

2h

And again, En and E1 combined give:

1

Now, this expression can be combined with Bohr’s second postulate to derive an expression for the

difference in energy between the two stationary states, and hence the energy of the photon emitted

or absorbed. If an electron jumps from an initial state, Ei to a different final state, Ef, the change in

energy in the electron is:

∆

1 1

1 1

This is the energy of the emitted photon, hf.

It is now possible to derive an equation for the frequency and wavelength of the photon:

1 1

1 1


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1 1

1hc

1 1

This bears the same form as Balmer’s equation, and when calculated, is equal to the Rydberg

constant, remembering E1 is negative. The Bohr model also predicted the correct value of the ground

state1 energy of the Hydrogen atom, ‐13.6 eV2 (Giancoli, 2005).

The ability of the Bohr model to theoretically replicate an empirical formula was a major

achievement and offered very strong support for the Bohr model. The largest effect this model had

on Physics was to question the validity of classical theory in regards to the miniscule world of the

atom. Bohr predicted that the new quantum theory did not invalidate classical theories. If the

energies for orbits with principle quantum number n=1,000,000 and n=1,000,001 are calculated,

along with their radius, they would be seen to be approaching the macroscopic scale, and the

difference in energies would be barely perceptible. This shows that the classical theory is quite

applicable in ‘normal’ situations, but an alternative theory must be adopted for the tiny values

associated with individual atoms. Another example of this is Einstein’s relativity, which is only used

for speeds approaching c. In situations ‘normal’ speeds (v << c), the relativity values are very nearly

identical to the classical ones. The insistence that a more general theory such as quantum mechanics

give the same results as a more restricted theory such as classical theory (which only works with

macroscopic dimensions) is called the correspondence principle. The two theories must correspond

where their realms of validity overlap. Therefore, classical mechanics is not contradicted by quantum

theory, it is just relevant to a more restricted area.

The Rutherford‐Bohr Model‐Limitations The Rutherford Bohr model was not, however, entirely successful. Its hybrid nature and inability to

explain certain phenomena meant that further development was still required. Some of the major

occurrences unable to be explained were:

The spectra of larger atoms, as the Bohr model could only successfully explain the Hydrogen

atom, and could not accurately calculate the spectral lines of larger atoms. This was later

found to be a result of the interactions between the multiple electrons of the atom.

The relative intensity of brightness of spectra lines could not be explained, as they suggested

that some transitions were more favoured than others. This was later shown to be a result

of certain quantum laws.

The hyperfine structure of spectra lines, where several fine lines were clustered around the

main spectral line, could also not be predicted, even for Hydrogen. Later found to be a result

of the interaction of angular momentum.

The Zeeman effect, discovered by Pieter Zeeman, involved a spectral line splitting into three

or more lines when the emission body was subjected to a magnetic field. A similar result was

found when an electric field is applied, and is known as the Stark effect. Later shown to be a

result of the tiny magnetic moment of an atom.

1 The stationary orbit with the lowest orbit and principle quantum number n=1 2 1 eV(electron volt) = 1.602 x 10‐19 J, and is the amount of energy gained by an electron passing through a potential difference of 1 Volt.


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The ad hoc nature of the Bohr theory, which made assumptions to make theory agree with

experiment, still left the model as an ambiguous theory awaiting further proof and development.

That development came in the form of a complete quantum mechanics model, through

scientists such as Louis de Broglie (1892‐1987), Erwin Schrödinger (1887‐1961), Werner Heisenberg

(1901‐1976) and Wolfgang Pauli (1900‐1958).

Louis de Broglie was the first to build onto Bohr’s theory and to theoretically prove why

stationary states existed. He extended a theory known as wave‐particle duality, which was originally

theorised to explain the dual nature of a photon. De Broglie applied the concept to matter as well,

and stated that the wavelength of a particle would be related to its momentum in the same way as a

photon:

Where h is Planck’s constant, p is the linear momentum mv and λ is the wavelength in m.

De Broglie then applied this theory to the electrons orbiting a nucleus. He predicted that an electron

in orbit around a nucleus is matter wave that consists of a circular standing wave (see Below). The

circumference of a Bohr orbit of radius rn is 2πrn, and so:

2 , 1,2,3, …

And if de Broglie’s formula ( / ) is substituted in:

2

2

Which is the quantum condition proposed by Bohr, on an ad hoc basis, and was the first explanation

for the quantised orbits and energy levels, and implied that the wave‐particle duality is part of all

atomic structure (Giancoli, 2005).

The later discoveries of Heisenberg and Pauli proved even greater advances, and will be

covered in the following sections.


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Werner Heisenberg

Brief History Werner Heisenberg was born on the 5th

December 1901 in Wurzburg, Germany, and

died on the 1st of February 1976 from cancer.

He attended the University of Munich to

study Physics. He then went to Gottingen

with Max Born (1882‐1970) to study, then to

the Institute of Theoretical Physics in

Copenhagen with Niels Bohr.

Contributions to the Atomic model Heisenberg’s two greatest contributions to

the arena of quantum mechanics were the

development of a complete quantum theory,

otherwise known as quantum mechanics, and

the uncertainty principle.

In 1924, when Heisenberg moved to

Copenhagen, it was widely accepted by

physicists that a new quantum mechanics

theory was needed to supersede the

limitations of the quantum theory.

Heisenberg’s contributed to the first complete quantum theory of the atom, which was described

mathematically through a complex system of matrices1By September 1925, Max Born, Pascual

Jordan and Heisenberg had completed a paper titled “Zur Quantenmechanik II” (“On Quantum

Mechanics II”2), subsequently becoming known as the “three man paper”, and is regarded as the

foundational document of a new quantum mechanics (Brittanica, 2015). This theory when equated

by Erwin Schrodinger (1887‐1961) to his own wave function theory, introduced the concept of

probability to quantum mechanics as opposed to the determinist view of classical mechanics. This

will be discussed further in a later section. The complex matrix mechanics developed by Heisenberg

solved the issues of hyperfine spectral structure, varying intensity of spectral lines and the Zeeman

effect (He was forced to introduce half‐integer principal quantum numbers to do so, which

conflicted with Bohr’s model. The concept of quantum numbers will be discussed later in regards to

Pauli, who explained these effects more correctly).

Heisenberg is perhaps most famous for his uncertainty principle, which he articulated within

his March 1927 paper, “Uber den anschulichen Inhalt der quantentheoretischen Kinematik und

Mechanik” (“On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics”)

(Brittanica, 2015). Heisenberg did not agree with the circular orbits of Bohr’s model. He argued that it

was not possible to predict the exact position of electrons at any point in time, reflecting the move

towards quantum mechanics. To describe the uncertainty principle, we will envisage an electron

moving independently through empty space. Physicists would describe the electron with four

properties: momentum, position, energy and time. These four properties are grouped into two pairs

known as ‘canonically conjugate’ variables. The classical theory held up until Bohr’s time assumed that

1 Plural of matrix, regarding matrix algebra 2 The first version of the paper was assembled by Born and Pascual only

Figure 7: Werner Heisenberg (Source:https://the‐history‐of‐the‐atom.wikispaces.com)


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the precision of an experiment relied solely on the accuracy and sensitivity of the measuring

instruments used, but Heisenberg theorised that quantum mechanics limits the precision possible

when measuring two canonically conjugate variables at a time. The two pairs are momentum and

position, energy and time, and Heisenberg developed a thought experiment for both. Outlined below

is Heisenberg’s “Gamma‐ray microscope” thought‐experiment.

Diagram 4: The 'Gamma‐Ray Microscope' Thought experiment

Heisenberg pictured a microscope, which obtained very high resolution through a use of high‐energy

gamma rays for illumination. While no such microscope has been constructed, it would be possible to

construct in principle. As in the diagram above, an electron sits beneath the centre of the microscope

lens, which form a cone of angle 2θ from the electron. Gamma rays (the highest energy

electromagnetic waves with a tiny wavelength, <10 pm) are emitted from a source to the left of the

electron. The microscope can then ‘see’ objects down to the size ∆x, which is related to the wavelength

A through the expression:

∆2 sin

Where L is the wavelength of the gamma ray in m, and is half the angle of the cone formed from the

edges of the lens to the electron, as shown in the above diagram. In quantum mechanics, where light

acts as a particle (as mentioned above) the gamma ray striking the electron will give it a ‘kick’. As the

ray is diffracted by the electron into the microscope lens, the electron is thrust to the right. The total

momentum of the gamma ray is given by the formula:

h


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Where p is the momentum, h is Planck’s constant and L is the wavelength. If the gamma ray is

diffracted to the right edge of the lens, the total momentum in the x direction would be the sum of

the electron’s momentum p’x in the x‐direction and the gamma rays momentum in the x direction: sin

Where px is the total momentum, p’x is the momentum of the electron (along the x‐axis), and L’ is the

wavelength of the deflected gamma ray. If the ray is diffracted in the other extreme, and it strikes the

left edge of the lens, the total momentum in the x‐direction, where p”x is the momentum of the

electron:

sin

Where L” is the wavelength of the diffracted gamma ray.

As momentum is conserved, these two final momentums must be equal to the initial momentum, and

therefore to each other:

sin sin

If is small, then the wavelengths of the diffracted rays are similar, ( ′~1 "~ ) and therefore:

∆2 sin

Since ∆ , a reciprocal relationship can be obtained between the minimum uncertainty in the

measured position, ∆ , along the x‐axis, and the uncertainty in its momentum, ∆ , in the x‐direction.

That is:

∆∆

or ∆ ∆ h

To incorporate more than minimum uncertainty, the “greater than” sign is added. This forms the

basic shape of Heisenberg’s uncertainty relationship, a more careful derivation produces:

∆ ∆ ≳2

This microscope demonstration is not fully valid, as all experiments have proven there to be no ‘real’

microscopic interaction between the photon and the electron, but it is useful to demonstrate

Heisenberg’s relation. The actual result is derived working through the formal mathematics, which

calculates probabilities for abstract quantum states.

Note that his does not limit individual precise measurements, but as the standard deviation (scope for

error) in one value approaches zero, the standard deviation for the other value approaches infinity.

A simplified derivation of the uncertainty/indeterminacy relation between energy and time is as

follows. As above, the object to be detected has an uncertainty in position ∆ . The photon that

detects the object travels at speed c, and takes a time ∆∆

to travel through the distance of

1 Note: ~ means ‘approximately’


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uncertainty. Hence, the uncertainty in the measured time when the object is at given position is

about:

∆

The photon can contribute some or all of its energy ( ) to the object, and the resulting

uncertainty in the energy of the object as a result is:

∆

The product of the two uncertainties is then:

∆ ∆

Again, when the formal mathematics is used, Heisenberg’s actual result is derived:

∆ ∆ ≳2

So Heisenberg’s uncertainty principles regarding canonically conjugate values are as follows:

∆ ∆ ≳ 1and ∆ ∆ ≳

Impact of Heisenberg’s contribution to the atomic model There was profound consequences linked to the introduction of probability to the atomic

model and quantum mechanics. The link between Heisenberg’s matrix mechanics and Schrodinger’s

equation was mentioned briefly earlier. That relationship will be made clearer now.

Independently of Heisenberg’s matrix mechanics, Erwin Schrodinger had developed a wave

theory to mathematically portray quantum mechanics. Schrodinger disliked the probabilistic features

of the matrix mechanics and was trying to reconcile classical and quantum theory. He thought that he

was representing the wave nature of matter with his ‘matter waves’. The ‘Schrodinger equation’ is as

follows (Bosi, et al., 2010):

2

The wave function produced by this equation contains all the measurable information about

a particle. It was, therefore, a great disappointment to Schrodinger when Max Born, who had worked

with Bohr on developing his matrix mechanics, was the first to interpret its true nature. Born realised

that if the function described a collection of many electrons, then at any point would be

proportional to the number of electrons expected to be found at that point. The disappointment to

Schrodinger was that his wave function was not a determinist theory. This is, as Heisenberg’s

interpretations were, a consequence of the wave‐particle duality of matter. If we treat electrons, for

example on a wave basis, then describes the amplitude over time, whereas if we treat them as a

particle, then represents the probability of finding the particle at a given point.

The classical theories of Newtonian physics are known as deterministic, as they can be used to

determine the position of an object at any point in the future. For example, if you shoot an arrow

several times, with the exact same forces on it, the final position of the arrow can be predicted if the

1 The symbol is known as h‐bar, and represents the value , because it occurs so often in Physics.


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forces are known, and the arrow will strike that predicted final position every time. When it was

developed, the quantum mechanics theory was completely radical, as it stated that you could not

even know the position and velocity of an object precisely, at the same time. As above, this is linked

to the fact that matter is not simple waves or particles, but is both. The resulting proposition of

quantum mechanics is that there is some inherent unpredictability in nature. Neils Bohr was an avid

supporter of this aspect of the quantum theory, and aspects of his arguments were touched on in the

section regarding his model. The interpretation of the probabilistic nature of quantum mechanics and

the deterministic nature of classical mechanics developed by Bohr and other prominent physicists

became known as the ‘Copenhagen interpretation’ in honour of Bohr.

This began, as above, with Heisenberg’s uncertainty principle. The determination of future

events through the knowledge of present values (respecting a particle or object) is known as causality.

Heisenberg stated that the uncertainty principle denies causality, because since present values cannot

be precisely determined simultaneously, then the future of that particle cannot be determined.

The Copenhagen interpretation states that even ordinary sized objects are subject to

probability, as opposed to determinism. Quantum mechanics predicts that there is an extremely high

probability that an object will obey classical laws of physics. Referring to the above analogy, this does

have the potential consequence that there is a negligibly small probability that the arrow will suddenly

curve upwards, rather than following a parabolic arc. These quantum laws, however, apply to

individual particles, and as the situation is made macroscopic, with vast quantities of particles, the

probability of the object not obeying classical laws is effectively reduced to zero. This gives rise to the

apparent determinism. It is only when very small numbers of particles are present that these

probabilistic effects are relevant. There is therefore no deterministic principles in quantum mechanics,

only statistical laws based on probability.

It should also be pointed out that these probabilities are different to those associated with

thermodynamics and the behaviour of gases. The latter probabilities occur due to there being such an

enormous number of particles and the inability to keep track of each. Quantum probabilities,

however, are inherent in nature, and not a result of calculation and measuring limitations (Giancoli,

2005).

It should also be outlined that these probabilities are highly precise, such as the probabilities

involved with card games, and rolling a dice, as opposed to the probabilities associated with sporting

events and weather, which are merely estimates.

Impact on Society While society is not necessarily physically impacted immediately by such philosophical

interpretations as those outlined above, often highly controversial and radical theories do impact the

psychological aspects of society, and in this instance, quite profoundly. Prominent physicists such as

Einstein and Schrodinger did not accept the Copenhagen interpretation until very late in their lives.

Einstein believed that the inability to calculate exact values represented that the quantum theory was

incomplete, but Heisenberg and Born stated in a paper delivered to the Solvay conference:

“We regard quantum mechanics as a complete theory for which the fundamental

physical and mathematical hypotheses are no longer susceptible of modification.”

‐Heisenberg and Max Born, paper delivered to Solvay Congress of 1927 (Cassidy,

2002)


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Conflict between such prominent scientists flows through in some way to increased discussion and

conversation in society.

The concepts described above have, however, allowed large developments in areas affecting

all of society. Equations of quantum mechanics are used to understand and improve computer

components, metals, lasers, properties of chemicals etc. (Cassidy, 2002) Things such as the shape of

snowflakes and the operation of fluorescent tubes require quantum mechanics to understand.

Electrons have been observed to pass between energy states at random, a phenomenon that would

not occur unless the energy is uncertain at that point in time. This is an effect known as “tunnelling”

and is what allows the nuclear reactions that power the sun to occur. These processes have been used

by physicists in micro‐electronics. One application is the delicate superconducting instruments that

use electron tunnelling to detect tiny magnetic fields, which are used to safely scan the human brain.


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Wolfgang Pauli

Brief History Wolfgang Pauli was born in Vienna, Austria

on 25 April 1900, and died in Zurich, Switzerland on

15 December 1958. Within twelve months of

beginning his studies at the University of Munich, he

had submitted three original papers on the theory of

relativity, and all were published before his

twentieth birthday (Bloor, Wolfgang Pauli, n.d.).

Pauli received his doctorate in 1921 and became an

assistant to Max Born at Gottingen. In 1923 he

moved to a new position at the University of

Hamburg. He became professor of theoretical

physics at the Federal Institute of Technology in

Zurich in 1928. During WWII, he worked at the

Institute for Advanced Studies at Princeton in New

Jersey, afterwards returning to Zurich where he

died. Note that a large part of Pauli’s work and

discoveries occurred before Heisenberg’s.

Contributions to the Atomic Model Pauli received his initial doctorate for theoretical work on the hydrogen ion. When he moved

to Gottingen to become assistant to Born, he began to work on the anomalous Zeeman effect. He had

decided that Bohr’s model needed to be modified, because each electron was only defined by two

quantum numbers, and multiple electrons could possess the same two numbers. He found that the

theory of the time to explain the Zeeman effect (involving a magnetic moment in the core of the atom),

was incorrect. Pauli was disconcerted by the fact that there was still no complete theory for the

structure of the periodic table of the elements, and no explanation for why all the electrons did not

crowd into the lowest orbital. He perceived that this problem and the Zeeman effect he was

investigating. Pauli is most famous for his exclusion principle, which states that within an atom, every

electron has a unique set of four quantum numbers. The concept of quantum numbers must be

addressed before this principle can be discussed.

Quantum numbers were first conceived in Bohr’s quantum‐classical hybrid theory of the

atom. The principal quantum number, n originally signified the number energy level, with the ground

state or lowest energy level corresponding to n=1. The second quantum number introduced by Bohr

was angular momentum, but this is described a little differently today. The third quantum number

was introduced by an English physicist, Edmond C. Stoner. This number was the magnetic quantum

number. Pauli, to complete the theory of quantum numbers, proposed a fourth number. Unlike the

other three numbers, which had a physical explanation, Pauli could not explain his reasoning for the

fourth number, describing it as “a two‐valuedness not describable classically”. A summary of the four

quantum numbers and what they represent is as follows:

The principal quantum number, n, can have any value from 1 to ∞. This number specifies the

energy of an electron and the size of the orbital (the distance from the nucleus of a peak in a radial

probability distribution plot). All electrons with the same value of n are regarded as in the same ‘shell’

or level. The total number of orbitals is given by n2. The other three quantum numbers are derived

from the principal quantum number.

Figure 8: Wolfgang Pauli (Source: https://www.nobelprize.org)


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The orbital/azimuthal/secondary quantum number, l, is related to the angular momentum

of the electron. This number can have values of 0, 1, 2, 3… 1 1. This number is related to the

actual angular momentum by2:

1

In hydrogen, the energy is governed nearly entirely by n, but in atoms with two or more electrons, the

energy depends on both n and l. The orbital quantum number also specifies the shape and type of

sub‐orbital or subshell the electron is in. Sub‐orbitals, within the primary orbital, are given a letter that

is linked to the secondary quantum number:

l 0 1 2 3 4 5 … Letter3 s p d f g h …

So an electron in the third shell, in the d subshell, would be in 3d. As l increases, there is a slight

increase in the energy of the subshell (s<p<d<f<…). The number of orbitals in a subshell is equal to

2l+1.

The magnetic quantum number, ml is related to the direction of the electron’s angular

momentum, and can any value within the inequality . The value ml is a vector, and a

consequence of this is that both it’s magnitude and direction are quantised. This is known as space

quantisation, and determines the orientation in space of the orbital. In quantum mechanics, angular

momentum is usually specified by giving its component along the z‐axis (an arbitrary choice). Lz is

subsequently defined2:

Lx and Ly are not definite, however. The name of this quantum number is derived from experiment,

namely that regarding the Zeeman effect. It was found that the energy levels must be split, which

implied that the energy of a state must depend not only on n but on ml when a magnetic field is

applied, hence its name (Giancoli, 2005).

The fourth number, first introduced by Pauli as a mathematical convenience, is the spin

quantum number, ms. The inability to explain its existence disturbed Pauli, as it could not be derived

from Schrodinger’s original theory as the others could. P.A.M. Dirac made a subsequent modification

to show that electron ‘spin’ was a relativistic effect. ms can only equal two quantities, .

These two states are generally referred to as ‘spin‐up’ and ‘spin‐down’, although the concept of

electron spin is an intrinsic property, as electrons cannot even be considered as localised objects,

much less spinning ones. A need for this number was observed when it was found through

experimentation that the spectral lines of hydrogen were each made up of multiple lines, and it was

hypothesised to be due to an angular momentum associated with the electron ‘spinning’. The tiny

variations in energy levels is due to the interaction between the tiny current of the spinning electron

interacting with the magnetic field due to the orbiting charge.

When an atom has more than one electron, the energy levels are different to Hydrogen, as the

electrons interact with each other, as well as with the nucleus. Each electron in a complex atom still

1 Bohr originally assigned l = 1 to the ground state. 2 Remember, 3 The letters s, p, d and f were originally abbreviations of “sharp”, “principal”, “diffuse” and “fundamental”, experimental terms referring to the spectra


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possesses its own set of four numbers, and with atoms larger than hydrogen, the energy levels depend

on both n and l. In the years following 1925, the new quantum theory was successful in dealing with

complex atoms, although the mathematics became increasingly complicated, since the electrons are

attracted to the nucleus and repelled by each other.

The Pauli exclusion principle helped to simplify and provide a basis for understanding complex

atoms, bonding and molecules better, among other phenomena. The principle can be stated as

follows:

No two electrons in an atom can occupy the same quantum state

Impact of Pauli’s contribution on the atomic model This principle provided the first theoretical basis to understand the structure of the periodic table. The

nature of the periodic table groupings can be discussed referencing shells and sub‐shells. The noble

gases in column VIII have filled shells and subshells. Due to the outermost subshell being full, the

electron distribution is spherically symmetric. This symmetry means that no other electrons are

attracted, and none are lost. Reciprocally, the outer s shell in alkali metals (column I) spends most of

its time in the inner shells and is ‘shielded’ from the effect of the charge on the nucleus, and is only

subjected to a net charge of about +1e. This allows the outer electron to be quite readily removed.

Pauli’s exclusion principle was also later found to apply to any system of fermions, which have half‐

integer spin, but not bosons, which have integer spin. These two categories of particles are part of the

continued exploration of the atom which followed Bohr, Heisenberg and Pauli. Pauli did continue to

contribute to the atomic model, and predicted the existence of the neutrino, to compensate for the

apparent lack of energy conservation in beta decay.

A spectacular consequence of the exclusion principle in astrophysics is demonstrated in white dwarfs

and neutron stars. A star that is defined as ‘active’ is in equilibrium between the gravity pulling

everything towards its centre and the pressure created through nuclear fusion. When this fusion

ceases, the star collapses. If the star is below a certain mass, the collapse of the star will be halted by

the exclusion principle, known as electron degeneracy pressure, and the star is called a white dwarf.

If the star is above a certain mass, the exclusion principle works in relation to the neutrons, and the

collapse is halted due to neutron degeneracy pressure. These degeneracy pressures establish a new

equilibrium with gravity, and if this force is also overcome, a black hole is formed 9 (Fundamentals of

Quantum Mechanics, n.d.)

Furthermore, no two electrons in a solid can occupy the same state, which leads to the band theory

of solids, applicable to semiconductors, conductors and insulators.

Impact on Society Pauli’s discovery, similarly to Heisenberg’s, did not impact directly on society. The exclusion theory

rather allowed for a rapid progression of quantum mechanics and hence the structure of the atom.

The introduction of the principle allowed for a greater understanding of molecular bonding, the

periodic table, band theory of solids (as above) and when later applied to fermions and bosons, to

superconductivity. These advancements are among the thousands of others that would have

occurred through an understanding of Pauli’s exclusion principle. These scientific and technological

advancements have subsequently allowed a rise in living standards and increase in the productivity

of economies. Considering broad effects such as these helps to understand the true import of

momentous breakthroughs in quantum mechanics. Understanding of the small leads to

development of the big.


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Future possible developments Since the time of Heisenberg and Pauli, there has been significant developments of the atomic model

and associated research. In 1930, Paul Dirac proposed the existence of anti‐particles (for example, a

positron is the anti‐particle of an electron, and has a positive charge equal in magnitude), James

Chadwick discovered neutrons in 1932, nuclear fission was discovered by Lise Meitner, Otto Hahn,

and Strassman in 1938 and Glenn Seaborg discovered eight trans‐uranium elements in 1941‐51. The

proposal of the quark model by Murray Gell‐Mann in 1964 was the next significant development of

the atomic model. This was also developed independently of Mann by George Zweig.

The Quark model/The Standard Model By the mid‐1930s, atoms were regarded as being made up of neutrons (neutral charge),

protons (positive charge, +e), and electrons (negative charge, ‐e), and atoms were no longer the basic

constituents of matter. There was six ‘elementary particles’ that were known, as well as the positron

(positive electron), the neutrino (proposed by Pauli) and the particle (photon). In the 1950s and

1960s, there were many new particles discovered, which were similar to the proton and neutron.

Other particles, of masses between the nucleons (proton and neutron) and the electron were also

discovered, and were named mesons. Physicists theorised that these were made up of smaller

fundamental particles, and named these quarks.

In the Standard Model, what we use today, there is quarks (make up the protons, neutrons and

mesons) and leptons (a classifications including electrons, positrons and neutrinos). These are then

complemented by what are known as ‘force carriers’ and include the photon as well as what are

known as photons. The discovery of these particles occurred as the equipment available to

experimental physicists developed further. Particle accelerators were made more powerful,

instruments more sensitive, and particles were probed deeper and deeper.

An extended discussion on the formulation of this theory will not be included, but instead a brief

summary of the current theory is provided to facilitate a discussion of the further development of the

theory.

The Standard model consist of fundamental particles which fall under two categories. These are

fermions, to which the Pauli exclusion principle applies, and bosons, which are force carriers.

The fermions consist of quarks and leptons, each arranged in three generations. Quarks are

generally referred to as fundamental, point‐like particles which interact via the strong nuclear force.

Quarks, like electrons can be described with quantum numbers, but quarks have six to describe them

(Charge‐Q, Baryon Number‐B, Strangeness‐S, Charm‐c, Bottomness‐b, Topness‐t). There is six quarks

(Up(u), Down(d), Strange(s), Charmed(c), Bottom(b) and Top(t)) and each has an anti‐quark, indicated

by a bar above its symbol. A meson is made from a quark‐anti‐quark pair. A Baryon such as a proton

is made from three quarks. Scientists are yet to isolate a single quark, due to the enormous force

between them. Leptons are particles which interact via the weak nuclear force, and include the

electron ( ), muon ( ) and tau ( ), along with their respective neutrino (electron neutrino, muon

neutrino and tau neutrino). Each of these also has an antiparticle. Their quantum numbers were

explored in the section analysing Pauli’s exclusion principle.

Quarks and Leptons, collectively known as fermions, all have half‐integer spin and obey the

Pauli exclusion principle. Gauge Bosons, however, have integer spins and do not obey the principle.

The gauge bosons are known as ‘force carriers’, or ‘field particles’. The gluon (g) is responsible for

carrying the strong nuclear force, and is what binds quarks and anti‐quarks together. The weak nuclear

force is carried by the Z boson ( ), and is responsible for radioactivity. The photon ( ) carries the


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electromagnetic force, responsible for interactions between charged particles. The ‘electro‐weak’

force, a combination of the weak nuclear force and electromagnetic force, is carried by the W ( ∓)

bosons, and is also responsible for radioactivity. A fourth questionable boson, the graviton, has been

proposed, to carry the gravitational force. Another more unique boson was also discovered in 2012

(although it was proposed much earlier), the Higgs Boson particle. As can be seen in figure 8 below,

the W and Z bosons have a mass, as opposed to the photon and the gluon. The Higgs boson particle

acts to slow the W and Z bosons down, so they travel at less than the speed of light, and this gives

them mass.

The fundamental particles of the standard model are summarised in the diagram below:

Figure 9: A summary of the fundamental particles in the Standard Model (Source: http://www.quantumdiaries.org)

The electroweak theory involves the weak and electromagnetic forces being regarded as a different

manifestation of a more fundamental electroweak interaction. Quantum chromodynamics (QCD) is a

whimsical theory where there are three different colour variations of each quark, red, green and blue.

Likewise, the anti‐quarks are anti‐red, anti‐green and anti‐blue. Three quarks of different colours

create a colourless or white particle, as does a colour with its anti‐colour. These colours act as a form

of charge regarding the strong nuclear force, like a negative electric charge is associated with the

electromagnetic force. The gluons therefore carry colour between quarks/anti‐quarks. The

combination of the electroweak and quantum chromodynamics forms the basis of the Standard

Model.


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Grand Unified Theories (GUTs) There a range of theories which attempt to integrate all forces with each other, much the same as the

electroweak interaction was formed. One theory suggests that the strong, weak and electromagnetic

forces become equal, and leptons and quarks become part of the same family. This only occurs within

distances of 10‐32m, and corresponding energies of around 1016 GeV1 (Giancoli, 2005). Once the

separation between the particles is greater than this distance, the different forces become

distinguishable, and the different particles begin obeying what is known as baryon and lepton number

conservation. These theories are very difficult to test experimentally, due to the enormous energies

involved, but hence present physicists with a challenge. One testable prediction is the decay of a

proton, which was not originally thought to occur. The summary given is of the more simplified GUTs,

and there is a range of more complex theories.

String Theory and Supersymmetry Two other ambitious theories are the String theory and Supersymmetry. These theories attempt to

incorporate gravity as well as all the other forces. String theories imagine the elementary particles

explained above as one‐dimensional strings, perhaps 10‐35m long (Giancoli, 2005). Supersymmetry

predicts that there is interactions that change fermions into bosons and vice versa, and that all

fermions would have supersymmetric boson partners. For example, each quark, which is a fermion,

would have a supersymmetric boson known as a squark. Photons would have supersymmetric

fermions known as photinos, and gluons, gluinos. The graviton is also part of the supersymmetric

model. Supersymmetric theories have been suggested to be able to explain “dark matter2”. One

reason that has been suggested to explain current inability to produce supersymmetric particles is

that they are much heavier than the particles able to be produced in our current particle accelerators.

1 GeV = Giga‐electron volt = 109 eV 2 Matter which does not emit radiation, this unobservable. Detected only through gravitational effects, and thought to make up a large part of the universe.


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Conclusion In summary, the atomic model has been restructured a number of times over the past 2500 years.

The period of swiftest advancement was in the 20th century, during which the Rutherford‐Bohr

model evolved, introducing quantum physics to the model for the first time. Heisenberg and

Schrodinger then paved the way with two different methods of displaying an identical, quantum

mechanical theory, which did away entirely with the classical theories. Heisenberg then published

his uncertainty principle, which had enormous implications for the study of quantum mechanics, and

led to the famous Copenhagen interpretation of physics, explaining the existence of classical

mechanics on a macroscopic level. Pauli’s exclusion principle then allowed a theoretical explanation

of the structure of the periodic table for the first time, and his subsequent contributions to the

particle model in the form of the neutrino again advanced our understanding of the elementary

particles.

These scientific advancements on scales which use femtometres and nanometres led to enormous

upheavals in the study and understanding of the atom, and quantum mechanics is now one of the

two great pillars of physics, the other being relativity. This allowed for much further development,

the effects of which flowed through to society, from philosophical discussions to advancements in

technology.

Todays standard model incorporates a much larger range of elementary particles, from quarks to

gluons to Higgs particles. Future development of these theories includes grand unified theories,

supersymmetric theories and string theories. The study of the construction of the atom is one of the

most promising aspect of modern physics, and will remain so for a long time into the future.


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Appendix‐Common Constants and Variables

Constants QUANTITY SYMBOL APPROXIMATE

VALUE CURRENT BEST VALUE1 (GIANCOLI, 2005)

Speed of light in a Vacuum c 3.00 x 108 ms‐1 2.99792458 x 108 ms‐1 Charge on electron e 1.60 x 10‐19 C 1.60217653(14) x 10‐19 C Planck’s Constant h 6.63 x 10‐34 Js 6.6260693(11) x 10‐34 Js Electron rest mass me 9.11 x 10‐31 kg 9.1093826(16) x 10‐31 kg Proton rest mass mp 1.6726 x 10‐27 kg 1.67262171(29) x 10‐27 kg Neutron rest mass mn 1.6749 x 10‐27 kg 1.67492728(29) x 10‐27 kg Magnetic force constant k 2.00 x 10‐7 N A‐2 2.00 x 10‐7 N A‐2

Variables SYMBOL VALUE SI UNITS

m Mass kg v Velocity ms‐1 Ek Kinetic Energy J Ep Potential Energy J F Force N Q Charge C r Radius m f Frequency Hz Wavelength m

p Momentum kg ms‐1

1 Brackets indicate a slight experimental uncertainty in final digits

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Physics AT3‐Part A The development of the quantum