This document summarizes the development of quantum mechanics from Bohr's early model of the atom to the fully quantum models developed in the 1920s. It discusses modifications made to Bohr's model, such as elliptical orbits and additional quantum numbers. Key figures who contributed include de Broglie, Heisenberg, Schrodinger, and Pauli. De Broglie hypothesized that particles have wave-like properties, which was confirmed by Davisson and Germer's electron diffraction experiment. Heisenberg formulated matrix mechanics and proposed the uncertainty principle. Schrodinger developed wave mechanics using wave functions, and his equation can be used to calculate energy levels.

1. CFI HSC Physics:
From Universe to Atom
Beyond Bohr:
The work of de Broglie, Heisenberg
and other key players
Towards a fully quantum model of the atom

2. Modifications to Bohr’s model
• The German scientist Arnold Sommerfeld proposed
that electron orbits were elliptical rather than
circular. He also introduced a second quantum
number. These ideas were used to explain the
hyperfine structure of the hydrogen spectral lines.
• Meanwhile, an experiment by James Franck and
Gustav Hertz with mercury atoms had indeed shown
that atoms would only accept energy in certain
minimum amounts, which was a key idea in Bohr’s
model.
• However, problems still remained (such as WHY
Bohr’s stationary states existed)

3. Background
• The photoelectric effect (to be studied in
module 7) established that light can be
thought of as a particle, and not just as a
wave.
• Also, these photons (discrete packets) of
light had momentum ! Even though the
photons had no mass!

4. Momentum of photons
E = h f and E = m c2
m c2 = h f
m c = h f / c
p = hf / c
As c = f λ,
p = h / λ
https://www.youtube.com/watch?v=WxoYr-Xe3wY
And:
https://physics.stackexchange.com/questions/2229/if-
photons-have-no-mass-how-can-they-have-momentum

5. Symmetry
• Is there some symmetry in this area of
physics?
• Can matter (traditionally classified as a
particle) be viewed as a wave?

6. De Broglie
• In 1924 Prince Louis de Broglie (France) noted
that light was now described in terms of the dual
nature model (wave-particle).
• Thus, why can’t particles (such as the electron)
also have a wave nature!
• He used E = mc2 , E = h f, and the notion of
momentum to derive an equation for the
wavelength of a particle.

8. De Broglie -continued
This equation was a radical step forward. It was
not immediately accepted. Although Einstein gave
it his public backing. What was needed was some
experimental evidence of the wave nature of
electrons

13. Diffraction - revision
• Recall that diffraction occurs when a wave
bends around obstacles or through gaps.

14. Diffraction and Interference

15. More Diffraction
The bright areas were places of
maximum intensity (constructive
interference), while the dark areas of
minimum intensity were places of
destructive interference

16. Davisson and Germer -1927
Accelerated electrons are fired onto nickel crystal. They
were then scattered off various planes of the crystal
(similar to the Braggs X-Ray experiment). This produced
a diffraction interference pattern (with positions of
maxima and minima). The electrons were acting as
waves!

19. Another way of finding wavelength
Try and derive this formula.

20. A little detour!
The double slit experiment showing
electrons acting as waves,
• https://www.youtube.com/watch?v=Q1YqgPAtzho
• According to a Physics World poll conducted in 2002, the
most beautiful experiment in physics is the two-​​slit experiment
with electrons (first performed in 1961). According to Richard
Feynman, this classic gedanken experiment “has in it the
heart of quantum mechanics” and “is impossible, absolutely
impossible, to explain in any classical way.”
http://physicsworld.com/cws/article/news/2013/mar/14/feynma
ns-double-slit-experiment-gets-a-makeover

21. Using de Broglie to explain Bohr’s orbits
• A short note on standing waves:
• Standing waves are produced when two identical waves
travelling in opposite directions pass through each other.
The two waves will interfere constructively and
destructively with each other in such a way that
produces a pattern that appears to be standing still
(hence the name standing or stationary waves).

22. Yet more on waves…
• Standing waves will occur for selected (‘resonant’)
frequencies within a given system.
• Energy is trapped within a standing wave, and is not lost
from the system. The energy is trapped between the
boundaries of the system. The wave does not
‘propagate’ (travel) outside of the system.
• standing wave animation

23. De Broglie and Matter Waves
• The circumference of an electron orbit will be an
integral (whole number) multiple of the electron
wavelength. In this way, standing waves (or
‘matter waves’) will occur and the electron will
exist in a stable state without radiating energy.
• If the circumference was not an integral multiple
of the electron wavelength, then no such pattern
would be formed, and any possible wave would
rapidly die out, as destructive interference took
place.

24. Orbit Circumference and de Broglie Wavelength
• For electrons in the first energy level (n = 1), the
circumference of the orbit is equal to ONE de
Broglie wavelength.
• For electrons in the second energy level, the
circumference is equal to TWICE the de Broglie
wavelength…….etc….
The amplitude of the standing
wave represents the probability
of the electron being at that
location. The electron should
not be visualised as following
that particular path along the
circumference

25. More on Orbit Circumference
• If the circumference is NOT an integral multiple
of electron wavelengths, then the beginning and
end of the wave will not be in phase, and
destructive interference will occur (and the
standing wave will not be maintained).

26. Linking de Broglie and Bohr
• For a particular electron orbit, n, the
following applies: n λ = 2 π r
Using this, Bohr’s third postulate on
angular momentum could now be derived.
Consider: λ = h / m v and n λ = 2 π r
Let’s substitute for λ into n λ = 2 π r
Therefore: n h / m v = 2 π r
Therefore: m v r = n h / 2 π (This is Bohr’s Third postulate)
In other words, Bohr’s postulate is simply a consequence of
the wave nature of matter.

27. Professor Dave Video
• Confused?
• This short clip summarises some key
ideas so far:
• https://www.youtube.com/watch?v=O6g-
7rUgrdg

28. Quantum Mechanics
• The work of de Broglie formed the foundations of
quantum mechanics.
• Quantum mechanics can be defined as a set of
mathematical laws that apply to the atomic and
subatomic world, the uncertainty principle and
wave-particle duality lie at the core of quantum
mechanics.
• The HSC syllabus requires you to comment on the
specific contribution of Erwin Schrodinger.
• But he was just one of many key players including
Werner Heisenberg and Wolfgang Pauli.

29. Werner Heisenberg
• By 1927 Bohr’s model (1913) was now superseded.
• In 1925 Heisenberg developed a purely mathematical
model of the atom, using complex mathematical tables
called matrixes. He stated that it was not useful to try
and visualise electron orbits. His theory was originally
called matrix mechanics.
• Heisenberg was still concerned with understanding the
hydrogen spectral lines. Later on, Wolfgang Pauli was
able to use Heisenberg’s work to derive the Balmer
equation and Rydberg’s constant (which, you may recall,
was an achievement of the Bohr model).
• Heisenberg could also solve issues such as why spectral
lines had different intensities, by calculating the
probabilities of certain electron transitions occurring.

30. All Measurements Have Uncertainty
• By now you appreciate that there is an error value associated with
measurements. Even if you use a very precise instrument like a micrometer
there will still be a finite limit to its precision.
• Also, when you take a reading (such as temperature of a liquid), the very
act of measuring can disturb the material you are observing.
• If you use photons of light to observe the position of something tiny like an
electron, the photon will transfer some momentum to the electron and
change its position (and momentum in unpredictable ways.
• BUT: quantum physics tells us there is uncertainty in observing nature
over and beyond the above described effects. The uncertainty is not
just an effect of limitations of technology or user competence.

31. Heisenberg Uncertainty Principle
• This is a consequence of the wave-particle duality, and is
NOT just an ‘observer’ or ‘instrument’ effect.
• It is not possible to know all the information about a particle
at a particular instant.
• This principle links conjugate pairs of variables: namely
position and momentum and energy and time.
• “It is impossible to measure simultaneously the momentum
and precision of a particle with infinite or indefinite
precision” .
• This was first set down by Werner Heisenberg in 1927
(written in German and translated into English). It has since
been extended and researched by many others.

32. Uncertainty Principle -1927
• Heisenberg’s other great contribution was
the uncertainty principle.
• Put simply, it states that it is not possible
to measure the position and momentum
(ie, velocity) of an object precisely at the
same time.
• From TED:
• https://www.youtube.com/watch?v=TQKE
LOE9eY4

33. Uncertainty Continued
(∆ x) (∆p) ≥ h / 4π
The uncertainty in position (∆ x) multiplied by the
uncertainty in momentum (∆p) is always equal to or
greater than Planck’s constant divided by four pi.
The more accurately one knows the position, the less
accurately you will know the momentum, and vice versa.
This is a radical break with classical or ‘clockwork
universe’ (Newtonian) physics, and any ideas about
absolute measurement.
Note: the use of Planck’s constant yet again!!!

34. Example of an uncertainty calculation
(Not in the HSC syllabus)
• The position of an electron is measured to
the nearest 0.50 x 10-10 , find the minimum
uncertainty in its momentum.
• (∆ x) (∆p) ≥ h / 4π
• Answer: 1.1 x 10-24 kg m/s

35. The Uncertainty Principle also
applies to time and energy
• Measurements of energy and time are
also linked by an uncertainty principle.
Another way of presenting this formula is:
Note: ħ = h/(2π)
Ħ is pronounced as ‘h bar’

36. Pauli
• Wolfgang Pauli used Heisenberg’s work to derive the
Balmer equation for Hydrogen spectral lines.
• He introduced a fourth quantum number associated with
electron ‘spin’.
• In 1925, he put forward the exclusion principle. This
states that no two electrons in the same atom can have
the same set of four quantum numbers.
• Note: There were now four quantum numbers assigned
to each electron: the principal quantum number (n), the
orbital number (l), magnetic number (mI), and magnetic
spin (ms).
• Pauli’s work could be used to explain the patterns of
elements in the periodic table, and give reasons why
electron configurations (‘electron shells’) exist in the way
that they do. Thus chemistry is now put on a secure
theoretical footing (which can guide future research).

37. Schrödinger
• Erwin Schrödinger (1926) put forward another model
called wave mechanics.
• It was a direct descendent of the work of de Broglie. He
introduced the wave function (Ψ), which predicted the
probability of finding an electron at a particular place (in
a three-dimensional region known as an orbital).
• The wave function (Greek letter psi) is not directly
observable, but is sort of like the sum of the properties of
what is known about that particle.
• Physicists liked Schrodinger’s work, as it spoke to them
in familiar language of waves, rather than esoteric maths
like Heisenberg’s matrices.

38. The Wave Function
• In 1926 Erwin Schrodinger proposed a model of the atom known as
quantum mechanics.
• This model could be used to explain the behavior of electrons, particularly
with reference to the hydrogen atom.
• A highly complex differential equation known as Schrodinger’s equation is
at the heart of this:
• This equation can be used to find things like the energy of electron orbits (just
as Bohr’s model could do). It can furthermore, predict of the probability of
certain electron transitions. It also yields a quantity known as the wave function.

39. More…..
• The Schrodinger model had to incorporate the wave
nature of the electron.
• The de Broglie equation assigned a wavelength, λ, to
an electron.
• As electron is thus known as a matter wave.
• The wave function, ψ (‘psi’) from Schrodinger’s
equation is a measure of the amplitude of the electron
matter wave. The value of ψ will vary depending on
where the electron is and what it is doing.
• BUT; this is a simplistic definition. The wave function is
not something that can be directly observed.
• BUT, it is a useful quantity that can allow us to find
interesting information……

40. Using the Wave Function
• The wave function squared, ψ2 is used to
predict the probability of finding an
electron at a particular place and time.

41. Some Wave Function Clips
From Up and At Them:
• https://www.youtube.com/watch?v=QeUMFo8sODk
• From Scientific American’s Instant Egghead series:
https://www.youtube.com/watch?v=aowYf44gDRY
• From the Science Channel:
• https://www.youtube.com/watch?v=7GTCus7KTb0
• From Bozeman Science:
• https://www.youtube.com/watch?v=Xh6NRenkdRk

42. Link to a Site Discussing the
Wave Function
• http://www.physlink.com/education/askexp
erts/ae329.cfm

43. Schrodinger – Part 3
• Schrodinger’s work was more widely accepted than
Heisenberg’s complicated mathematics, as it spoke to
scientists in the familiar language of waves.
• Eventually Schrödinger showed that his work and
Heisenberg’s model were, essentially, the same thing
(as each model could be used to derive formulae from
the other model).

44. Schrodinger’s Cat

45. YouTube and Schrodinger’s Cat
• YouTube: From Minute Physics
• https://www.youtube.com/watch?v=IOYyCHGWJq4
• YouTube: Big Bang Theory and Sheldon:
• https://www.youtube.com/watch?v=pNTMYNj2Ulk
•

46. Quantum mechanics - uses

48. Complementarity
• Niels Bohr said that if we are to have a full
understanding of matter, then you need to
consider both its wave and particle
properties…..but not both of these at the
same time!
(The same complementarity principle
applies to light).

49. Copenhagen Interpretation
• The Copenhagen interpretation is an expression of the
meaning of quantum mechanics that was largely devised from
1925 to 1927 by Niels Bohr and Werner Heisenberg. It
remains one of the most commonly taught interpretations of
quantum mechanics.
• According to the Copenhagen interpretation, physical systems
generally do not have definite properties prior to being
measured, and quantum mechanics can only predict the
probability distribution of a given measurement's possible
results. The act of measurement affects the system, causing
the set of probabilities to reduce (or ‘collapse’) to only one of
the possible values immediately after the measurement. This
feature is known as wave function collapse This collapse is
irreversible. (adapted from Wikipedia June 12th 2019).

50. Just to add to the confusion…
• The term Copenhagen Interpretation is one, although
coined by Heisenberg himself in the 1950s, lacks an
agreed set of rules. There is not and has never been a
definitive statement on just what the term means!
Furthermore, Heisenberg and Bohr disagreed on much
about quantum theory. For example, Bohr said a particular
experiment or formula could only be used to show the wave
or particle nature. But Heisenberg said that these formulae
are open to both wave and particle formulations.

51. Many Worlds Interpretation
• Not everyone likes the Copenhagen interpretation.
• https://www.youtube.com/watch?v=Um0AvHrPp_w
• A common alternative is the many worlds interpretation.
• The many-worlds interpretation asserts the objective
reality of the universal wavefunction and denies the
actuality of wavefunction collapse.The existence of the
other worlds makes it possible to remove randomness and
action at a distance from quantum theory and thus from all
physics. Many-worlds implies that all possible alternate
histories and futures are real, each representing an actual
"world" (or "universe").

52. Many worlds - continued
• In layman's terms, the hypothesis states there is a very
large—perhaps infinite number of universes, and everything
that could possibly have happened in our past, but did not,
has occurred in the past of some other universe or universes.
The theory is also referred to as MWI, the relative state
formulation, the Everett interpretation, the theory of the
universal wavefunction, many-universes
interpretation, multiverse theory or just many-worlds.
• Yet other physicists simply ultimately discard the very
question of realism and posing scientific theory as a tool to
help humans make predictions, not to attain an understanding
of the world. This view is summed by the famous quote
of David Mermin, "Shut up and calculate“ (Wikipedia June 12th
2019)