Origin and discovery of quantum mechanics
Interplay of eye and mind
Physics look at nature.
Ask question about nature and try to give answer them, imagine answers.
For instance, why does the sun shine? Why do stars shine? Why is the sky blue? Why do metals emit
light when heated to very high temperature?
In physics one can make mistakes but one cannot cheat!
There are many reasons to learn quantum physics.
All physics is quantum physics, from elementary particles to the big bang, semiconductors, and solar
energy cells.
Our world is filled with advanced technologies. Many of these new technologies come from the
fundamental research within the framework of quantum theories.
In order to understand modern physics, three fundamental links are necessary: quantum mechanics,
statistical physics and relativity.
Quantum mechanics play a key role in engineering. It will become increasingly relevant in
nanotechnology, semiconductors, polymer technology, nuclear/photonic devices, magnetic devices,
optics and many other things.
New ideas come only from the minds of creative thinkers. Physicist learn to use their intelligence and
can explain their findings.
Quantum theory is subtle.
Mysteries of light: Blackbody radiation
In physics, two great discoveries of the 20th century is based on properties of light: Relativity (𝐸 =
𝑚𝑐 2 ) and quantum physics with black body theory (𝐸 = ℎ𝜈).
In the 18th century, Newton decided that light was made of corpuscles (particles), because only
particles can travel along straight lines. However, since the end of the 17th century, interference and
diffraction phenomena were known and the 19th century saw the success of wave optics.
Nobody could imagine the incredible answer of quantum theory.
It is a matter of experiences that a hot object can emit radiation.
A pieces of metal stuck into a flame can become red hot. At high temperature it can become white hot
then red hot then blue hot.
The discovery of quantum mechanics could have happened by analyzing frequency distribution of
radiation inside an oven (black body) at temperature T.
A blackbody is an object that is a perfect absorber (emitter) of
radiation (in ideal case).
Figure shows experimental measurements of the thermal
radiation at several temperatures.
What is the origin of this radiation? This was the major topic
of 19th century physics.
Please carefully review the following:
Figure 1. Measured distribution of thermal radiation at several temperatures.
Consider a cubic cavity of volume V and length L. The electrons or atoms on
the surface of the cavity act as harmonic oscillators. When the material is
heated then electrons or atoms gain kinetic energy and they begin to oscillate.
Meanwhile we mention here that energy of the classical harmonic oscillator is
1
𝐸 = 2 𝑚𝜔2 𝐴2 . Oscillating charged particles emits radiation (light). The
emitted radiation in the hot cavity produce standing wave and number of
modes per unit frequency per unit volume (number of degrees of freedom for
frequency ν) is given by:
8𝜋𝜈 2
𝑐3
(For evaluation of number of modes visit
astr.gsu.edu/hbase/quantum/rayj.html#c2)
the
web
page:
http://hyperphysics.phy-
In order to calculate energy density of emitted radiation from cavity we can use:
𝑢(𝜈, 𝑇) = [
𝑚𝑜𝑑𝑒𝑠 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑒𝑛𝑒𝑟𝑔𝑦
]×[
]
𝑝𝑒𝑟 𝑚𝑜𝑑𝑒
According to the classical theories, energy of each oscillator is continuous and average energy per
mode (per degree of freedom) can be calculated as follows:
∞
𝐸̅ =
∫0 𝐸𝑒 −𝐸/𝑘𝑇 𝑑𝐸
∞
∫0 𝑒 −𝐸/𝑘𝑇 𝑑𝐸
Where E is energy of the oscillator, k is Boltzmann constant and T is temperature. We evaluate this
integral and we obtain:
𝐸̅ = 𝑘𝑇
Then energy density of emitted radiation from a cavity can be written as:
8𝜋𝜈 2
𝑘𝑇
𝑐3
This is Rayleigh-Jeans classical formula. This formula can also be expressed interms of wavelength by
using: 𝑢(𝜆, 𝑇)𝑑𝜆 = 𝑢(𝜈, 𝑇)𝑑𝜈 and 𝜆𝜈 = 𝑐, we obtain:
𝑢(𝜈, 𝑇) =
𝑢(𝜆, 𝑇) =
8𝜋𝑐 3
𝑘𝑇
𝜆4
It is obvious that this formula is
not
compatible
with
the
experimental results at high
frequencies.
Planck made the assumption that
an exchange of energy between
the electrons in the wall of the
cavity
and
electromagnetic
radiation can only occur in
discrete amounts. Basic quantum
of energy can be written as
𝜀 = ℎ𝜈
Where the constant ℎ = 6.62 ×
10−34 𝐽. 𝑠𝑒𝑐 is called Planck’s
constant. Furthermore, energy can only come in amounts that are integer multiples of the basic
quantum:
𝐸 = 𝑛𝜀 = 𝑛ℎ𝜈,
𝑛 = 0,1,2,3, …
An immediate mathematical consequence of this assumption is that the integrals in the average energy
equation turn into discrete sums. So when we calculate the average energy per degree of freedom, we
must change all integrals to sums
∞
𝐸̅ =
∫0 𝐸𝑒 −𝐸/𝑘𝑇 𝑑𝐸
∞
∫0 𝑒 −𝐸/𝑘𝑇 𝑑𝐸
→
−𝑛ℎ𝜈/𝑘𝑇
∑∞
𝑛=0 𝑛ℎ𝜈𝑒
−𝑛ℎ𝜈/𝑘𝑇
∑∞
𝑛=0 𝑒
To evaluate this formula we use analogy of the geometric series
∞
𝑎
= ∑ 𝑎𝑟 𝑛
1−𝑟
𝑛=0
Then average energy can be written as:
𝐸̅ =
ℎ𝜈
ℎ𝜈
𝑒 𝑘𝑇
−1
Then energy density is given by
𝑢(𝜈, 𝑇) =
8𝜋𝜈 2
𝑐3
ℎ𝜈
ℎ𝜈
𝑒 𝑘𝑇
−1
This worked brilliantly! It provide a good fit with the experimental results.
Classically the emission and absorption of energy to be continuous. Then, Planck suddenly changed
the story, moving to a totally nonclassical concept, that the oscillators could only gain and lose energy
in chunks, or quanta. (Incidentally, it didn’t occur to him that the radiation itself might be in quanta:
he saw this quantization purely as a property of the wall oscillators.) As a result, although the
exactness of his curve was widely admired, and it was the Birth of the Quantum Theory (with
hindsight), no-one—including Planck—grasped this for several years!
Other equations governing blackbody radiation
Wien’s displacement law
Experimentally, the peak of the spectrum was found to obey with the following relation:
𝑚𝑎𝑥 𝑇 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 2.898 × 10−3 𝑚·𝐾
Wavelength of maximum peak of a black body radiation can be obtained from this relation.
Stefan-Boltzmann Law
This law states that the power emitted per unit area of the surface of a black body is directly
proportional to the fourth power of its absolute temperature.
The total radiation energy perunit volume in the cavity:
∞
𝑈(𝑇) = ∫
0
8ℎ𝜋𝜈 3 𝑑ν
ℎ𝑣
=
𝑐 3 (−1 + 𝑒 𝑘𝑇 )
8𝑘 4 𝜋 5 𝑇 4
= 𝑎𝑇 4
15𝑐 3 ℎ3
Where a=7.566210-16 J/m3.K4. We can relate this energy density to the energy I emitten per second
from the surface of the black body. Without further discussion
1
𝐼 = 𝑎𝑐𝑇 4 = 𝜎𝑇 4
4
Where the fundamental constant 𝜎 = 5.67 × 10−8 𝑊𝑚−4 𝐾 −4 . This expression had been derived
earlier by Boltzmann using thermodynamics arguments. This expression is Stefan-Boltzmann
expressions.
Energy of the photon
Planck’s assumption also change our understanding about energy and intensity of electromagnetic
radiation. The term intensity has a particular meaning here: it is the number of waves or photons of
light reaching your detector; a brighter object is more intense but not necessarily more energetic.
M ore energetic
M ore intense
Photon's energy depends on the frequency only, not the intensity. The photons in a beam of X-ray light
are much more energetic than the photons in an intense beam of infrared light.
Particles of Light: Photoelectric effect
In 1887, the photoelectric effect was discovered by Heinrich Hertz. He
observed that the metal plates emits electrons depends on wavelength of
the light. Only light with a frequency greter than a given treshold
frequency will produce a current through the circuit.
Lénard (1888) found the energies of the emitted electrons to be
independent of the intensity of the incident radiation.
Planck’s photon model explained black boody radiation. Einstein thought
he saw an inconsitency in the way Planck used Maxwell’s wave theory of
electromagnetic radiation in his derivation.
The photoelectric effect is perhaps the most direct and convincing evidence of the existence of photons
and the 'corpuscular' nature of light and electromagnetic radiation. That is, it provides undeniable
evidence of the quantization of the electromagnetic field and the limitations of the classical field
equations of Maxwell.
Mathematical Formulation and Experimental Procedure of Photoelectric effect
The photoelectric effect exhibits the following:
1)
There is a minimum frequency, 𝜈𝑐 , called the threshhold frequency (or cutoff frequency)
required for the effect to occur.
2)
The maximum kinetic energy of the photoelectrons does not depend on the intensity of the
light.
3)
The maximum kinetic energy of the photoelectrons increases as the frequency of the light
increases.
4)
There is no appreciable time delay between the illumination of the surface and the emission of
the photoelectrons.
Observation of the photoelectric effect is accomplished with the arrangement shown. Ejection of
photoelectrons causes a current to be registered in the ammeter A. Increasing the voltage V repels the
electrons from the cathode C. The value of V that reduces the current to zero is called the stopping
voltage Vs.
Then the work done on the photoelectron to keep it from reaching the cathode (collector) is 𝑒𝑉𝑠 .
1
𝑚 𝑣2
2 𝑒 𝑚𝑎𝑥
= 𝐾𝑚𝑎𝑥 = 𝑒𝑉𝑠 ;
The electrons is bounded to the metal surface with a potential energy. 𝑈  − 𝜑, where U = potential
energy of an electron, and -φ is the highest value of U. Energy conservation requires:
(𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝑝ℎ𝑜𝑡𝑜𝑛) = (𝑤𝑜𝑟𝑘 𝑡𝑜 𝑒𝑗𝑒𝑐𝑡 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛) + (𝐾𝐸 𝑜𝑓 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛).
ℎ = 𝜑 + 𝐾
φ is called the photoelectric work function of the metal.
The particle-particle collision concept explains the immediate ejection of photoelectrons. Since K max
cannot be less than zero, the minimum frequency is explained: for
𝐾𝑚𝑎𝑥 = 0,
ℎ𝜈𝑐 = 𝜑
Where νc is “cutoff” frequency and the result shows no dependence on light intensity for Kmax.
The other experiments shows particle properties of light are:
Compton scattering
Raman scattering
Wave-Particle Duality
Pair Production :
Pair production is the formation or materialization
of two electrons, one negative and the other
positive
(positron),
from
a
pulse
of
electromagnetic energy traveling through matter,
usually in the vicinity of an atomic nucleus.
Pair production is a direct conversion of radiant
energy to matter.
It is one of the principal ways in which highenergy gamma rays are absorbed in matter.
For pair production to occur, the electromagnetic
energy, in a discrete quantity called a photon, must
be at least equivalent to the mass of two electrons.
The mass m of a single electron is equivalent to
0.51 million electron volts (MeV) of energy E as calculated from the equation formulated by Albert
Einstein, E = mc2, in which c is a constant equal to the velocity of light. To produce two electrons,
therefore, the photon energy must be at least 1.02 MeV. Photon energy in excess of this amount, when
pair production occurs, is converted into motion of the electron-positron pair. If pair production occurs
in a track detector, such as a cloud chamber, to which a magnetic field is properly applied, the electron
and the positron curve away from the point of formation in opposite directions in arcs of equal
curvature. In this way pair production was first detected (1933). The positron that is formed quickly
disappears by reconversion into photons in the process of annihilation with another electron in matter.
Wave Behavior of Particle
What is this wave? (Review diffraction and intereference phenomena)
And why is this result so extraordinary?
After particle behavior of wave accepted, the question became whether this was true only for light or
whether material objects also exhibited wave-like behavior.
De Broglie's Hypothesis
In his 1923, Louis de Broglie made a bold assertion. Considering Einstein's relationship of wavelength
to momentum p, de Broglie proposed that this relationship would determine the wavelength λ of any
matter, in the relationship:
𝜆=
ℎ
𝑝
This wavelength is called the de Broglie wavelength. This equation and energy of the photon can be
written as:
𝑝 = ℏ𝑘 𝑎𝑛𝑑 𝐸 = ℏ𝜔
Where 𝑘 =
2𝜋
𝜆
is angular wavenumber and ω is angular frequency.
Significance of the de Broglie Hypothesis
The de Broglie hypothesis showed that wave particle duality was not merely an aberrant behavior of
light, but rather was a fundamental principle exhibited by both radiation and matter. As such, it
becomes possible to use wave equations to describe material behavior, so long as one properly applies
the de Broglie wavelength. This would prove crucial to the development of quantum mechanics.
Experimental Confirmation
Electron diffraction
In 1927, physicists Clinton Davisson and Lester Germer, of Bell Labs, performed an experiment
where they fired electrons at a crystalline nickel target. The resulting diffraction pattern matched the
predictions of the de Broglie wavelength. Electron diffraction refers to the wave nature of electrons.
Electrons are incident on a crystal. The periodic structure of a crystalline solid acts as a diffraction
grating. Interference of electrons shows that electron act as wave.
Electrons are accelerated in an electric potential 𝑈, then their velocities are:
2𝑒𝑈
𝑣=√
𝑚
Then de Broglie relation takes the form:
𝜆=
ℎ
ℎ
ℎ
ℎ
=
=
=
𝑝 𝑚𝑣 √2𝑚𝑒𝑈 √2𝑚𝐸
Where 𝐸 = 𝑒𝑈 is energy of fired electrons.
Neutral atoms
Experiments with diffration and reflection of neutral atoms confirm the application of the de Broglie
hypothesis to atoms, i.e. the existence of atomic waves which undergo diffraction, interference and
allow quantum reflection by the tails of the attractive potential.
This effect has been used to demonstrate atomic holography, and it may allow the construction of an
atom probe imaging system with nanometer resolution. The description of these phenomena is based
on the wave properties of neutral atoms, confirming the de Broglie hypothesis.
Waves of molecules
Recent experiments even confirm the relations for molecules and even macromolecules, which are
normally considered too large to undergo quantum mechanical effects. In 1999, a research team in
Vienna demonstrated diffraction for molecules as large as fullerenes. The researchers calculated a De
Broglie wavelength of the most probable C60 velocity as 2.5 picometer.
In general, the De Broglie hypothesis is expected to apply to any well isolated object.
Macroscopic Objects & Wavelength
Though de Broglie's hypothesis predicts wavelengths for matter of any size, there are realistic limits
on when it's useful. A baseball thrown at a pitcher has a de Broglie wavelength that is smaller than the
diameter of a proton ... by about 20 orders of magnitude. The wave aspects of a macroscopic object are
so tiny as to be unobservable in any useful sense.
Bohr Atom
In 1911, Rutherford introduced a new model of the atom in which cloud of negatively charged
electrons surrounding a small, dense, positively charged nucleus. This model is result of experimental
data and Rutherford naturally considered a planetary-model atom. The laws of classical mechanics
(i.e. the Larmor formula, power radiated by a charged particle as it accelerates.), predict that the
electron will release electromagnetic radiation while orbiting a nucleus. Because the electron would
lose energy, it would gradually spiral inwards, collapsing into the nucleus. This atom model is
disastrous, because it predicts that all atoms are unstable.
To overcome this difficulty, Niels Bohr proposed, in 1913, what is now called the Bohr model of the
atom. He suggested that electrons could only have certain classical motions:
1. The electrons can only travel in special orbits: at a certain discrete set of distances from the
nucleus with specific energies.
2. The electrons of an atom revolve around the nucleus in orbits. These orbits are associated with
definite energies and are also called energy shells or energy levels. Thus, the electrons do not
continuously lose energy as they travel in a particular orbit. They can only gain and lose
energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic
radiation with a frequency ν determined by the energy difference of the levels according to the
Planck relation:
Δ𝐸 = 𝐸2 − 𝐸1 = ℎ𝜈
3. Kinetic energy of the electron in the orbit is related to the frequency of the motion of the
electron:
1
1
𝑚𝑣 2 = 𝑛ℎ𝜈
2
2
For a circular orbit the angular momentum L is restricted to be an integer multiple of a fixed
unit:
𝐿 = 𝑚𝑣𝑟 = 𝑛ℏ
where n = 1, 2, 3, ... is called the principal quantum number. The lowest value of n is 1; this gives a
smallest possible orbital radius of 0.0529 nm known as the Bohr radius.
Bohr's condition, that the angular momentum is an integer multiple of ħ was later reinterpreted by de
Broglie as a standing wave condition: the electron is described by a wave and a whole number of
wavelengths must fit along the circumference of the electron's orbit:
𝑛𝜆 = 2𝜋𝑟
The Bohr model gives almost exact results only for a system where two charged points orbit each
other at speeds much less than that of light.
To calculate the orbits requires two assumptions:
1. (Classical Rule)The electron is held in a circular orbit by electrostatic attraction. The
centripetal force is equal to the Coulomb force.
𝑚𝑣 2
𝑍𝑒 2
=
𝑟
4𝜋𝜖0 𝑟 2
It also determines the total energy at any radius:
1
𝑍𝑒 2
𝑍𝑒 2
𝐸 = 𝑚𝑣 2 −
=−
2
4𝜋𝜖0 𝑟
8𝜋𝜖0 𝑟
The total energy is negative and inversely proportional to r. This means that it takes energy to
pull the orbiting electron away from the proton. For infinite values of r, the energy is zero,
corresponding to a motionless electron infinitely far from the proton.
2. (Quantum rule) The angular momentum 𝐿 = 𝑚𝑣𝑟 = 𝑛ℏ, so that the allowed orbit radius at
any n is:
𝑛2 ℏ2
𝑟𝑛 = 4𝜋𝜖0 2
𝑍𝑒 𝑚
The energy of the n-th level is determined by the radius:
2
𝑍𝑒 2
𝑍𝑒 2
𝑚
𝑍 2 13.6
𝐸=−
= −(
=−
𝑒𝑉
)
8𝜋𝜖0 𝑟𝑛
4𝜋𝜖0 2ℏ2 𝑛2
𝑛2
An electron in the lowest energy level of hydrogen (n = 1) therefore has 13.6 eV less energy than a
motionless electron infinitely far from the nucleus.
The combination of natural constants in the energy formula is called the Rydberg energy (RE):
2
𝑒2
𝑚
𝑅𝐸 = (
)
4𝜋𝜖0 2ℏ2
This expression is clarified by interpreting it in combinations which form more natural units. We
define 𝑚𝑐 2 is rest mass energy of the electron (511 keV) and
𝑒2
4𝜋𝜖0 ℏ𝑐
= 𝛼 is the fine structure constant
then
1
𝑅𝐸 = (𝑚𝑐 2 )𝛼 2
2
Bohr Atom and Rydberg formula
The Rydberg formula, which was known empirically before Bohr's formula, is now in Bohr's theory
seen as describing the energies of transitions or quantum jumps between one orbital energy level, and
another. When the electron moves from one energy level to another, a photon is emitted. Using the
derived formula for the different 'energy' levels of hydrogen one may determine the 'wavelengths' of
light that a hydrogen atom can emit.
The energy of a photon emitted by a hydrogen atom is given by the difference of two hydrogen energy
levels:
1
1
𝐸 = 𝐸𝑖 − 𝐸𝑓 = 𝑅𝐸 ( 2 − 2 )
𝑛𝑓 𝑛𝑖
where nf is the final energy level, and ni is the initial energy level.
Since the energy of a photon is 𝐸 =
ℎ𝑐
,
𝜆
the wavelength of the photon given off is given by
1
1
1
= 𝑅( 2 − 2 )
𝜆
𝑛𝑓 𝑛𝑖
This is known as the Rydberg formula, and the Rydberg constant R is RE / hc. This formula was
known in the nineteenth century to scientists studying spectroscopy, but there was no theoretical
explanation for this form or a theoretical prediction for the value of R, until Bohr. In fact, Bohr's
derivation of the Rydberg constant, as well as the concomitant agreement of Bohr's formula with
experimentally observed spectral lines of the Lyman (nf = 1), Balmer (nf = 2), and Paschen (nf = 3)
series, and successful theoretical prediction of other lines not yet observed, was one reason that his
model was immediately accepted.
Improvement of Bohr Model
Several enhancements to the Bohr model were proposed; most notably the Sommerfeld model or
Bohr-Sommerfeld model, which suggested that electrons travel in elliptical orbits around a nucleus
instead of the Bohr model's circular orbits. This model supplemented the quantized angular
momentum condition of the Bohr model with an additional radial quantization condition, the
Sommerfeld-Wilson quantization condition
𝑇
∫ 𝑝𝑟 𝑑𝑞𝑟 = 𝑛ℎ
0
where pr is the radial momentum canonically conjugate to the coordinate q which is the radial position
and T is one full orbital period. The Bohr-Sommerfeld model was fundamentally inconsistent and led
to many paradoxes. The Sommerfeld quantization can be performed in different canonical coordinates,
and sometimes gives answers which are different. In the end, the model was replaced by the modern
quantum mechanical treatment of the hydrogen atom, which was first given by Wolfgang Pauli in
1925, using Heisenberg's matrix mechanics. The current picture of the hydrogen atom is based on the
atomic orbitals of wave mechanics which Erwin Schrödinger developed in 1926.
However, this is not to say that the Bohr model was without its successes. Calculations based on the
Bohr-Sommerfeld model were able to accurately explain a number of more complex atomic spectral
effects.
Quantum Tunneling and Quantum Uncertainty
Tunneling is a fascinating phenomena both in its own rights and for its many applications. Tunneling
refers to the quantum mechanical phenomenon where a particle tunnels through a barrier that it
classically could not surmount.
The uncertainty principle was first recognized by the German physicist Werner Heisenberg in 1926 as
a corollary of the wave-particle duality of nature. He realized that it was impossible to observe a subatomic particle like an electron with a standard optical microscope, no matter how powerful, because
an electron is smaller than the wavelength of visible light.
Roughly stated, this is the mathematical origin of the uncertainty principle. The particle position and
momentum cannot be “known” simultaneously to arbitrary precision.
Mathematically, Heisenberg's result looks like this:
ℏ
2
Now the uncertainty principle is not something we notice in everyday life. For example, we can weigh
an automobile (to find its mass), and all automobiles have speedometers, so we can calculate the
momentum. But doing so will not make the position of the car suddenly become hazy (especially if
we're inside it). So measuring the momentum of the car seems to produce no uncertainty in the car's
position.
Δ𝑥Δ𝑝 ≥
The reason we don't notice the uncertainty principle in everyday life is because of the size of Planck's
constant. It's very small ℏ = 1.05 × 10−34 𝐽𝑜𝑢𝑙𝑒. 𝑆𝑒𝑐𝑜𝑛𝑑𝑠
The Copenhagen Interpretation
If you ask ten different physicists what the Copenhagen interpretation is, you'll get nine similar (but
not exactly the same) answers, and one "Who cares?"
The Copenhagen interpretation of quantum physics can be summarized as:
1. The wave function is a complete description of a wave-particle.
2. When a measurement of a wave-particle is made, its wave function collapses.
3. If two properties of a wave-particle are related by an uncertainty relation (such as the
Heisenberg uncertainty principle), no measurement can simultaneously determine both
properties to a precision greater than the uncertainty relation allows.
References
Quantum Mechanics, David McMahon
Introduction To Quantum Mechanics, Harald J W Müller-Kristen
http://www.thebigview.com/spacetime/uncertainty.html
http://en.wikipedia.org/wiki/
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
http://galileo.phys.virginia.edu/classes/252/PlanckStory.htm
http://abyss.uoregon.edu/~js/glossary/
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/