Transcript Document 7267682

ON THE ROAD
AGAIN…
Final Destination – Quantum Mechanics
Taking the Trip From Classical
Physics to Quantum Mechanics
What is Quantum Mechanics one may ask?
Well, in a nutshell, quantum mechanics is:
– The FUNdamental branch of physics replacing
classical Newtonian Mechanics and
Electromagnetism at the atomic and subatomic
levels.
– Quantum mechanics provides explanations for
many phenomena that are unattainable through
classical mechanics
• Namely : quantization, wave – particle duality, the
uncertainty principle, quantum entanglement
Don’t be fooled, this will
be no short trip…
•
•
Quantum theory ideas did not originate overnight. There were numerous
experiments and observed phenomena that eventually led to this modern
way of thought.
The move from General Relativity to Quantum mechanics is a huge step,
since the two theories seemingly contradict each other. The two
fundamental issues yielding this contradiction are;
– Classical is essentially a deterministic approach while quantum
mechanics is essentially indeterministic
– General relativity relies mainly on gravity while quantum mechanics
relies on three fundamental forces, the strong, weak, and
electromagnetic.
• (though these forces are imperative in Quantum Theory, we will
leave out a discussion of them here, for they will be covered in a
latter presentation) For the remainder.. We will look at the major
players on the road to the discovery and development of this way
of thought.)
Why are we going here?
Two fundamental discrepancies between
experimental observations and classical
physics were noticed in the late 1800’s.
These two discrepancies (namely – the ultra
violet catastrophe and the discrete
spectral line phenomena) caused scientist
to re-examine several established ideas
and arrive at a new way of thought.
EVERYONE JUMP IN THE
VAN! We’re hittin the
road!
• Problem 1 : Energy and Blackbody radiation
• Intro;
– A black body is a surface which absorbs all radiation and at a
given temperature T, can also emit radiation
– Thus, using the ρ(λ,T) energy function, holding T constant, the
individual wavelengths λ of the radiation can be analyzed
– At the turn of the 20th century, this in fact was done and the
observations were first recorded by Otto Lummer and Ernst
Pringsheim and then again the following year by Heinrich
Rubens and Ferdinand Kurlbaum
• The results gathered are shown in the following figure
Results
Uh Oh!
•
According to the Equipartition Theorem (a non-quantum theorem)
every degree of freedom of a system must share equally in the
energy available to the system,
– Take for example, a blackbody radiation of dimension l , then
we know that wavelengths are given by λ = (2l )/ n, where n =
1,2,3 ….
– Each of these represent a degree of freedom, and therefore,
by the equipartition theorem, should share the energy equally.
– Thus, since there are infinitely many waves as the wavelength
gets shorter, it would be assumed that the majority of the
light (ie radiation) would be at the short wavelength end of the
spectrum, as shown by the dotted line;
• Notice the dotted line…
Why is this a problem?
Lets do the math…
•
A major problem arises, since we know from derivations that follow directly from the
equipartition theorem, done by Rayleigh and Sir James Jeans, that through classical
analysis, the energy density function can be expressed as:
where k is Boltzmann’s constant
•
•
But it is easily seen that by applying basic calculus the limit
of
as λ approaches 0, is infinity. Obviously
contradicting the observed results shown in the figure by the solid line. If this were
indeed the case, the results would be described by the dotted line in the previous
figure.
This discrepancy and “blow – up” at short wavelengths is often referred to as the
“ultra-violet catastrophe” and spurred much thought as to how to remedy it.
FLAT TIRE!!
Our classic car is getting
old, time to trade it in
• The discrepancy between the
observational results and this equation is
an intense problem. Since the energy
density equation derived by Jeans in 1905,
followed directly from the well established
equipartition theorem, which was a
consequence of general relatively. Thus
because these do not correlate, it implies
that the overlying theory must be
inadequate.
Problem 1 –Flat Tire
now we have Problem 2 –
running out of gas
• The second phenomena leading to quantum
theory stems from the pattern of spectral
lines emitted by an element
– When an electric arc passes through a sample
of gas, it is observed that only certain
frequencies of light are present.
– When observing the spectral lines by
separating them with a prism, it is observed
that the wavelengths are distinct to the
element emitting the spectrum
•
What’s so bad about
that?
This observation directly contradicts the predictions of classical
physics
– According to classical physics, accelerated particles must emit
electromagnetic radiation, and thus if they were moving
randomly within the atom, all the spectra would be emitted,
thus producing a continuous spectrum, however the results
showed a discrete distribution.
– Due to this phenomena, scientist began associating these
spectral emission with stable orbits, and an empirical formula
was even discovered through trial and error efforts, to fit the
observed reality.
This equation is:
v = R(1/m^2 +1/n^2) known as Balmer’s Eq.
where v is the velocity of frequencies of the lines, R is a
universal constant, m is a constant representing a particular
series, and n is an integer representing a particular line
What are we going to do?
• Many models of the atom where constructed to
try to describe this phenomena. However, they
nearly all assumed a uniform distribution of the
positive charge.
– However, in 1909, Hans Greiger and Ernst Marsden
disproved this theory through careful examination of
the scattering of a beam of charged particles. What
would be expected if the previous models were in fact
reality, would have been a uniform scattering of the
particles. However this was not the case
What happened?
•
•
•
•
•
Grieger and Marsden used an experiment suggested by Ernst
Rutherford, a premier physicist and often considered the Faraday
of nuclear physics.
The experiment consisted of using gold tin foil about 400 atoms
thick with He ++ particles as their bombarding projectiles.
Most of the particles went straight through with no deflection,
which seemed to support classical theory, since it is known that
electrons are much smaller and thus when all charge is equally
distributed, there would be little deflection
HOWEVER, this was not the case for all particle, there were a
given number that were scattered back at angles larger than 90
degrees, implying something else is going on.
Rutherford, examined the results and concluded that the bulk of
the atom and the majority of positive charge, HAD to be
concentrated at a single point in the atom.
• Thus, the concept of a nuclear atom model was
presented. However, this causes more problems.
• With this model, under classical theory, it would
necessarily yield a continuous spectrum, since
there is no partitioning of the energy, thus no
constraints on either the frequency or
wavelength.
• With classical theory again failing to explain the
observed phenomena, physicist began searching
for an explanation and theory that accurately
describes reality.
•
Now that we know the
problems… let fix them!
Lets meet the mechanics…
The Big Wigs:
-Max Plank
-Niels Bohr
-Albert Einstein
-de Broglie
Other notable participants (Max Born, John von
Neumann, Paul Dirac)
Who is Max Plank??
• Brief Background:
– Planck came from traditionally intellectual background,
his great grandfather and grandfather were both
theology professors, while his father was a law
professor and uncle was a judge
– He studied under Hermann Muller at Munich’s
Königliches Maximiliangymnasium, where he was taught
mechanics, astronomy, and mathematics
– An incredibly gifted child, he graduated at the age of 16
and in 1874 began studying at university of Munich
Background Continued
– While at university, he was advised by his professors not
to study physics, because they believed that nearly
everything had already been discovered and all that was
left to do was fill in a few holes
– However, this did not discourage Planck, rather he
claimed that he was not in the field to make
revolutionary discoveries, but rather to gain a fuller
understanding of those theories already established
– This desire for complete understanding, consequently
led him to arguably, one of the greatest discoveries in
modern physics, since he refused to accept that classical
theory just didn’t explain reality.
Career background
• Planck did very few experiments before entering into the
realm of theoretical physics. He was more concerned with
why things were happening, than searching for new
observational realities
• He went to Berlin to study with two of the premiere
physicists of the time, Hermann von Helmholtz and
Kirchhoff
• Studying entropy and thermodynamics, he was eventually
appointed to the position of Kirchhoff’s successor
• These events gave him the background needed to make his
revolutionary discoveries
What exactly were
Planck’s findings??
• Planck worked on the Blackbody radiation problem
– Plank was hired by electrical companies to find a way to
produce the most light (radiation) using the least amount
of energy possible.
– Since he had worked under Kirchhoff, Planck new that
Kirchhoff had already contemplated the question of how
the intensity of the radiation emitted by a blackbody
depends on the frequency of the radiation
– Thus, Planck decided to utilize a collection of radiating
harmonic oscillators in thermal equilibrium to describe
black body radiation
Quick Review…
– We know that classical methods had failed to
describe the observational reality as seen
through its discrepancy with the Ryleigh – Jean
equation, since it failed to work for short
wavelengths.
– There was also conjecture presented by
Willheim Weil that described the phenomena
for short wavelengths, but failed for long
wavelengths. Thus Planck decided to utilize
both ideas and interpolate between the two.
• To do this, Planck used a thermo dynamical
argument to produce a two parameter ad hoc
expression which we will see later.
– He did this through modification of classical
relations involving entropy of radiation
• His argument was incredibly complex, and too
intense to derive currently, however, it is
imperative to note that it was incredibly based on
phenomological curve fitting
• Basically, he was left with a curve that fit the
observed data perfectly, but had no solid
theoretical justification for his results.
Why Why Why…
• Thus, he returned to his studies in order to try to
derive a theoretical justification
• To do this, he found that he was required to
utilize statistical – mechanical techniques (which
allows for distributions to be describe on its
micro – state) which had been introduced by
Boltzmann.
• This was a big step, because he had been
extremely reluctant to accept these new
techniques, because he felt they were merely
axiomatic by nature
– However, he claimed “it was an act of despair… I was
ready to sacrifice any of my previous convictions about
physics.”
• Though he was originally reluctant, he
allowed himself to accept these new
techniques, which allowed him to partition
the total energy of the system into
discrete amounts
– Therefore, his oscillators could only absorb
and emit discrete amounts of radiation… which
consequently yielded the proper distribution.
• Through these methods, he arrived at the
notion that the energy absorption and
emission must be quantized into discrete
amounts ε (modernly referred to as
quanta)
Final result!! 
• Thus, Planck had a theoretical basis
and therefore showed that the
energy ε , is related to the frequency
v by:
• ε = hv. where h is Planck’s
constant
Still Hesitant
• Though he was certain that energy absorption and emission
had to be quantized, and was described by the formula, he
was still hesitant to accept energy quantization in
electromagnetic radiation.
– He felt that Maxwell’s electrodynamics, which claimed that an
electromagnetic field could carry continuously varying amounts
of energy, had been too successful to just disregard them
• He spent much time trying to fit his prior finding of ε = hv.
into classical electrodynamics, however, after many failed
attempts, he came to a final conclusion accepting the reality
of quanta, which has since been accepted by nearly all
physicists.
– This is evident today, as we readily use the notion of a photon,
which is merely the name for a quantized electromagnetic
field.
So What?
• How does this solve the problem which arises
from classical mechanics??
– Look at the equation for wavelength
v=c/
we see that Plancks equation ε = hv = hc / 
• Thus, if energy is finite, there must exist a
shortest and a longest wavelength, and thus, if
very few quanta are released when  is either
large or small.
• Further, it is obvious from the equation that the
peak will occur at the most probable frequency.
Mile Marker 1
• With Planck’s recognition that energy
could in fact be discretely quantized, an
entire new wave of physical thought arose
• Planck’s findings are often considered the
birth of quantum mechanics.
• However, this is merely the first step…
lets now look to the problem of discrete
emission of spectral lines.
AN ATOMIC TRANSMISSION
(uh transition)!
Niels Bohr
• Who is Niels Bohr?
– Bohr made many notable contributions to
physics, namely:
•
•
•
•
A model of the atomic structure
Electron orbital momentum is quantized by L=nħ
Notion that electrons travel in discrete orbitals
Notion that when electrons drop from higher to
lower energy, it emits a photon
• The principle of complementarity
Why we need him…
• Though he made notable contributions, we will
focus on his theory of atomic transitions
• Bohr attended the University of Copenhagen and
then went to Manchester to work under
Rutherford, who (as we saw previously) was
actively working on developing an atomic model
• This influenced Bohr greatly, and within four
months of working with Rutherford, he
formulated his theory.
Lets Derive the Theory
• Bohr began by assuming Rutherford’s model, ie an
electron of charge –e and mass m in circular orbit
of radius r about the nucleus (charge +e)
• Thus, if a stable nuclear orbit is to be attained,
the electrostatic force of attraction must yield
the imperative centripetal force.
Where is this going?
• Knowing this, and applying the law of conservation
of energy, Bohr derived the following expression:
which represents the frequency in relation to its
energy.
• However, if we were to apply classical theory
(which implies that an accelerated particle emits
radiation wit frequency equal to that as seen
above) problems obviously arise.
What’s Wrong?
• According to classical theory, the energy E could
be of any value and thus the atom should radiate
all frequencies, yielding a continuous spectrum.
However, we know this is not the case as seen
prior
• Therefore, Bohr began working to attain a set of
discrete orbits such that it is stable only when
the electron is within one the these distinct
orbitals, and thus only emits radiation when
transitioning between them.
Follow the Leader
•
•
Knowing that Planck had quantized energy emitted and absorbed in
oscillators, Bohr decided to quantize the energy of the photons
released when entering and leaving the stable orbits.
With this concept, he derived the equation:
He introduces the factor of ½ because if the the electron is initially
at rest and its final state is in the stable orbit, then it will have
velocity v, thus the average between the two is simply v/2.
However, it must be noted that he did not come up with this justification
until he examined Balmer’s equation (presented earlier) and found that it
was the ½ factor that allowed for an accurate fit)
• Therefore, he felt that the emitted radiation
would be some multiple of this, which led to the
n/2 factor.
• By combining this derivation with the expression
of frequency in terms of energy, Bohr obtained
the expression
Further, when combined with the conservation of
energy law E(final) – E(initial) = hv, Balmer’s
equation is obtained, and thus, the spectral lines
emitted by Hydrogen are accurately depicted by
Bohr’s expression.
Even Better!
• Bohr also stated that using the same
techniques and theory, his derived
expression, was equivalent to
quantizing angular momentum l = mvr.
• This is very efficient, but similar to
past explanations, thus we will not
derive here.
Does it Work?
• Bohr’s model works quantitatively for oneelectron atoms such as hydrogen, ionized helium,
and doubly ionized Lithium.
• Though his original model only worked for these
few elements, his work made immediate impact
and commanded much attention.
• After he presented his work in 1913, many
generalizations were made and a set of rules for
treating atoms was establishes (modernly
referred to as “old quantum theory”)
How we go from one to
another…
• A three step process was employed to
move from classical to quantum theory
when describing atomic structure…
– Initially, classical theory is used to determine
the possible motions of the system
– Secondly, quantum theory is employed to
depict the possible orbits
– And finally, the law of energy conservation is
employed to fix the frequencies during an
atomic transition
Who Cares?
• Does it really matter how Bohr arrived at his
model of the atom, or how Bohr determined that
the frequency of an atom depends on the negative
of its energy raised to the 3/2 power?
• Does anyone need to know this besides scientific
historians and PHIL/PHYS 30389 students?
• Probably not
• However, insight into the process helps make the
discoveries more understandable
Brilliance or
Backpedaling?
• We see Plank makes an ad hoc attempt to make
some argument to justify his curve fitting
• Then, against his will, he accepts the hypothesis
of quantization
• Energy quantization providing the necessary
limitation on the blackbody radiation curve at
short wavelengths was not a moment of brilliance
but instead a drawn out process Plank himself
wanted to avoid
Brilliance or Backpedaling
2?
• Bohr in a desperate panic to obtain a fit to the data (our good
friend the Balmer formula)
• By 1913 Planks quantization of energy is highly accepted so
Bohr is not as adamant about avoiding it
• Bohr is conservative in his methods, for example sought to
quantize energy and not angular momentum
• His attempts to avoid numerous new principles leads to the
acceptance of his theory
• Don’t you appreciate their work better now???
Fork in the Road
• At this point in history the theory of
indeterminism was a serious question for the
enlightened
• Poincare, Høffding, and Kierkegaard all
incorporated quantum theory into their work
– Høffding claimed that decisive events in life
proceed through discontinuities, or sudden ‘jerks’,
which faintly resembles atomic phenomena
• These ideas were in the minds of scientists,
which helps explain why some were more
inclined to accept certain models of quantum
theory
Team ‘New School’, turn
left
• Bohr, Heisenberg, Pauli, Jordon, and Born
• All found in Copenhagen
• Exclusive group who worked together and rarely
sought help from others
• New School were inclined towards a discontinuous
structure in nature at the most fundamental level and
to a doctrine of complementary between opposites
• Discontinuousness was the language used by these
men to describe atomic phenomena
• NOTE: Causality was not the central question in the
development of the theory
‘New School’ does
Philosophy
• Because of the failures of some classical
approaches team New School took up new
philosophical positions on what was possible
– Bohr believed the failure of the classical mechanics
explaining the electron theory of metals was due to an
insufficiency in the classical principles
– Pauli was convinced that a Continuum Field theory, with
particles as singularities, was not possible
– Pauli and Heisenberg decided electron orbitals were
meaningless because of the failure to apply old quantum
theory to molecular systems
– There existed a clear desire to revolutionize the
concepts of the time
•From 1924-1927 Heisenberg worked on his version of the
quantum model
•Of note are his ‘operators’, which have the unusual property
that the order of multiplication matters (A*B≠B*A usually)
•We now know this is usually the case when A and B are (n x
n) matrices, n>1, and * is standard matrix multiplication
•The new math seemed mysterious yet worked extremely well,
seemed promising….and for real there were no other
alternatives
•This new math is discrete (as opposed to continuous) which
appealed to and suited well New School’s understanding of the
nature of the atom
Team ‘Old School’, turn
right
• Einstein, Louis de Broglie, and Schrodinger
• Not as closed at Team New School
• Considered the continuous wave as the
basic physical entity subject to a causal
description
• Team Old School avoids the notion of
discontinuity and other radical
ideas…making them team Old School
Einstein’s influence
• Einstein views the foundational questions of
physics (such as relativity, quantum theory,
unified field theory) as the search for a rational,
causal reasoning which can be comprehended in
terms of objective reality…which suggests a
continuity of basic physical processes
• In 1909 Einstein used Blackbody radiation to
show the radiation exhibited wave and particle
behaviors
• However after showing that the direction a
molecule has after emitting radiation is left to
chance, Einstein proclaimed it a mistake of the
theory
De Broglie
• In 1923 he started the theory of wave mechanics
to try to understand the dual nature of a photon
• Interested in the dual nature of light, proposed a
model of a particle that followed the trajectory
determined by its associated waves
• De Broglie turned mathematical analogy between
waves and particles into theory
• Later Schrödinger gave a plausibility argument for
his wave equation
Review
• New School
– From Copenhagen school
– ‘Discontinuous’ school
– Saw need to revolutionize current principles
• Old School
– Not From Copenhagen school
– ‘Continuous’ school
– Committed to continuous wave as basic physical
entity subject to causal description
Fight!
• The 2 quantum theories did not coexist
long
• Their collision lead to the eventual
consistent interpretation of quantum
mechanics
• We will now see how the Copenhagen (Old
School) interpretation established
dominance over all other lesser versions of
quantum theory
Philosophy actual effects
something!
• Old school is from an older generation
which is less likely to accept philosophical
ideas such as indeterminism and
randomness in nature
• New School more willing to accept radical
theories, such as indeterminism and
unusual atomic motions
Subatomic Kombat
• Heisenberg’s Matrix mechanics had no physical
interpretation, but worked quite
well.
• Heisenberg thought classical mechanics could not be
changed in order to make sense of quantum phenomena
without destroying the theory, hence the need for a new
theory
• Heisenberg bothered by Schrödinger unique formalism
concerned with continuous,
causal, visible properties
• But it turns out
Matrix Mechanics
and Wave
Mechanics
Formalisms are
Mathematically
Equivalent!
The Race is On
• There became a great push to determine the
correct interpretation of Matrix Mechanics
• For calculation purposes many scientist were
adopting the Wave-Mechanics model
• Heisenberg had personal ambitions and could not
let Old School steal his subatomic thunder
• For the love of god the fate of the direction of
theoretical physics was at stake!!!
• New School worked together in
Copenhagen as a team to push the
matrix mechanics model
• Heisenberg’s uncertainty principle
helped establish New School as the
leading school of quantum thought
• Meanwhile Old School is doing their own
thing working on individual projects
• Thus Copenhagen is able to take charge
and influence a century’s worth of
thought
• At the 1927 Slovay Congress, de Broglie suggested
a synthesis of the wave and
particle nature of matter.
• Pauli used a specific example to criticize de
Broglie’s theory de Broglie and others responded
poorly or not at all, leaving the Copenhagen theory
to be accepted.
• Experience has shown the consistency of the
Copenhagen interpretation and that fundamental
atomic phenomena are discrete
• “Physical reality is whatever quantum mechanics is
capable of describing” –Cushing on Borh’s thoughts
• “The subjective epistemological criterion of the
need for classical concepts to describe the results
of measurements” –Cushing on Copenhagen’s
thoughts
• Copenhagen interpretation refers to a
common set of principles shared by a
group of scientists who follow Bohr.
• Differences between Old School should
be clear
• New School was by no means in total
agreement over quantum theory
• The next presentation will explain this!!
Works Cited:
• Wikipedia- The Free Encyclopedia
• Philosophical Concepts In Physics –
James Cushing
• From X-Rays to Quarks – Emilio
Segré