ON THE ROAD AGAIN

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ON THE ROAD AGAIN

• Final Destination Quantum Mechanics

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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

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Dont 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.)

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Why are we going here?

• Two fundamental discrepancies between
  experimental observations and classical physics
  were noticed in the late 1800s.
• 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.

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EVERYONE JUMP IN THE VAN! Were 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 ?(?,T) energy function, holding T
  constant, the individual wavelengths ? 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

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Results
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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 ? (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

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• Notice the dotted line

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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
  Boltzmanns constant
• But it is easily seen that by applying basic
  calculus the limit
• of as ?
  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.

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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.

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Problem 1 Flat Tirenow 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

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Whats 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/m2 1/n2) known as Balmers 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

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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

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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.

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• 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.

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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)

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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 Munichs
  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

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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
  didnt explain reality.

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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
  Kirchhoffs successor
• These events gave him the background needed to
  make his revolutionary discoveries

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What exactly were Plancks 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

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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.

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• 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.

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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.

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• 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 e (modernly
  referred to as quanta)

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Final result!! ?

• Thus, Planck had a theoretical basis and
  therefore showed that the energy e , is related
  to the frequency v by
• e hv. where h is Plancks constant

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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 Maxwells 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 e 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.

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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 e 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.

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Mile Marker 1

• With Plancks recognition that energy could in
  fact be discretely quantized, an entire new wave
  of physical thought arose
• Plancks 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.

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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 Lnh
• 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

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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.

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Lets Derive the Theory

• Bohr began by assuming Rutherfords 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.

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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.

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Whats 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.

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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
  Balmers equation (presented earlier) and found
  that it was the ½ factor that allowed for an
  accurate fit)

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• 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, Balmers
  equation is obtained, and thus, the spectral
  lines emitted by Hydrogen are accurately depicted
  by Bohrs expression.

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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.

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Does it Work?

• Bohrs model works quantitatively for
  one-electron 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)

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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

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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

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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

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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
• Dont you appreciate their work better now???

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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

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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

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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

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• 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 (AB?BA usually)
• We now know this is usually the case when A and B
  are (n x n) matrices, ngt1, 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
  Schools understanding of the nature of the atom

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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 ideasmaking them
  team Old School

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Einsteins 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 realitywhich
  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

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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

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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

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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

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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

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Subatomic Kombat

• Heisenbergs Matrix mechanics had no physical
  interpretation, but worked quitewell.
• 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

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Matrix Mechanics and Wave Mechanics Formalisms
are Mathematically Equivalent!
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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!!!

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• New School worked together in Copenhagen as a
  team to push the matrix mechanics model
• Heisenbergs 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 centurys worth of thought

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• At the 1927 Slovay Congress, de Broglie suggested
  a synthesis of the wave andparticle nature of
  matter.
• Pauli used a specific example to criticize de
  Broglies 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 Borhs
  thoughts
• The subjective epistemological criterion of the
  need for classical concepts to describe the
  results of measurements Cushing on Copenhagens
  thoughts

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• 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!!

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Works Cited

• Wikipedia- The Free Encyclopedia
• Philosophical Concepts In Physics James Cushing
• From X-Rays to Quarks Emilio Segré