On the road again… Final Destination – Quantum Mechanics

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 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”)

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

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

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

• 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.

• 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.

• 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