Tuesday, 6 March 2012

Quantum Mechanics: A First Encounter

By Luke Kristopher Davis

Classical physics

“Yes! Physics has given up. We do not know how to predict what would happen in a given circumstance, and we believe now that it is impossible, that the only thing that can be predicted is the probability of different events.”
Richard Feynman {1}

Explaining the world through certain principles and mathematics is the essence of physics. We observe events in the world and begin to try and understand their mechanisms through experiment, the data these experiments produce are then modelled by mathematics. A physicist then, with a great physical insight, begins to formulate general principles which underpin not just a specific physical phenomenon but are deemed mechanisms of the whole natural world.

The principles of classical physics, which was made intelligible by Sir Isaac Newton, have been the principles which we thought the whole natural world obeyed. The most significant of these principles is the principle of motion:

Eq{1}

For which the force acting on a mass m is the scalar product between m and the second derivative of its displacement with respect to a duration t. The mass is viewed as being a point particle in space and matter in general communicate through mutually interacting forces. Matter can be described in terms of its energy and momentum (E,p).

Classical physics contains a description of electric and magnetic phenomena which are conveniently described in terms of E and B which represent the electric and magnetic fields respectively. These two fields obey this equation:

Eq{2}

figure 1

Physically this means that these fields travel through space as waves with a constant velocity, c (see Figure 1). This velocity is the speed of light and hence we recall that these two fields act as electromagnetic radiation which embodies; light, UV, x-rays, radio waves, infrared etc.

It is with the Newtonian point-particle mechanics and the classical electromagnetic wave theory which have enabled humans to describe a great amount of physical phenomena. With time this has ingrained in us the idea that the world should be intrinsically deterministic i.e. we should be able to predict quantities in our equations with a very definite idea of its value[1].

This assumption that determinism is intrinsic to the world will be shown false through the failure of classical physics to describe certain phenomena. As scientists then, we must re-formulate the theory or arrive at a new theory built upon different foundations. It is with the efforts of Planck, De Broglie, Einstein, Bohr, Born, Schrodinger, Heisenberg, Pauli and Dirac that enabled physics to overcome the limitations of classical mechanics and gradually arrive at a new theory of the world which will encompass its small scales.

Breakdown of the Classical theory and origins of quantization

Classical theory struggles to describe the complicated phenomenon of black body radiation, which consists of the thermodynamics of the exchange of energy between radiation and matter. Classical theory supposes that this exchange of energy is continuous and that light of frequency v can give up any amount of energy on absorption. This theory also predicts that higher frequencies of radiation will be emitted as there are much higher frequencies (say above 1 million) than lower frequencies; also we are assuming that every frequency has the same probability for emission. Nature says this is not the case. A black body will radiate at lower frequencies.

In 1901 Planck formulated his law for the radiation rate of a black body by rejecting the assumption that the exchange of energy is continuous, the exchange of energy is discrete which means radiation is quantized.

Eq {3}

Eq {3} represents the radiation rate as the classical radiation rate found by Rayleigh-Jeans times by a factor which is deduced from the quantization of radiation.

Eq{4}                                                        Eq{5}

Here represents the energy E of a discrete packet of radiation, p is the momentum and h=h2pi where h is Planck’s constant. The quantities v and k are the frequency and wave number respectively and are wave quantities. Planck had to change an assumption of classical theory to fit experimental data, this was the first historical crack shown within the marble of classical physics. The reader may also notice the first indication of a wave-particle duality in a theory i.e. the quantified relationship between particle properties and wave properties.

Albert Einstein in 1905 produced a paper[2] on the photoelectric effect which with its empirical verification confirmed Planck’s quantization. The photoelectric effect describes how a light beam incident on a metal surface causes electron emission from that surface. The kinetic energy, E, of an emitted electron is related to the energy of the incident quantized radiation by:
Eq{6}

Here phi represents the threshold work needed to detach an electron and hv is the energy of the photon which collided with the electron. A modern verification of this is in solar panels which uses photons from the sun (hence solar) to generate electrical energy. The photoelectric effect simply confirms Planck’s hypothesis.

Bohr in later years used quantization to describe how electrons exist within the Hydrogen atom; he had shown that electrons exist in discrete energy levels. If an electron moves to a lower energy state, it will have lost energy and does so by an emission of a particle. This particle was shown to be a photon with frequency v.

Eq{7}

We have established then the particle aspects of radiation and the need for it. This then necessitates a falsehood within classical theory which restricts its validity to the motion of macroscopic objects (including gases).

Electron diffraction, Wave-particle duality, Probability and Bohr’s Complementarity

Davisson and Germer[3] (also Sir George Paget Thomson) had revealed the wave nature of electrons through the diffraction patterns of a beam of electrons reflected from the surface of a nickel crystal. The diffraction pattern still existed even when the electrons went through one at a time. This showed that small scale particles can exhibit wave phenomena such as diffraction. De Broglie suggested that the electron could be described by the same equation which relates the particle and wave aspects of radiation. Davisson, De Broglie and Thomson received the Nobel prize in physics (De Broglie in 1929, Davisson and Thomson in 1937).

So the reader may be slightly puzzled by the fact that there are two views of matter on the small scale which are both empirically correct for certain situations. A paradox has arisen but a solution exists which requires us to change our fundamental viewpoint of matter yet again and change our conceptual thinking of the wave.

Consider Young’s interference experiment (see figure 2) which consists of a source of light, S, which passes through a lens, L. This beam will go through two slits A and B and produce interference patterns with bright and dark fringes on a sheet. Let us change the sheet to a photoelectric emitter such that electrons will be emitted if light of threshold frequency is incident on it. We witness wave-particle duality in such an experiment, in order to explain the bright and dark fringes we must describe light as a wave but to understand the electron emissions we must describe light as a particle.

Figure 2

The locations of the bright and dark fringes on P depend upon the separation between A and B. A photon sufficiently small enough to eject an electron from P could not go through both A and B. If one places photon detectors at A and B one only finds whole photons or the absence of a photon, it never finds partial photons. The question of how a photon passing through A ‘could know’ about the presence of B is perplexing.

One possibility is that photons through A and photons through B act on each other in such a way as too produce the interference patterns at P. However this is incorrect as we decrease the intensity of the beam in such a way to only allow one photon pass through either A or B, the pattern still exists!

This experiment highlights the unpredictability of the destiny of any given photon. It will appear on a bright fringe at P but one cannot know which fringe. Furthermore, the intensity distribution over a fringe (see figure 2 on P) serves as a probability distribution for a photon going there. This distribution does not allow for exact prediction of where the emitted electron from P will show up.

Here we find a resolution to the wave-particle duality paradox. Instead of thinking of the wave of radiation which propagates through real space, we interpret the wave function as a probability type function. The wave intensity is the probability density of finding a photon at a particular location.

It is also possible to formulate quantum mechanics by beginning with the classical wave theory and quantizing its equations. This approach is known as quantum-mechanical field theory and its formulation pays homage to Paul Dirac.

In the experiment in figure 2 the wave that falls on P is not a pure plane wave. It can however be separated into plane waves. It is important to consider the state of a particle which has a wave function in the form of a super-position of two or more plane waves. The wave-function of a particle is:

Eq{8}

This is in agreement with De Broglie’s equation. The probability is therefore the absolute value of phi squared:
Eq{9}
Notice that at the end of the equation we have the interference term which is localized (not being averaged over all space). However if Eq{9} is averaged over all space, the interference term averages to zero and the square of absolute value of phi is a measure of the probability of finding the particle at somewhere, without regard to location. It must also be said that the position and momentum of a particle is uncertain when it is described by Eq{8}.

Also the equations we have just written down are wave functions prior to observation. If the particle is observed in some restricted region of space then this observation has disrupted the quantum-mechanical system hence its wave function. This is called the collapse of the wave function and after observation the wave-function behaves appropriately to the observed momentum.

If the characteristic wavelength of a particle relating to the particle momentum is given by:

Eq{10}
Then a particle which is localised in a defined region of space must have a spread of momenta. It can be deduced that the smaller the region of space the greater is the spread of wavelengths and hence momenta. This is an example of the idea of complementary observables in quantum mechanics in which an exact specification of the value of one quantity can be obtained only at the expense of uncertainty of the other complementary quantity.

This is the basis of the Complementarity principle formulated by Bohr, which states that a physical system can only be described in terms of imprecise specification of a pair of complementary quantities. This is expressed as a commutation relation:

Eq{11}
Where x hat and p hat are position and momentum observables respectively.

The uncertainty in the position, x, and the uncertainty in the momentum, p, of a particle are related. It can be shown that this uncertainty relation is an explicit consequence of Bohr’s Complementarity principle shown in Eq{11}. The general form of this uncertainty relation for any wave function which specifies the position and momentum of a particle is:
Eq{12}

This means that if the position of a particle is known very well then as a consequence of this the momentum of the particle will be uncertain by a degree determined by delta(x) and h/2pi.

The atomic and subatomic world implied by these principles and equations goes against all our intuitions about nature. Why should it adhere to our intuitions however? We as biological beings have evolved to function in the macroscopic world (everyday world) why should we assume nature to conform to this evolution? To me it seems that strangeness and the probabilistic nature of quantum mechanics is exciting. It shows the complexity and ingenuity of nature, as if we were trying to find the rules to a grand game, one which as quantum mechanics shows, we play a part.

“If I have seen further than others, it is by standing upon the shoulders of giants.”
Isaac Newton

The Solvay Conference 1927 which shows the like of Einstein (centre), Dirac, Bohr, Heisernberg, Pauli, Shrodinger, Lorentz and many more great minds.

Bibliography
Feynman{1} - Feynman, Leighton, Sands: The Feynman Lectures on
Physics Volume 1 first printed in 1965 Addison-Wesley Publishing Company,
INC. 
Eqs{1-12} – P.T.Matthews, F.R.S.: Introduction to Quantum Mechanics
(Chapters I-III) 1974 McGraw-Hill Publishing Company Limited/ R.H. Dicke,
J.P Wittke: Introduction to Quantum Mechanics Addison-Wesley Publishing
Company, INC/ P.W.Atkins: Molecular Quantum Mechanics second-edition (1:
Historical Introduction) Oxford University Press Figure 2- R.H. Dicke, J.P
Wittke: Introduction to Quantum Mechanics Addison- Wesley Publishing
Company, Inc. The actual drawing was done by Luke Kristopher Davis on
Microsoft Paint. 
Isaac Newton{2} – James Gleick: Sir Isaac Newton
(biography) 
General references (of which aided my general understanding
and knowledge) – Jagdish Mehra, Helmut Rechenberg: The Historical
Development Of Quantum Theory Springer Volume 6 part 1/ P.T.Matthews,
F.R.S.: Introduction to Quantum Mechanics (Chapters I-III) 1974 McGraw-Hill
Publishing Company Limited/ R.H. Dicke, J.P Wittke: Introduction to Quantum
Mechanics Addison- Wesley Publishing Company, INC/ P.W.Atkins: Molecular
Quantum Mechanics second-edition (1: Historical Introduction) Oxford
University Press/ Stehle: Quantum Mechanics/
http://cvitae.org/images/stories151/history_of_quantum.pdf (thank you
author)/ http://noberlprize.org/nobel_prizes/physics/ :Max Born: The
statistical interpretations of Quantum Mechanics/ Murray Gell-Mann: The
Quark and the Jaguar (popular science)

[1] Of course this strict determinism fails within molecular dynamics (thermodynamics) as we have to take averages of certain temperatures and densities. However this statistical mechanics is due to our inability to specify all positions and momenta of small particles, the statistics are not assumed to be intrinsic to the molecular system.
[2] This paper granted Albert Einstein the nobel prize in physics 1921.
[3] C.Davisson and L. Germer, Nature, 119, 558 (1927)

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