Origin and discovery of quantum mechanics

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1 Origin and discovery of quantum mechanics Interplay of eye and mind Physics look at nature. Ask question about nature and try to give answer them, imagine answers. For instance, why does the sun shine? Why do stars shine? Why is the sky blue? Why do metals emit light when heated to very high temperature? In physics one can make mistakes but one cannot cheat! There are many reasons to learn quantum physics. All physics is quantum physics, from elementary particles to the big bang, semiconductors, and solar energy cells. Our world is filled with advanced technologies. Many of these new technologies come from the fundamental research within the framework of quantum theories. In order to understand modern physics, three fundamental links are necessary: quantum mechanics, statistical physics and relativity. Quantum mechanics play a key role in engineering. It will become increasingly relevant in nanotechnology, semiconductors, polymer technology, nuclear/photonic devices, magnetic devices, optics and many other things. New ideas come only from the minds of creative thinkers. Physicist learn to use their intelligence and can explain their findings. Quantum theory is subtle. Mysteries of light: Blackbody radiation In physics, two great discoveries of the 20 th century is based on properties of light: Relativity ( ) and quantum physics with black body theory ( ). In the 18th century, Newton decided that light was made of corpuscles (particles), because only particles can travel along straight lines. However, since the end of the 17th century, interference and diffraction phenomena were known and the 19th century saw the success of wave optics. Nobody could imagine the incredible answer of quantum theory. It is a matter of experiences that a hot object can emit radiation. A pieces of metal stuck into a flame can become red hot. At high temperature it can become white hot then red hot then blue hot. The discovery of quantum mechanics could have happened by analyzing frequency distribution of radiation inside an oven (black body) at temperature T. A blackbody is an object that is a perfect absorber (emitter) of radiation (in ideal case). Figure shows experimental measurements of the thermal radiation at several temperatures. What is the origin of this radiation? This was the major topic of 19th century physics. Please carefully review the following: Figure 1. Measured distribution of thermal radiation at several temperatures.

2 Consider a cubic cavity of volume V and length L. The electrons or atoms on the surface of the cavity act as harmonic oscillators. When the material is heated then electrons or atoms gain kinetic energy and they begin to oscillate. Meanwhile we mention here that energy of the classical harmonic oscillator is Oscillating charged particles emits radiation (light). The emitted radiation in the hot cavity produce standing wave and number of modes per unit frequency per unit volume (number of degrees of freedom for frequency ν) is given by: (For evaluation of number of modes visit the web page: In order to calculate energy density of emitted radiation from cavity we can use: According to the classical theories, energy of each oscillator is continuous and average energy per mode (per degree of freedom) can be calculated as follows: Where E is energy of the oscillator, k is Boltzmann constant and T is temperature. We evaluate this integral and we obtain: Then energy density of emitted radiation from a cavity can be written as: This is Rayleigh-Jeans classical formula. This formula can also be expressed interms of wavelength by using: and, we obtain: It is obvious that this formula is not compatible with the experimental results at high frequencies. Planck made the assumption that an exchange of energy between the electrons in the wall of the cavity and electromagnetic radiation can only occur in discrete amounts. Basic quantum of energy can be written as Where the constant is called Planck s

3 constant. Furthermore, energy can only come in amounts that are integer multiples of the basic quantum: An immediate mathematical consequence of this assumption is that the integrals in the average energy equation turn into discrete sums. So when we calculate the average energy per degree of freedom, we must change all integrals to sums To evaluate this formula we use analogy of the geometric series Then average energy can be written as: Then energy density is given by This worked brilliantly! It provide a good fit with the experimental results. Classically the emission and absorption of energy to be continuous. Then, Planck suddenly changed the story, moving to a totally nonclassical concept, that the oscillators could only gain and lose energy in chunks, or quanta. (Incidentally, it didn t occur to him that the radiation itself might be in quanta: he saw this quantization purely as a property of the wall oscillators.) As a result, although the exactness of his curve was widely admired, and it was the Birth of the Quantum Theory (with hindsight), no-one including Planck grasped this for several years! Other equations governing blackbody radiation Wien s displacement law Experimentally, the peak of the spectrum was found to obey with the following relation: Wavelength of maximum peak of a black body radiation can be obtained from this relation. Stefan-Boltzmann Law This law states that the power emitted per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature. The total radiation energy perunit volume in the cavity: ν Where a= J/m 3.K 4. We can relate this energy density to the energy I emitten per second from the surface of the black body. Without further discussion

4 Where the fundamental constant. This expression had been derived earlier by Boltzmann using thermodynamics arguments. This expression is Stefan-Boltzmann expressions. Energy of the photon Planck s assumption also change our understanding about energy and intensity of electromagnetic radiation. The term intensity has a particular meaning here: it is the number of waves or photons of light reaching your detector; a brighter object is more intense but not necessarily more energetic. More energetic More intense Photon's energy depends on the frequency only, not the intensity. The photons in a beam of X-ray light are much more energetic than the photons in an intense beam of infrared light. Particles of Light: Photoelectric effect In 1887, the photoelectric effect was discovered by Heinrich Hertz. He observed that the metal plates emits electrons depends on wavelength of the light. Only light with a frequency greter than a given treshold frequency will produce a current through the circuit. Lénard (1888) found the energies of the emitted electrons to be independent of the intensity of the incident radiation. Planck s photon model explained black boody radiation. Einstein thought he saw an inconsitency in the way Planck used Maxwell s wave theory of electromagnetic radiation in his derivation. The photoelectric effect is perhaps the most direct and convincing evidence of the existence of photons and the 'corpuscular' nature of light and electromagnetic radiation. That is, it provides undeniable evidence of the quantization of the electromagnetic field and the limitations of the classical field equations of Maxwell. Mathematical Formulation and Experimental Procedure of Photoelectric effect The photoelectric effect exhibits the following: 1) There is a minimum frequency,, called the threshhold frequency (or cutoff frequency) required for the effect to occur.

5 2) The maximum kinetic energy of the photoelectrons does not depend on the intensity of the light. 3) The maximum kinetic energy of the photoelectrons increases as the frequency of the light increases. 4) There is no appreciable time delay between the illumination of the surface and the emission of the photoelectrons. Observation of the photoelectric effect is accomplished with the arrangement shown. Ejection of photoelectrons causes a current to be registered in the ammeter A. Increasing the voltage V repels the electrons from the cathode C. The value of V that reduces the current to zero is called the stopping voltage V s. Then the work done on the photoelectron to keep it from reaching the cathode (collector) is. ; The electrons is bounded to the metal surface with a potential energy., where U = potential energy of an electron, and -φ is the highest value of U. Energy conservation requires: φ is called the photoelectric work function of the metal. The particle-particle collision concept explains the immediate ejection of photoelectrons. Since K max cannot be less than zero, the minimum frequency is explained: for Where ν c is cutoff frequency and the result shows no dependence on light intensity for K max. The other experiments shows particle properties of light are: Compton scattering Raman scattering Wave-Particle Duality Pair Production : Pair production is the formation or materialization of two electrons, one negative and the other positive (positron), from a pulse of electromagnetic energy traveling through matter, usually in the vicinity of an atomic nucleus. Pair production is a direct conversion of radiant energy to matter. It is one of the principal ways in which highenergy gamma rays are absorbed in matter. For pair production to occur, the electromagnetic energy, in a discrete quantity called a photon, must be at least equivalent to the mass of two electrons. The mass m of a single electron is equivalent to 0.51 million electron volts (MeV) of energy E as calculated from the equation formulated by Albert Einstein, E = mc 2, in which c is a constant equal to the velocity of light. To produce two electrons, therefore, the photon energy must be at least 1.02 MeV. Photon energy in excess of this amount, when

6 pair production occurs, is converted into motion of the electron-positron pair. If pair production occurs in a track detector, such as a cloud chamber, to which a magnetic field is properly applied, the electron and the positron curve away from the point of formation in opposite directions in arcs of equal curvature. In this way pair production was first detected (1933). The positron that is formed quickly disappears by reconversion into photons in the process of annihilation with another electron in matter. Wave Behavior of Particle What is this wave? (Review diffraction and intereference phenomena) And why is this result so extraordinary? After particle behavior of wave accepted, the question became whether this was true only for light or whether material objects also exhibited wave-like behavior. De Broglie's Hypothesis In his 1923, Louis de Broglie made a bold assertion. Considering Einstein's relationship of wavelength to momentum p, de Broglie proposed that this relationship would determine the wavelength λ of any matter, in the relationship: This wavelength is called the de Broglie wavelength. This equation and energy of the photon can be written as: Where is angular wavenumber and ω is angular frequency. Significance of the de Broglie Hypothesis The de Broglie hypothesis showed that wave particle duality was not merely an aberrant behavior of light, but rather was a fundamental principle exhibited by both radiation and matter. As such, it becomes possible to use wave equations to describe material behavior, so long as one properly applies the de Broglie wavelength. This would prove crucial to the development of quantum mechanics. Experimental Confirmation Electron diffraction In 1927, physicists Clinton Davisson and Lester Germer, of Bell Labs, performed an experiment where they fired electrons at a crystalline nickel target. The resulting diffraction pattern matched the predictions of the de Broglie wavelength. Electron diffraction refers to the wave nature of electrons. Electrons are incident on a crystal. The periodic structure of a crystalline solid acts as a diffraction grating. Interference of electrons shows that electron act as wave. Electrons are accelerated in an electric potential, then their velocities are: Then de Broglie relation takes the form: Where is energy of fired electrons. Neutral atoms Experiments with diffration and reflection of neutral atoms confirm the application of the de Broglie hypothesis to atoms, i.e. the existence of atomic waves which undergo diffraction, interference and allow quantum reflection by the tails of the attractive potential.

7 This effect has been used to demonstrate atomic holography, and it may allow the construction of an atom probe imaging system with nanometer resolution. The description of these phenomena is based on the wave properties of neutral atoms, confirming the de Broglie hypothesis. Waves of molecules Recent experiments even confirm the relations for molecules and even macromolecules, which are normally considered too large to undergo quantum mechanical effects. In 1999, a research team in Vienna demonstrated diffraction for molecules as large as fullerenes. The researchers calculated a De Broglie wavelength of the most probable C 60 velocity as 2.5 picometer. In general, the De Broglie hypothesis is expected to apply to any well isolated object. Macroscopic Objects & Wavelength Though de Broglie's hypothesis predicts wavelengths for matter of any size, there are realistic limits on when it's useful. A baseball thrown at a pitcher has a de Broglie wavelength that is smaller than the diameter of a proton... by about 20 orders of magnitude. The wave aspects of a macroscopic object are so tiny as to be unobservable in any useful sense. Bohr Atom In 1911, Rutherford introduced a new model of the atom in which cloud of negatively charged electrons surrounding a small, dense, positively charged nucleus. This model is result of experimental data and Rutherford naturally considered a planetary-model atom. The laws of classical mechanics (i.e. the Larmor formula, power radiated by a charged particle as it accelerates.), predict that the electron will release electromagnetic radiation while orbiting a nucleus. Because the electron would lose energy, it would gradually spiral inwards, collapsing into the nucleus. This atom model is disastrous, because it predicts that all atoms are unstable. To overcome this difficulty, Niels Bohr proposed, in 1913, what is now called the Bohr model of the atom. He suggested that electrons could only have certain classical motions: 1. The electrons can only travel in special orbits: at a certain discrete set of distances from the nucleus with specific energies. 2. The electrons of an atom revolve around the nucleus in orbits. These orbits are associated with definite energies and are also called energy shells or energy levels. Thus, the electrons do not continuously lose energy as they travel in a particular orbit. They can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency ν determined by the energy difference of the levels according to the Planck relation: 3. Kinetic energy of the electron in the orbit is related to the frequency of the motion of the electron: For a circular orbit the angular momentum L is restricted to be an integer multiple of a fixed unit: where n = 1, 2, 3,... is called the principal quantum number. The lowest value of n is 1; this gives a smallest possible orbital radius of nm known as the Bohr radius. Bohr's condition, that the angular momentum is an integer multiple of ħ was later reinterpreted by de Broglie as a standing wave condition: the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit:

8 The Bohr model gives almost exact results only for a system where two charged points orbit each other at speeds much less than that of light. To calculate the orbits requires two assumptions: 1. (Classical Rule)The electron is held in a circular orbit by electrostatic attraction. The centripetal force is equal to the Coulomb force. It also determines the total energy at any radius: The total energy is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of r, the energy is zero, corresponding to a motionless electron infinitely far from the proton. 2. (Quantum rule) The angular momentum so that the allowed orbit radius at any n is: The energy of the n-th level is determined by the radius: An electron in the lowest energy level of hydrogen (n = 1) therefore has 13.6 ev less energy than a motionless electron infinitely far from the nucleus. The combination of natural constants in the energy formula is called the Rydberg energy (R E ): This expression is clarified by interpreting it in combinations which form more natural units. We define is rest mass energy of the electron (511 kev) and is the fine structure constant then Bohr Atom and Rydberg formula The Rydberg formula, which was known empirically before Bohr's formula, is now in Bohr's theory seen as describing the energies of transitions or quantum jumps between one orbital energy level, and another. When the electron moves from one energy level to another, a photon is emitted. Using the derived formula for the different 'energy' levels of hydrogen one may determine the 'wavelengths' of light that a hydrogen atom can emit. The energy of a photon emitted by a hydrogen atom is given by the difference of two hydrogen energy levels: where n f is the final energy level, and n i is the initial energy level. Since the energy of a photon is the wavelength of the photon given off is given by

9 This is known as the Rydberg formula, and the Rydberg constant R is R E / hc. This formula was known in the nineteenth century to scientists studying spectroscopy, but there was no theoretical explanation for this form or a theoretical prediction for the value of R, until Bohr. In fact, Bohr's derivation of the Rydberg constant, as well as the concomitant agreement of Bohr's formula with experimentally observed spectral lines of the Lyman (n f = 1), Balmer (n f = 2), and Paschen (n f = 3) series, and successful theoretical prediction of other lines not yet observed, was one reason that his model was immediately accepted. Improvement of Bohr Model Several enhancements to the Bohr model were proposed; most notably the Sommerfeld model or Bohr-Sommerfeld model, which suggested that electrons travel in elliptical orbits around a nucleus instead of the Bohr model's circular orbits. This model supplemented the quantized angular momentum condition of the Bohr model with an additional radial quantization condition, the Sommerfeld-Wilson quantization condition where p r is the radial momentum canonically conjugate to the coordinate q which is the radial position and T is one full orbital period. The Bohr-Sommerfeld model was fundamentally inconsistent and led to many paradoxes. The Sommerfeld quantization can be performed in different canonical coordinates, and sometimes gives answers which are different. In the end, the model was replaced by the modern quantum mechanical treatment of the hydrogen atom, which was first given by Wolfgang Pauli in 1925, using Heisenberg's matrix mechanics. The current picture of the hydrogen atom is based on the atomic orbitals of wave mechanics which Erwin Schrödinger developed in However, this is not to say that the Bohr model was without its successes. Calculations based on the Bohr-Sommerfeld model were able to accurately explain a number of more complex atomic spectral effects. Quantum Tunneling and Quantum Uncertainty Tunneling is a fascinating phenomena both in its own rights and for its many applications. Tunneling refers to the quantum mechanical phenomenon where a particle tunnels through a barrier that it classically could not surmount. The uncertainty principle was first recognized by the German physicist Werner Heisenberg in 1926 as a corollary of the wave-particle duality of nature. He realized that it was impossible to observe a subatomic particle like an electron with a standard optical microscope, no matter how powerful, because an electron is smaller than the wavelength of visible light. Roughly stated, this is the mathematical origin of the uncertainty principle. The particle position and momentum cannot be known simultaneously to arbitrary precision. Mathematically, Heisenberg's result looks like this: Now the uncertainty principle is not something we notice in everyday life. For example, we can weigh an automobile (to find its mass), and all automobiles have speedometers, so we can calculate the momentum. But doing so will not make the position of the car suddenly become hazy (especially if we're inside it). So measuring the momentum of the car seems to produce no uncertainty in the car's position. The reason we don't notice the uncertainty principle in everyday life is because of the size of Planck's constant. It's very small

10 The Copenhagen Interpretation If you ask ten different physicists what the Copenhagen interpretation is, you'll get nine similar (but not exactly the same) answers, and one "Who cares?" The Copenhagen interpretation of quantum physics can be summarized as: 1. The wave function is a complete description of a wave-particle. 2. When a measurement of a wave-particle is made, its wave function collapses. 3. If two properties of a wave-particle are related by an uncertainty relation (such as the Heisenberg uncertainty principle), no measurement can simultaneously determine both properties to a precision greater than the uncertainty relation allows. References Quantum Mechanics, David McMahon Introduction To Quantum Mechanics, Harald J W Müller-Kristen