# Quantum Phenomena and the Theory of Quantum Mechanics

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1 Quantum Phenomena and the Theory of The Mechanics of the Very Small Waseda University, SILS, Introduction to History and Philosophy of Science

2 . Two Dark Clouds In 1900 at a Friday Evening lecture at the Royal Institution in London, W. Thompson, then Lord Kelvin, discussed two dark clouds that hung over 19th century physics. One was the failure of Michelson and Morley to detect ether drift. This was the experiment that attempted to measure the speed of two beams of light traveling at right angles to one another. The other was a technical assumption of the Maxwell- Boltzmann theory of heat radiation, which would be involved in the development of quantum dynamics. This was a simplifying assumption which set particles vibrating at discrete intervals.

3 . Planck and Blackbody Radiation Max Plank ( ) was a theoretical physicist working on thermodynamics. He founded, almost inadvertently, a new branch of physics which came to be called quantum physics and went on to become one of the leaders of German science. Blackbody Radiation A blackbody absorbs light of every wavelength and grows warmer as function of this radiation. When a blackbody is heated it should emit light at all wavelengths. Plank studied the spectrum of radiant heat at the Physikalisch-Technische Reichsanstalt for its application in the lighting and heating industries. There were significant discrepancies between the prediction based on continuous radiation and the experimental values.

4 . Energy Quanta Planck introduced energy quanta discrete packets of energy as a purely theoretical device, to explain the experimental values of blackbody radiation. Using a statistical model, he modeled the energy of the body as a statistical characteristic of set of unknown resonators. The important equation is: ϵ nhν, energy, ϵ, is equal to frequency of vibration, ν, times Plank s constant h 6.6 ˆ J/s, times some whole number n 0, 1, 2, 3... This quantum discontinuity was at first not considered physically important. It was a just a simplifying assumption that produced an accurate radiation law.

5 . Einstein s Contribution The Photoelectric Effect, When light shines on a metal plate, the plate emits electrons. The rate of emission is a function of the wavelength of the light. The function has a series of maxima around certain wavelengths. Einstein s paper was a simple argument that this experimental fact could be explained on the basis of Planck s quanta of energy radiation. It implied, however, that the quanta must no longer be considered a simplifying assumption, but must be a physical characteristic of the light. The only way to understand the effect seemed to be to assume that the light acted as quanta that is, that light it made up of particles. This was contrary to a long tradition of viewing light as a type of wave in an assumed ether.

6 . Bohr s Atomic Theory Niels Bohr ( ) came from an academic family and became one of the founders of the study of quantum dynamics and nuclear physics. Bohr began working on atomic theory when he was in Manchester with Rutherford. The lab was full of characters from all parts of the world working with joy under the energetic and inspiring influence of the great man. Bohr was interested in the theoretical conditions for the stability of the atom. Since negative electrons were orbiting the positive protons (and neutrons), it was a question of what arrangements and velocities would keep the whole structure from collapsing.

7 . Rutherford s Atom Rutherford s atom was a mechanical system like planets in orbit around a central star. As the electrons radiated electromagnetic energy (light) they should lose speed and eventually collapse into the nucleus. That is, they should emit energy of slowly varying wavelengths to the surrounding systems, atoms, and so on. This would also mean atoms would run down over time and collapse, but neither of these effects appeared to happen. Bohr realized he could use Planck s quanta to stabilize these orbits. During this process, a colleague pointed out that his model should also account for spectral lines of chemical elements.

8 . Bohr s Atomic Model On the Constitution of Atoms and Molecules, Bohr set the electrons orbiting around the nucleus only at certain determined intervals. When they were in those prescribed positions, he thought they obeyed the laws of classical mechanics but when they absorbed or emitted electromagnetic radiation they did so in quantum jumps. Bohr, 1913 The dynamical equilibrium of the systems in the stationary states is governed by the ordinary laws of mechanics, while those laws do not hold for the transition from one state to another. We no longer hold that the stationary state is governed by ordinary mechanics, but this quote shows a realization that stable and transitionary states are different, and that the laws of mechanics will have to be rewritten for subatomic particles.

9 .

10 . The Implications of the Model Bohr was able to use his model to given an explanation of the red spectral line for hydrogen. He predicted further lines in the ultraviolet and infrared ranges. These were found the next year. That is, the model made novel predictions, which were later confirmed. (The confirmation required Einstien s theory of relativity.) The model, however, indicated that atoms have fundamental behaviors that are unlike anything we encounter with ordinary objects. Both light radiation and electrons seem to exhibit some wave characteristics and some particle characteristics. But the ordinary mathematics and mechanics of waves and particles is very different. Furthermore, there seems to be no way to visualize these atoms.

11 . The Limits of Visualization Bohr to Heisenberg There can be no descriptive account of the structure of the atom; all such accounts must necessarily be based on classical concepts which no longer apply. You see that anyone trying to develop such a theory is really trying the impossible. For we intend to say something about the structure of the atom but lack a language in which to make ourselves understood... In this sort of situation, a theory cannot explain anything in the strict scientific sense of the word. All it can hope to do is reveal connections and, for the rest, leave us to grope as best we can.

12 . Quantum Objects Quantum objects, like electrons and photons and all subatomic particles, have a strange kind of behavior that we never could have imagined by studying the macroscopic world around us. At the end of the 19th century it was a long established fact that light was a wave. Now, Einstein and Bohr were proposing that light is a kind of particle. But the behaviors and effects of ordinary particles and waves are very different. Key Point Quantum objects are neither waves nor particles. They are a new class of objects, unknown in the macro-world that we inhabit, which have some wave-like properties and some particle-like properties.

13 . The Wave Theory of Light One of the simplest confirmation arguments of the wave theory of light was that light which is passed through a double slit displays patterns of wave interference.

14 Wave interference in a bay

15 . One-Slit Experiment When quantum objects are passed through a single-slit apparatus, they behave as we would expect particles to behave. The quantum objects go through the slit and are received on the screen like particles. Actually, there is some wave-like interference, but it is very hard to detect.

16 . Double-Slit Experiment When quantum objects are passed rapidly through a double-slit apparatus, they behave as we would expect waves to behave. We can clearly see the wave-interference patterns. There are a number of bright bands and the brightest are not directly in front of the two slits.

17 . Double-Slit Experiment, with Detectors The strange thing is that when we put detectors near the slits of the two-slit set up, the pattern changes. We get two single-slit results with no wave interference. The overall pattern has been changed, merely by detecting the particles.

18 . Double-Slit Experiment, in Slow Motion We can even set up the apparatus to fire particles slowly. Say, once every few seconds. Then there can be no question that the particles go through individually. Nevertheless, after a while, we see the wave pattern. What is causing the interference?

19 Photographic images of the particles collected over time on a optical screen, using the two-slit apparatus with no detectors.

20 . Randomness Quantum objects have a certain indeterminacy, but this is not total randomness. We can never know exactly both the position and momentum of a quantum object, but we can know their product with great accuracy. We can never know the path that a quantum object took to get to the detector, but we can predict the probability that it will arrive at a detector. We can never know exactly when a quantum object will decay, but we can model the probability of it. (This leads to the concept of half-life.) We can know the probabilities of all these events with great accuracy.

21 . Beam splitter with two detectors If we set up a beam splitter that passes half the particles (photons, electrons, etc.) and reflects the other half, then we get half the particles going one way, half the other. We can set up multiple splitters, such that we have detectors at 50%, 25% and 25%. But what about for any individual particle? We can use the theory of quantum mechanics to predict the probability that it will arrive at any detector. But where is the particle before it arrives at a detector? The theory doesn t say and maybe the theory shouldn t say.

22 . Beam Splitter Experiment Laser t 1 Detector 3 Detector 2. Detector 1 In t 1, the photon leaves the laser and travels to the first semi-mirror.

23 . Beam Splitter Experiment Laser t 2 t 2 Detector 3 Detector 2. Detector 1 In t 2, the photon travels to the second mirror 50% of the time and towards Detector 1 the other 50% of the time.

24 . Beam Splitter Experiment Laser t 3 Detector 3 t 3 Detector 2 t 3. Detector 1 In t 3, the photon travels to Detector 1 50% of the time, to Detector 2 25% of the time, and to Detector 3 the final 25% of the time. It is possible to know where the photon is before it has been detected? How can we say where the photon is at time = 2.5?

25 . Quantum mechanics was developed over a long period of time by many physicists such as Bohr, Born, Heisenberg, Pauli, Schrödinger, Dirac, etc. It is a set of mathematical principles and rules that apply to quantum objects. It is a highly abstract theory that attempts to formulate the quantum behavior of sub-atomic processes through mathematical models that can be used to predict the probabilities of various outcomes. Heisenberg argued that these mathematical rules are the only understanding we will ever have that the underlying structure can never be discovered.

26 . The Uncertainty Principle In 1927, Werner Heisenberg ( ) approached subatomic uncertainty from a different perspective. He used a thought experiment involving a microscope which measures the position of electrons in atoms by firing photons at them and measuring the reaction. We want to know the position and the momentum of the electron. When the photon collides with the electron, however, it disturbs it in such a way that we no longer know its momentum precisely. What we know is x p ě h{4π, where x is change in position and p is change in momentum, and h is Plank s constant. This means, again, we can only develop a statistical idea of the electron s path. In fact, the more precisely we know the position the less precisely we know the momentum and the converse.

27 . Schrödinger s Contribution Erwin Schrödinger ( ) developed a wave mechanics that was meant to be a unified field theory in the tradition of Einstein. In 1926, he published a quantum theory based on his wave equation. It could solve many problems and explain the known phenomena but it had no intuitive physical interpretation. It could predict future states of the quantum system, but the equation gave only probabilistic predictions. Moreover, it implied that a particle had to have more than one possible path at the same time. Born and Pauli concluded that the wave equation simply gave a probability of finding a particle at or near a given place. For them, quantum mechanics was a form of statistical mechanics.

28 . Schrödinger s Cat Schrödinger, however, believed that there was something missing in this statistical account. He tried to exemplify this with the famous thought experiment about a cat. Schrödinger, 1935 One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device... : in a Geiger counter, there is a tiny bit of radioactive substance, so small that perhaps in the course of the hour, one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges, and through a relay releases a hammer that shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.

29 Schrödinger s Cat

30 . Fundamental Indeterminacy The statistical path, or orbital, of a quantum object can be exhibited with x-ray diffraction, but they are technically difficult to produce. Schrödinger s wave equations, however, can be used to calculate the probability distributions. The fundamental particles exhibit either the properties of waves or particles, depending on how they are observed. Bohr Quantum mechanics is about one thing: What can we do with our instruments?

31 . Resistance The paradoxical features of the quantum theory were unacceptable to many of the older generation. Einstein to Max Born, personal communication Quantum mechanics is very worthy of regard, but an inner voice tells me that it is not the true Jacob. The theory yields much, but it hardly brings us closer to the secrets of the Ancient One. In any case, I am convinced that he does not play dice. Wolfgang Pauli ( ) called QM young boy s physics. In 1925, Heisenberg was 23, Pauli 25, Jordan 22, Dirac 22. More than half were under 30 years old and they wrote 65% of the papers in the field. Nevertheless, QM went on to become the most important physical theory of the 20th century and lead to all of the advances in physical chemistry, solid state physics, molecular biology, etc., that have produced our modern world.

32 . What kind of world? Our natural inclination is to believe that the world is something that will make intuitive sense to us. But why do we believe this? How are our intuitions formed? In the 20th century, we had to question our intuitions of space and time; we learned that some attributes of objects appear to be developed by an interaction between observer and observed. Our tools for understanding the world are our ideas our conceptual metaphors and models, our mathematics. But the world may be something unlike anything we have ever thought of and as we build more developed models and new mathematical theories we may find new areas of application for them, and we will have to develop new models to conceptualize new experimental discoveries.

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