*Reference: A Logical Approach to Theoretical Physics*

## THE WORLD OF ATOM by Boorse

**PART XII – WAVE MECHANICS **

## Chapter 62: The Principle of Least Action (William Rowan Hamilton 1805 – 1865)

**On a General Method of Expressing the Paths of Light, and of the Planets, by the Coefficients of a Characteristic Function. **Hamilton’s mathematics was based on the analogy between the behavior of light of very short wavelength and the behavior of ordinary particles of matter. He demonstrated that the dynamical problem may be solved by considering the motion of a system as though it were a gradual unfolding of a series of states, each one derived from the preceding one by an infinitesimal transformation similar to what we have when a ray of light advances from one wave front to the next.

## Chapter 63: The Wavelengths of Particles (Prince Lois V. de Broglie 1892 – 1987)

**The Undulatory Aspects of the Electron. **At small scales, there is dualism in Nature between waves and corpuscles. The wavelength associated with a corpuscle is as concrete a physical quantity as its mass. The velocity of the corpuscle is equal to the group velocity of its associated wave. Planck showed that energy is connected to frequency, *E = hv*. It can also be shown that momentum is connected to wavelength, *p = h/λ. *This is a fundamental relation of the Theory.

## Chapter 64: A Wave Equation for Particles (Erwin Schrodinger 1887 – 1961)

**Derivation of the fundamental idea of wave mechanics from Hamilton’s analogy between ordinary mechanics and Geometrical Optics**. At close range, the particle appears to be a wave motion represented mathematically by a continuous manifold of wave functions. Using this approach, Schrodinger could derive the arbitrary integers assigned to electron’s energy-levels in the Bohr’s Atomic model.

## Chapter 65: Statistics and Waves (Max Born 1882 – 1970)

**Wave Corpuscles. **The waves do not represent physical vibrations but rather the unfolding of the probabilities of future events from a given initial state. The complex amplitudes of the waves obtained as solutions to Schrodinger’s wave equation can be interpreted better as the probability of finding the electron at a location.

## Chapter 66: The Uncertainty Principle (Werner Karl Heisenberg 1901 – 1976)

**Critique of the Physical Concepts of the Corpuscular Theory of Matter. **Heisenberg discovered the foundation of quantum mechanics—the uncertainty principle.The uncertainty principle refers to a limit on the accuracy with which we can measure certain pairs of quantities simultaneously. Heisenberg introduced his square arrays or matrices, which depict the electron as existing simultaneously in all possible Bohr orbits.

## Chapter 67: The Barrier around the Nucleus (George Gamow 1904 – 1968)

**Quantum Theory of the Atomic Nucleus. **While Gamow’s paper explains the spontaneous emission of particles (alpha decay) from radioactive nuclei, it also validates Schrodinger’s wave equation inside the nucleus and demonstrates the correctness of Born’s concept that the Schrodinger wave function is the probability amplitude for finding a particle in a given small neighborhood of space.

## Chapter 68: Electron Waves (Clinton J. Davisson 1881 – 1958, George Paget Thomson 1892 – 1975)

**Diffraction of Cathode Rays by a Thin Film. **Davisson: Electrons reflect from crystalline surfaces. The relationship between the angle of maximum intensity, the speed of the electrons, and the lattice spacing in the crystal are the same as that for a wave. Thomson: The rings arising from diffraction show the wave properties of the electron. The radius of the rings is inversely proportional to the velocity of the electrons. These experiments provided evidence supporting the De Broglie equation.

## Chapter 69: The Electron and Relativity (Paul Adrian Maurice Dirac 1902 – 1984)

**The Principle of Superposition. **All one needs to know about the observable properties in order to understand their physics is the algebra that governs them (theory of operators). A system in quantum mechanics must be looked upon as simultaneously being in a whole set of states rather than in some particular state. The electron is forced in one of these states due to the perturbation of measurement. Dirac’s idea of states very brilliantly connected the Schrodinger wave function with the probability concept ascribed by Born.

## Chapter 70: “Holes” in the Dirac Theory (J. Robert Oppenheimer 1904 – 1967)

**On the Theory of Electrons and Protons. **Dirac’s theory predicts the existence of an infinite continuum of negative energy. Dirac proposed that all but few of these negative-energy states are filled with electrons with negative energy; negative-energy states that do not have electrons represent protons. Oppenheimer pointed out that this hole theory gave a different mass dependence and led to insurmountable difficulties. He proposed to retain the picture of the electron and the proton as two independent particles of opposite sign and dissimilar mass and to picture all of Dirac’s negative-energy states as filled.** **

## Chapter 71: Complementarity (Niels Bohr 1885 – 1962)

**Discussion with Einstein on Epistemological Problems in Atomic Physics.** The discussion led to the Principle of Complementarity: There are always two complementary and mutually exclusive ways of looking at a physical phenomenon, depending on how we arrange our apparatus to measure the phenomenon. When we deal with an electron, we must use both the wave picture and the particle picture; one is complementary to the other in the sense that the more our apparatus is designed to look for the electron as a particle, the less the electron behaves like a wave and vice versa.

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**MAIN POINTS**

- The behavior of light of very short wavelength approximates the behavior of ordinary particles of matter.
- The wavelength associated with a corpuscle is as concrete a physical quantity as its mass.
- The velocity of the corpuscle is equal to the group velocity of its associated wave.
- Energy is connected to frequency, and momentum is connected to wavelength.
- At close range, the particle appears to be a wave motion.
- The “waves” represent the unfolding of the probabilities of future events from a given initial state.
- The electron seems to exist simultaneously in all possible Bohr orbits.
- The mathematics applied to electrons seems to work inside the nucleus as well.
- A quantum system is simultaneously in a whole set of states rather than in some particular state.

**THEORY****The mathematical development seems to confirm that electrons are a wavelike flow that has variable consistency. Even the nucleus is a pattern of a very condensed flow.**

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