# Modern Physics

1. How did Rutherford determine the size of a nucleus? What is the reasoning behind it? Rutherford made use of the principle of conservation of energy to determine the size of the nucleus. He made an indirect use of energy to calculate the distance of closest approach as a measure for the size of a nucleus. When the alpha particle is brought to rest, the work done will be equal to its initial kinetic energy. When the speed and hence the kinetic energy is zero, the energy is now electrostatic potential energy.

If the alpha particle momentarily stops when at a distance d from the centre of the nucleus of charge Ze, its electrical potential energy is E = (1 / 4?? ) 2e (Ze / d) The potential energy equals the initial kinetic energy of the particle. Given that Rutherford had values for alpha energy, charge Ze, he was able to solve for distance d. The distance was used to compare the diameter of the atom to the nucleus, which was at least 10,000 times smaller than an atom. 2. What is De Broglie’s matter wave hypothesis? What is De Broglie’s reasoning for such hypothesis? Derive Bohr’s angular momentum quantization using the hypothesis.

Louis de Broglie proposed that matter, consisting of particles, may also exhibit wave-like behavior, like light. Following so, he made a hypothesis that the relation between a particle’s energy E and the frequency of the associated wave is the same as the energy of a photon and the frequency of light radiation. We can follow that: E = hv and p = E / c but we can also find that p = h / ? so we have ? = h / mv wherein ? is the wavelength, h is Planck’s constant, and mv is the momentum (product of mass and velocity). We should take note that de Broglie used the photon to describe his study.

A photon may not have a mass, but it has energy, and thus it has momentum. It possesses wave-particle duality which is also inherent in light. To derive Bohr’s angular momentum, we use ? = h / mv but we still consider the circular orbit that the wave has to fit, so we use the circumference instead of the wavelength (we note that r is the radius and n is a positive integer): n? = 2? r 2? r = nh / mv mvr = nh / 2? For a circular orbit, the angular momentum L is L = mvr so that if we assume that the electron is a wave, we are able to get Bohr’s angular momentum to be:

L = nh / 2? where n could be denoted as the energy level. 3. a) Using X-rays as an example, explain how X-rays occur. X-rays can be produced by means of the electrons (emitted from the negative electrode called the cathode), applied with a voltage that causes them to accelerate, collide with the positive electrode (called the anode) and in the process emit an electromagnetic radiation. The emitted electromagnetic radiation is what we call as the X-ray. X-rays have been used to explore the atomic structure of matter as well as in the development of the quantum theory.

The L? is produced by transitions from higher energy states to a vacated place in the n = 2 (L) shell. b) Using X-rays as an example, explain how X-rays occur. Similarly, the K? line arises from transitions from the n = 2 (L) shell to the n = 1 (K) shell. c) Explain what an Auger effect is. Use specific example of Auger emission to demonstrate why each element has characteristic Auger spectrum? (That is, why is it unique to each element? ) Auger electrons are produced when a sample is bombarded with electrons and a characteristic X-ray produced by inner shell ionization is reabsorbed, ejecting an electron. For example, a Si-K?

(K-L1) X-ray (energy of 1690 eV) may be emitted from a sample or transfer its energy to the L2,3 shell (binding energy ~70 eV), ejecting a Si KL1L2,3 Auger electron (energy 1620 eV). Auger electron production is quantified by fluorescent yield (? ) which is the fraction of inner shell ionization that produce X-rays (thus, 1 – ? gives the fraction of Auger electrons). Auger electron have energies characteristic of their atom of origin, ranging from ~280 eV (C) to 2. 1 keV (S). Given these low energies, Auger electrons only escape from the surface of a sample. 4. How is probability density related to wave function?

There is a direct correspondence between the places where electrons are most likely to be and the places where two waves added constructively. The locations where the probability of finding the electrons is zero also correspond to locations where the waves added to zero. These waves each represent part of an electron’s wave function. A large amplitude is related to a large probability while a small amplitude goes with a small probability. We should also take into consideration that waves can have amplitudes that are either positive or negative, but probabilities can only be positive numbers.

But then again, we could square the wave function, and we obtain the probability density. Solving classical equation give the exact prediction of physical observables. What does solving Schrodinger’s equation tell you about the physical observables? The wave function itself is said to be not observable in the sense that it can be complex-valued. However, the absolute square of the wave function is observable, taking note that the absolute square of the wave function is a probability density.

How does probability nature and uncertainty of quantum mechanics differ from that of tossing a coin (or throwing a dice)? In classical perspective, probabilities add; yet in quantum mechanical perspective, the probability amplitudes add. This lead to the presence of the extra product terms in the quantum case. We can deduce that in quantum theory, even though only one of the various outcomes is obtained in any given observation, some aspect of the non-occurring events can make a difference in the overall probabilities. 5. What is Copenhagen interpretation on quantum wave function?

The Copenhagen interpretation involves Heisenberg’s uncertainty principle which states that the more precisely the position is determined, the less precisely will the momentum be known at this instant made use of matrix mechanics, and was deemed incomplete by Bohr such that he also had his own version of explanation. Bohr’s complementarity principle included wave and particle representation and study. Also, statistical interpretation of Schrodinger’s wave function proved to be part of the Copenhagen interpretation. What are the differences between the results predicted by quantum mechanics and classical physics?

In classical mechanics, a particle can behave with any speed or energy, but in quantum mechanics, the particles must be quantized. Classical mechanics also determines the exact location and velocity of a particle at some future time, but quantum mechanics only determines the probability for a particle to be in a certain location with a certain velocity at some future time. Quantum mechanics permits the superposition of states wherein a quantum particle can be in two different states at the same time, as exhibited in entanglement, while in classical mechanics, only the locality of particles is possible.

What are the similarity & the difference between the probability nature and uncertainty of quantum mechanics and that of tossing a coin? The probability and uncertainty principles employed in quantum mechanics may be similar to classical mechanics in the sense that their probabilities both add. The difference comes with the quantum probability that obtains extra product terms which is different from the classical mechanics’ tossing of a coin. In classical mechanics, tossing of a coin will only yield exactly two probabilities: head and tail.

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