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The reason why most alpha particles pass through the gold foil undeflected is that the nucleus occupies a very small volume compared to the entire atom's volume. As alpha particles are positively charged, they are repelled by the positively charged nucleus, but the tiny nuclear size makes direct hits unlikely.

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In a uni-electron system like the hydrogen atom, the wave number of any spectral line is proportional to $ \frac{1}{n_1^2} - \frac{1}{n_2^2} $, where n_{1} and n_{2} are the principal quantum numbers of the initial and final states involved in the transition.

The Aufbau principle states that electrons fill orbitals in order of increasing energy. It is based on the observation that electrons occupy the lowest available energy levels first.

Matter waves are associated with particles, not electromagnetic fields. The velocity of matter waves is generally less than the speed of light, according to de Broglie's relation.

The emission spectra of atoms, which consist of discrete lines, provided experimental evidence for the quantization of energy in atoms, as proposed by Niels Bohr's atomic model.

In the Schrödinger wave equation, $ \phi $ represents the wave function, which describes the behavior and properties of a particle in quantum mechanics. The wave function contains all the information about the particle's state and its probability distribution.

The alpha ($ \alpha $) particle scattering experiment, performed by Ernest Rutherford in 1911, provided evidence for the existence of a dense, positively charged nucleus at the center of an atom. The unexpected deflection of some alpha particles led Rutherford to propose the nuclear model of the atom.

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Isotopes are atoms of the same element with different numbers of neutrons. Isobars are atoms of different elements with the same mass number (number of protons plus neutrons). Isotones are atoms of different elements with the same number of neutrons.

According to Bohr's model of the hydrogen atom, the total energy of the electron is quantized (allowed only certain discrete values), and the angular momentum of the electron is also quantized and given by the expression $ \sqrt{l(l+1)}\frac{h}{2\pi} $, where l is the angular momentum quantum number and h is Planck's constant.

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No explanation available.