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Unit 12: Part 3 Quantum Mechanics and Atomic Physics

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Overview Matter Waves: The de Broglie Hypothesis The Schrödinger Wave Equation Atomic Quantum Numbers and the Periodic Table The Heisenberg Uncertainty Principle Particles and Antiparticles

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Matter Waves: The de Broglie Hypothesis The momentum of a photon is given by: The de Broglie hypothesis is that particles also have wavelengths, given by:

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Matter Waves: The de Broglie Hypothesis It can be shown that the allowed orbits in Bohr’s theory of the hydrogen atom are those that contain an integral number of de Broglie wavelengths around their circumference.

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Matter Waves: The de Broglie Hypothesis Electrons can be observed to diffract from crystals whose plane spacing is close to the electron wavelength, just as X- rays do. Diffraction is purely a wave phenomenon.

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The Schrödinger Wave Equation The Schrödinger wave equation describes the electron wave and how it propagates in space and time. The solution to the Schrödinger equation is called the wave function. It gives the probability of finding the electron at any location.

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The Schrödinger Wave Equation The most probable place to find the electron is at the Bohr radius, but the probability is significant near the radius as well.

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The Schrödinger Wave Equation Another prediction of this equation is tunneling; a particle has some probability of appearing on the “wrong” side of a barrier. This effect is used in the operation of a scanning tunneling microscope, and is responsible for some forms of radioactive decay.

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Atomic Quantum Numbers and the Periodic Table The energy of an electron orbit is characterized by its quantum number, n. Within the same energy level, different values of the orbital angular momentum are possible.

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Atomic Quantum Numbers and the Periodic Table Electrons with the same energy and angular momentum can have different values of the z component of angular momentum.

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Atomic Quantum Numbers and the Periodic Table Finally, each electron possesses a quantity called “spin,” which can be in either of two directions.

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Atomic Quantum Numbers and the Periodic Table This table summarizes the different quantum numbers and their allowed values.

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Atomic Quantum Numbers and the Periodic Table For multielectron atoms, the electrons fill the quantum states in order of increasing energy. For each “shell” (energy level) there are subshells corresponding to different values of l.

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Atomic Quantum Numbers and the Periodic Table This diagram illustrates the different energy levels.

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Atomic Quantum Numbers and the Periodic Table The Pauli exclusion principle states that no two electrons in an atom can have the same values of all four quantum numbers; that is, no two electrons can be in the same quantum state.

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Atomic Quantum Numbers and the Periodic Table This shows how energy levels fill up, one electron per state:

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Atomic Quantum Numbers and the Periodic Table As n increases, the total number of states per shell increases as well.

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Atomic Quantum Numbers and the Periodic Table The periodic table was originally organized by putting elements with similar chemical behavior in columns of increasing mass. Comparing with electron configurations, we see that the elements in each column have the same number of electrons in their outer shell, called valence electrons.

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Atomic Quantum Numbers and the Periodic Table

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The periods in the periodic table correspond to the outermost shell being filled.

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The Heisenberg Uncertainty Principle The Heisenberg uncertainty principle: It is impossible to know simultaneously an object’s exact position and momentum. In order to understand this, we need to consider how we would measure an object’s position and momentum.

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The Heisenberg Uncertainty Principle Any observation is going to involve interacting with the object in one way or another. If we shine light on a book, the book’s momentum doesn’t really change very much. If we shine light on an electron, though, the electron’s momentum changes a lot.

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The Heisenberg Uncertainty Principle There is a lower limit on how small the product of the uncertainty in position and the uncertainty in momentum can be: The smallness of Planck’s constant means that uncertainty effects are observable only in atomic particles.

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The Heisenberg Uncertainty Principle The uncertainty principle can also be expressed in terms of energy and time; the faster you measure energy (or the shorter the lifetime of an excited state), the more uncertain it is. This leads to natural broadening of spectral lines.

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Particles and Antiparticles When quantum mechanics was expanded to include relativity (Paul A. M. Dirac), it predicted that there should be an antiparticle to the electron. This particle has been observed, and is called the positron. If an X-ray passes sufficiently close to a nucleus, an electron–positron pair may be created. The X-ray must have at least as much energy as the rest energies of the electron and positron:

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Particles and Antiparticles In this picture, the magnetic field serves to deflect the electron and positron in opposite directions, making them easier to detect.

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Particles and Antiparticles Positrons don’t last long once they’ve been produced, though; they annihilate almost immediately with nearby electrons, producing X-rays in the process.

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Review Photon momentum: de Broglie hypothesis: particles also have a wavelength: Atomic electrons are described by a wave function, which gives the probability of finding the electron at a particular position.

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Review Electron orbital energies are determined mainly by the principal quantum number, n. The orbital quantum number gives the electron’s angular momentum. The magnetic quantum number gives the z component of the electron’s angular momentum. The spin quantum number gives the direction of the electron’s spin.

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Review Exclusion principle: no two electrons in the same atom can have exactly the same set of quantum numbers. Uncertainty principle: position and momentum, or energy and time, cannot be simultaneously measured with arbitrary accuracy.

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Review Pair production is the creation of a particle and its antiparticle; pair annihilation is the opposite.

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