Introduction to Quantum Mechanics. and Quantum Numbers

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1 Introduction to Quantum Mechanics and Quantum Numbers

2 The Quantum Mechanical Model quantum mechanics: the application of quantum theory to explain the properties of matter, particularly electrons in atoms

3 Schrödinger s Standing Wave Erwin Schrödinger and Louis de Broglie found that an electron bound to a nucleus in an atom resembled a standing wave, so they began research on a description of the atom based on wave behaviour instead of particle behaviour.

4

5 Schrödinger and de Broglie took the idea of standing waves and applied it to the electron in a hydrogen atom. In their model, the electron is a circular standing wave around the nucleus. This circular standing wave consists of wavelengths that are multiples of whole numbers (n = 1, 2, 3, 4,...). Only certain circular orbits have a circumference into which a whole number of wavelengths can fit.

6 Any other orbits of the standing electron wave are not allowed because they would cause the standing wave to cancel out or collapse.

7 Orbitals and Probability Distributions Schrödinger s work on quantum mechanics led to his development of a mathematical equation, called the Schrödinger wave equation, that could be used to calculate electron energy levels. Orbital: the region around the nucleus where an electron has a high probability of being found

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9 Werner Heisenberg came up with a statistical approach for locating electrons. To measure the location and speed of an object, you must be able to observe it. Heisenberg s Uncertainty Principle: the idea that it is impossible to know the exact position and speed of an electron at a given time

10 The best we can do is to describe the probability of finding an electron in a specific location. wave function: the mathematical probability of finding an electron in a certain region of space Quantum mechanics does not describe how an electron moves or even if it moves. It only tells us the statistical probability of finding the electron in a given location in an atom. The area or region where we are likely to find an electron is an orbital.

11 Using wave functions, physicists have created a three-dimensional electron probability density, which is a plot that indicates regions around the nucleus with the greatest probability of finding an electron. The electron probability density plot for a hydrogen electron in the ground state (lowest energy state) is spherical and is called the 1s orbital.

12 The greatest probability of finding the electron occurs at a distance r max from the nucleus. This distance is the same as the distance Bohr calculated for the radius of the first circular orbit of hydrogen s electron.

13 The two main ideas of the quantum mechanical model of the atom are that electrons can be in different orbitals by absorbing or emitting quanta of energy, and that the location of electrons is given by a probability distribution.

14 Quantum Numbers There are 4 quantum numbers (numbers that describe the quantum mechanical properties of orbitals; from the solutions to Schrödinger s wave equation)

15 The Principal Quantum Number (n) The integer, n, that Bohr used to label the orbits and energies describes a main shell of electrons, and is referred to today as the principal quantum number. Bohr s theory used only one quantum number, which is the main reason that it worked well for hydrogen but not for other atoms.

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17 The Secondary Quantum Number, (l) Arnold Sommerfeld (1915) boldly employed elliptical orbits to extend the Bohr theory and successfully explain that the main lines of the bright-line spectrum for hydrogen were actually composed of more than one line.

18 He introduced the secondary quantum number, l, to describe additional electron energy sublevels, or subshells, that formed part of a main energy level. Using the analogy of a staircase for an energy level, this means that one of Bohr s main energy steps is actually a group of several little steps.

19 Notice that the number of sublevels equals the value of the principal quantum number.

20

21 The Magnetic Quantum Number, m l The scientific work of analyzing atomic spectra was still not complete. If a gas discharge tube is placed near a strong magnet, some single lines split into new lines that were not initially present. This observation was first made by Pieter Zeeman in 1897 and is called the normal Zeeman Effect.

22 He observed, for example, triplets where only one line existed without the magnetic field. The Zeeman effect was explained using another quantum number, the magnetic quantum number, m l, added by Arnold Sommerfeld and Peter Debye (1916). Their explanation was that orbits could exist at various angles. The idea is that if orbits are oriented in space in different planes, the energies of the orbits are different when the atom is near a strong magnet.

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24 Shapes and Orientations of Orbitals

25 The Spin Quantum Number, m s Paramagnetism is another kind of magnetism of substances and is recognized as a relatively weak attraction to a strong magnet. Paramagnetism refers to the magnetism of individual atoms; ferromagnetism is due to the magnetism of a collection of atoms.

26 Samuel Goudsmit and George Uhlenbeck, found that a fourth quantum number was necessary to account for the details of the emission spectra of atoms due to paramagnetism. Since they knew from classical physics that a spinning charge produces a magnetic moment, it seemed reasonable to assume that the electron could have two oppositely directed spin states

27 In 1925, Wolfgang Pauli, suggested that each electron spins on its axis. For an electron, the two spins are equal in magnitude but opposite in direction, and these are the only choices; i.e., the spin is quantized to two and only two values. This fourth quantum number is called the spin quantum number,m s, and is given values of either +1/2 or -1/2. Qualitatively, we refer to the spin as either clockwise or counterclockwise or as up or down.

28 Pauli s Exclusion Principle In a given atom, no two electrons can have the same set of four quantum numbers (n, l, m l, and m s ).

29 Summary of Quantum Numbers

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