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  • Introduction to Quantum

    Mechanics

    and Quantum Numbers

  • The Quantum Mechanical Model

    quantum mechanics: the application of

    quantum theory to explain the

    properties of matter, particularly

    electrons in atoms

  • Schrdingers Standing Wave

    Erwin Schrdinger 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.

  • Schrdinger 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.

  • Any other orbits of the standing electron

    wave are not allowed because they

    would cause the standing wave to

    cancel out or collapse.

  • Orbitals and Probability Distributions

    Schrdingers work on quantum

    mechanics led to his development of a

    mathematical equation, called the

    Schrdinger 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

  • http://hyperphysics.phy-

    astr.gsu.edu/%E2%80%8Chbase/quant

    um/schr2.html

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  • 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.

    Heisenbergs Uncertainty Principle: the

    idea that it is impossible to know the

    exact position and speed of an electron

    at a given time

  • 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.

  • 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.

  • The greatest probability of finding the

    electron occurs at a distance rmax from

    the nucleus. This distance is the same

    as the distance Bohr calculated for the

    radius of the first circular orbit of

    hydrogens electron.

  • 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.

  • Quantum Numbers

    There are 4 quantum numbers

    (numbers that describe the quantum

    mechanical properties of orbitals; from

    the solutions to Schrdingers wave

    equation)

  • 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. Bohrs theory used only one

    quantum number, which is the main

    reason that it worked well for hydrogen

    but not for other atoms.

  • 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.

  • 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

    Bohrs main energy steps is actually a

    group of several little steps.

  • Notice that the number of sublevels equals

    the value of the principal quantum number.

  • The Magnetic Quantum

    Number, ml

    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.

  • 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, ml , 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.

  • Shapes and Orientations of Orbitals

  • The Spin Quantum Number,

    ms

    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.

  • 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

  • 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,ms, 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.

  • Paulis Exclusion Principle

    In a given atom, no two

    electrons can have the same

    set of four quantum numbers

    (n, l, ml, and ms).

  • Summary of Quantum Numbers

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