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									Quantum Mechanics

     Adapted from:
Classical Mechanics v. Quantum

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  What are Quantum Numbers?
• An electron’s unique “fingerprint” that
  describes it position and behavior

• Quantum Mechanics = explains the
  behavior of very SMALL, FAST moving
     Quantum Mechanics overview
• We will see: electrons have discrete energies,
  not because they are in shells but because
  they can only have certain wavelengths
• Line spectra are not due to electrons jumping
  from shell to shell (as in Bohr’s model)…
• Instead they’re due to electrons transforming
  from one wavelength (waveform) to another
• Each electron is a wave that can be described
  by a series of “quantum numbers”
• There are four quantum numbers: n, l, ml, ms
• The combination of the first 3 defines an
                Quantum Mechanics

I don't like it, and I'm sorry I ever had anything
to do with it. -- Erwin Schrodinger talking
about Quantum Physics
     Classifying electron waves
• The waves of electrons are similarly class-
  ified according to certain variables (n, l, ml)
• The rationale for the numbers is not always
  clear. These numbers come from some
  pretty advanced math. You don’t have to
  know why we use certain formulas for
  determining quantum numbers.
• You do have to know what the formulas are,
  when to use them, and what the resulting
  quantum numbers represent.
 Quantum Numbers
There are four numbers that come into the theory of
electron clouds as waves called quantum numbers.

The first quantum number, n, is the principle energy
level. This is the 1 in 1s2. It can have the values 1,
2, 3, …

The second quantum number, l, is the sublevel.
The nth principle energy level has n sublevels. We
refer to these sublevels by letters: s, p, d, f, g, h, i, j,
k, … Sometimes numbers are used too: 0, 1, 2, 3,
l : The secondary quantum number
• Each value of l is associated with a letter:
• 0 = s, 1 = p, 2 = d, 3 = f
• after 3, the associated letters go
  alphabetically from f up, so 4 = g, 5 = h, etc.
• Normally, we don’t talk about electrons
  beyond l = 3 (the f subshell)
• Whereas n represents size and energy, l
  tells us of the shape (a.k.a. sublevel) of an
  electron (more detail later).
• We often identify electrons by shell and
  subshell: e.g. 1s, 3d, 2s, and 5d subshell
 l : The secondary quantum number
• If n can be thought of as shells, l can be
  thought of as “subshells” dividing each shell
  into subsections … (l = 0  n - 1)

         l = 0 (s)
        n=2                        l = 0 (s)
     l = 0 (s)                     l = 1 (p)
     l = 1 (p)                     l = 2 (d)

                     Use QN WS as study tool
 Quantum Numbers
The third quantum number, ml, is the orbital. Every
sublevel has one or more orbitals. The s sublevel
has 1 orbital, the p sublevel has 3 orbitals, the d
sublevel has 5 orbitals, etc. These orbital can be
indicated by the number ml = l, l-1, …0, -1, … -l

The fourth quantum number, ms, is the spin of the
electron. Electrons can be either spin up or spin
down. ms can be either +½ or -½

Spintronics: This is a new type of electronics which is based on the spin of the
It is possible to filter electrons which have different spins using very thin magnetic
A. Principal QN (n = 1, 2, 3, . . .)
   1. Related to size of the atomic orbital (distance from the
   2. Larger n value indicates higher energy
   3. Larger n value means electrons are less strongly bound
      to nucleus
B. Angular Momentum (sublevel) QN (l = 0 to n  1)
   1. Relates to shape of the atomic orbital.
   2. Each l number is assigned a letter
   3. n = 3, l = 0, 1, 2 (s, p, and d orbitals in the third shell)
C. Magnetic QN (ml = l to  l)
   1. Relates to orientation of the orbital in space relative to
      other orbitals.
   2. 2.      For l = 2, ml = -2, -1, 0, 1, 2 (Five d-orbitals)
D. Electron Spin QN
  (ms = +1/2, 1/2)
  • Relates to the spin states of the electrons.

  2. Electrons are –1 charged and are spinning

  3. Spinning charge creates a magnetic field

  4. You can tell the direction of the spin by which
     way the magnetic moment lines up in an external
     magnetic field
E. Pauli Exclusion Principle
    1. In a given atom, no two electrons can have the
       same set of four quantum numbers (n, l, ml, ms).
    2.    Therefore, an orbital can hold only two
  electrons, and they must have opposite spins.
    3.    Electrons can have the same n, l, and
    ml values
 a) n = 3, l = 2 (d-orbital), ml = -2 (a single d-
 b) That single d-orbital can only hold 2 e-, one
 ms = +1/2, and one with ms = 1/2
   http://newsbureau.upmc.com/TX/Nanotubes04.htm
   http://www.news.utoronto.ca/bin6/050110-832.asp
   http://www.vega.org.uk/series/lectures/feynman/
   http://informationweek.com/story/showArticle.jhtml?articleID=59300089
II. Orbital Shapes and Energies
A. Atomic orbital shapes are surfaces that
   surround 90% of the total probability of
   where its electrons are
  1. Look at l = 0, the s-orbitals
  2. Basic shape of an s-orbital is spherical
  3. centered on the nucleus
  3. Basic shape is same for same l values
  4. Nodes = areas of zero probability
  5. Number of nodes changes for larger n
  6. We will usually just use outer surface
  7. to describe the shape of an orbital
B. p-orbitals
  1. There are no 1p orbitals (n = 1, l = 0 only)
  2. 2p orbitals (n = 2, l = 1) have 2 lobes with a
     node at the nucleus
  3. There are three different p-orbitals (l = 1, ml = -
     1, 0, 1)
     a. 2px lies along the x-axis
     b. 2py lies along the y-axis
     c. 2pz lies along the z-axis
  4. All three 2p orbitals have the same energy =
  5. 3p, 4p, 5p, etc… have the same shape and
     number, just larger
C. d-orbitals
  1. There are no 1d or 2d orbitals (d needs l = 2, so
     n = 3)
  2. 5 degenerate d-orbitals (ml = -2, -1, 0, 1, 2)
  3. 4 of the d-orbitals have 4 lobes which lie in
     planes on or between the xyz axes:   3dxy,
     3dxz, 3dyz, 3dx2-y2
  4. 1 is composed of 2 lobes and a torus-shaped
     area:   3dz2
  5. The 4d orbitals etc…are the same shape, only
D. f-orbitals
  1. n = 4, l = 3, ml = -3, -2, -1, 0, 1, 2, 3
  2. 7 f-orbitals in the fourth shell are degenerate
  3. The f-orbital are only used for the lanthanides
     and actinides and are complex shapes. We
     won’t use them.
E. Orbital Energies
  1. Orbital energies are largely determined by
  2. the n value: 3 > 2 > 1 for H atom (s = p)

  2. But, for polyelectron atoms, the different
  • l values are not all degenerate (s ≠ p)
  • a. 2s is larger than 2p orbital
  • b. 2s “penetrates” the 2p, so is lower energy
  • c. Penetration effects help explain energy
III. The History of the Periodic Table
  A. Patterns in element properties were recognized
     1. Dobereiner (1780-1849) found “triads” of similar elements:
        Cl, Br, I
     2. Newlands suggested in 1864 that elements should be
        arranged in “octaves” because similarities occurred every
        8th element

  B. The Modern Periodic Table
     1. The German Meyer (1830-1895) and Russian Mendeleev
        (1834-1907) independently developed the current
        arrangement of elements
     2. Mendeleev predicted the properties of “missing” elements
Hund’s Rule

Orbitals of equal energy are each
 occupied by ONE electron before any
 orbital is occupied by a SECOND electron

All electrons in a single occupied orbital
 must have the same spin.
Principal Quantum Number

Symbol = n
Represents the main energy level of the
Range = 1- 7
Ex. = 3s
          Principal Quantum number = 3
Angular Momentum Quantum Number

Symbol = l (small letter L)
Represents the shape of the orbital (also
 called sublevel)
Range = 0 – n-1 (whole number)
0 = s (sphere)           1 = p (petal)
 2 = d (double petal)     3 = f (flower)
Magnetic Quantum Number
Symbol = m
Represents the orientation of the orbital
 around the nucleus
Each line holds 2 electrons
              ___ = s
         ___ ___ ___ = p
         -1    0 +1
Magnetic Quantum Number (cont.)

       ___ ___ ___ ___ ___ = d
       -2 -1 0     +1 +2

   ___ ___ ___ ___ ___ ___ ___ = f
   -3 -2 -1     0 +1 +2 +3
    So what is ms (or just “s”)?
• The spin (clockwise or counterclockwise)
  on the electron
• It describes which of the 2 possible
  electrons in any orbital is being described
• Values: +/- ½
Spin Quantum Number

2 Spin States
Clockwise spin = +1/2 (upward arrow)
Counterclockwise spin = -1/2 (downward

A Single orbital can hold two electrons, but
  they must have opposite spins
Unit 2 – Electrons and Periodic
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