Download 3 Nov 08 - Seattle Central College

Plan for Mon, 3 Nov 08
• Exam 2, Quiz 5, and Quiz 4 regrades returned
Wednesday
• Bring a USB drive to lab on Wednesday
• Lecture
– Cheer up, it’s only quantum mechanics! 
– Wavefunctions, energies, and the Hamiltonian for the
H atom (not in book)
– Quantum numbers (7.6)
– Orbital shapes and energies (7.7)
– Electron spin and the Pauli Principle (7.8)
Particles and Waves
• EM radiation can behave as either a wave or a particle
depending on the situation
• “Light has properties that have no analogy at the
macroscopic level, and thus, we have to combine two
different ideas to describe its behavior.” (Cracolice &
Peters, 3/e)
Famous Opinions of QM
“A scientific truth does not triumph by convincing its opponents
and making them see the light, but rather because its opponents
eventually die and a new generation grows up that is familiar
with it.”
(Max Planck, 1920)
“All these fifty years of conscious brooding have brought me no
nearer to the answer to the question, 'What are light quanta?‘”
(Albert Einstein, 1954)
“Those who are not shocked when they first come across
quantum physics cannot possibly have understood it.”
(Niels Bohr, 1958)
Famous Opinions of QM
The one great dilemma that nails us…day and
night is the wave-particle dilemma.
(Erwin Schrodinger, 1959)
I think I can safely say that nobody understands
quantum mechanics.
(Richard Feynman, 1965)
Wavefunctions
• A wavefunction is a probability amplitude.
The “square” of a wavefunction gives the
probability density…the likelihood of finding
the particle in region of space.
• The wavefunctions and kinetic energies
available to a quantum particle are
quantized if the particle is subject to a
constraining potential.
• We can determine the wavefunctions and
KEs available to our system by considering
the field of force (the PE) our system is
subject to.
The Hamiltonian
• Erwin Schrodinger developed a mathematical formalism
that incorporates the wave nature of matter.
• H, the “Hamiltonian,” is a special kind of function that
gives the energy of a quantum state, which is described
by the wavefunction, Y.
• H contains a KE part and a PE part:
2
2
2
2


h
d
d
d
ˆ
H   2  2  2  2   V ( x, y, z )
8 m  dx dy dz 
• By solving the Schrodinger equation (below) with a
known Hamiltonian, we can determine the
wavefunctions and energies for quantum states.
Ĥ  E
H-atom wavefunctions
• In the H atom, we are interested in describing the regions in
space where it is likely we will find the electron, relative to the
nucleus…we want the wavefunction for the electron.
• We can model the attraction of the H atom’s single electron to
its single proton with a “Coulombic” potential curve:
er
r
0
P+
e 2
V (r) 
r
• The V(r) potential becomes part of the Hamiltonian for
the electron.

H-atom wavefunctions (cont.)
• The radial dependence of the potential suggests
that we should from Cartesian coordinates to spherical
polar coordinates.
r = interparticle distance
(0 ≤ r ≤ )
e-
p+
 = angle from “xy plane”
(/2 ≤  ≤ - /2)
 = rotation in “xy plane”
(0 ≤  ≤ 2)
H-atom wavefunctions (cont.)
• Then the Schrodinger equation Ĥ  E for the
hydrogen atom becomes:
3-dimensional KE operator in
spherical polar coordinates
   2  
1  
 
1  2 
 2 2  r

 sin 
 2
8  r  r  r  sin   
  sin   2 
Ze2

  E
r
h2
Radial Coulombic
PE operator
H-atom wavefunctions (cont.)
• If we solve the Schrodinger equation using this
potential, we find that the energy levels are
quantized:
2 

Z 2  me4 
Z
E n   2  2 2  2.178x1018 J 2 
n 80 h 
n 
n is the principle quantum number, and can have
integer numbers ranging from 1 to infinity.

The higher n, the greater the distance between the
nucleus (+) and the electron (-).
H-atom wavefunctions (cont.)
• In solving the Schrodinger Equation, two other
quantum numbers become evident:
…the orbital angular momentum quantum number.
Ranges in value from 0 to (n - 1 ).
ml … the “z component” of orbital angular momentum.
Ranges in value from - to 0 to .
• We can characterize the hydrogen-atom orbitals
using the quantum numbers: n, , ml
Orbitals and Quantum Numbers
• Naming the electron orbitals is done as follows
– n is simply referred to by the quantum number
– (0…n - 1) is given a letter value as follows:
•0 = s
•1 = p
•2 = d
•3 = f
- ml (- …0… ) is usually “dropped”
For example: for n = 3, = 2  “3d orbital”
Quantum Mechanical Model
The Bohr model is deterministic…uses fixed “orbits” around a central
nucleus to describe electron structure of atoms.
The QM model is probabilistic…uses probabilities to describe electron
structure.
A probabilistic electron structure is much more difficult to visualize.
HOWEVER, the electronic energy levels are still quantized.
Deterministic vs. Probabilistic
• In the Bohr model, you can
always find the electron in an
atom, just like you can always
find the moon as it orbits the
earth.
• You can always determine the
relative location of the nucleus
and electron in Bohr’s model.
• This is because the electron
follows a particular orbit
around the nucleus.
• In the QM model, the electron
does not travel along a
particular path around the
nucleus.
• You can never determine the
electron’s exact location…you
can only find where it is likely
to be.
• The Bohr orbit is replaced by
orbital which describes a
volume of space in which the
electron is likely to be found.
Quantum Numbers and Orbitals
(cont.)
• Table 7.2: Quantum Numbers and Orbitals
1
2
3
Orbital
0
0
1
0
1
2
1s
2s
2p
3s
3p
3d
ml
0
0
-1, 0, 1
0
-1, 0, 1
-2, -1, 0, 1, 2
# of Orb.
1
1
3
1
3
5
Increasing
Energy
n
Orbital Shapes (cont.)
• Example: Write down the orbitals associated with n = 4.
l = 0 to (n - 1)
= 0, 1, 2, and 3
= 4s, 4p, 4d, and 4f
Ans: n = 4
4s
4p
4d
4f
(1 ml sublevel)
(3 ml sublevels)
(5 ml sublevels
(7 ml sublevels)
Which of the following sets of quantum numbers (n, l, m)
is not allowed?
A. (3, 2, 2).
B. (0, 0, 0).
C. (1, 0, 0).
D. (2, 1, 0).
Electron Orbitals
s orbitals
Orbitals represent a
probability space where
an electron is likely to be
found.
All atoms have all
orbitals, but many of
them are not occupied.
p orbitals
d orbitals
The shapes of the
orbitals are determined
mathematically...they are
not intuitive.
Electron Orbital Shapes
• The “1s” wavefunction has no angular dependence
(i.e., it only depends on the distance from the
nucleus).
3
1  Z 
1s 
  e
 ao 

2 Z r
a0
Probability =
3
1  Z  2 

  e
 ao 

*
• Probability is spherical

Electron Orbital Shapes (cont)
s (l = 0) orbitals
as n increases,
orbitals demonstrate
n - 1 nodes.
Node: an area of
space where the
electron CAN’T be,
ever, no matter how
much it wants to.
Aside: What’s a node??
• Remember the guitar-string standing wave analogy?
• A standing wave is a motion in
which translation of the wave does
not occur.
• In the guitar string analogy
(illustrated), note that standing
waves involve nodes in which no
motion of the string occurs.
Electron Orbital Shapes (cont.)
2p (l = 1) orbitals
not spherical, but lobed.
2pz
2py
2px
2 p
z
1

4 2
3
Z  2 
  e 2 cos
ao 
labeled with respect to orientation along x, y, and z.

Electron Orbital Shapes
(cont.)
3p (l = 1) orbitals
3
3 p
z

2  Z  2
2

  6   e 3 cos
81  ao 

• more nodes as compared to 2p (expected.).
• still can be represented by a “dumbbell” contour.
Orbital Shapes (cont.)
3d (l = 2) orbitals labeled as dxz, dyz, dxy, dx2-y2 and dz2.
Orbital Shapes (cont.)
4f (l = 3) orbitals
We will not
show the
exceedingly
complex
probability
distributions
associated
with f orbitals.
Electron Orbital Energies in
the H-atom
• energy increases as 1/n2
• orbitals of same n, but different
l are considered to be of equal
energy (“degenerate”).
• the “ground” or lowest energy
orbital is the 1s.
Orbital Summary
• Orbital E increases with n.
– At higher n, the electron is farther away from the nucleus...this is a
higher energy configuration.
• Orbital size increases with n.
– There is a larger area of space where you are likely to find an
electron at higher E’s.
• Orbital shape is the same no matter the value of n.
– 3s looks like 1s, except it’s bigger and has more nodes. Same for p,
d, f, etc.
• Number of nodes in an orbital goes as n - 1.
–
–
–
–
1s has zero nodes, 2s has one node, 3s has two nodes...
2px, 2py, 2pz each have one node, 3px, 3py, 3pz each have two nodes
the 3d orbitals each have two nodes, 4d have three, etc.
Note that number of nodes indicates relative energy!
• All atoms have all orbitals…but in an unexcited atom, only
those closest to the nucleus will be occupied by electrons
Orbital Quiz
F•
The shape of a given type of orbital changes as
n increases.
T•
The number of types of orbitals in a given
energy level is the same as the value of n.
T•
The hydrogen atom has a 3s orbital.
F•
The number of lobes on a p-orbital increases
as n increases. That is, a 3p orbital has more
lobes than a 2p orbital.
F•
The electron path is indicated by the surface of
the orbital.