Electrons as Waves
Bohr’s model seems to only work for one electron atoms, or ions.
i.e. H, He+, Li2+, ….
It can’t account for electron-electron interactions properly
The wave “like” properties of electron need to be explored to
do job properly.
Louis de Broglie :
All matter has a corresponding wave character, with a wavelength
determined by its momentum p (= mv)
l = h/p = h/mv
Electrons and Baseballs as Waves
Example: Electron moving at 0.1000 C:
l = (6.626*10-34 Js)/(9.109*10-28 kg)*(0.1000*2.998*108 m/s)
= 2.423*10-14 m = 242.3 pm
Example: A 0.100 kg baseball moving at 150. km/hr:
v = (150000 m)/(3600 s) = 41.7 m/s
l = (6.626*10-34 Js)/(0.1 kg)*(41.7m/s)
=1.59*10-34 m
The Correspondence Principle
Macroscopic bodies don’t feel the effect of quantum mechanics due to their
large masses and slow motion
Ex) The wavelength of the base ball is insignificant on the scale of the
base ball
Microscopic bodies do feel the effect of quantum mechanics strongly due to
their small masses and fast motion
Ex) The wavelength of the electron ball is very large on the scale of the
size of the electron i.e. 10-30 m
A Little Math
A function, y = g(x) describes a relationship between y with the variable x.
ex) g(x) = cx , g acts on x to giving x multiplied by some constant c
Consider: F[g(x)] → h (x)
- F is like a function acting on g(x), where g(x) is a function of x.
- F is referred to as an operator, and changes g(x) into another function h(x)
ex) g(x) = cx , and
F[g(x)] =
h(x)
[g(x)]2
Then F[g(x)] = h(x) = [g(x)]2
g(x)
y
= [cx]2
= c2x2
x
Math in Science
In general, problems in science can be written as :
F[g(x)] = h(x)
F acting on g(x) gives back another function h(x).
- The operator
- The solution to this problem is the function g(x) itself.
- Often there are many solutions: g1(x), g2(x), g3(x)…. gn(x)
A special case:
H[gn(x)] = cn gn(x)
- The operator transforms the function to itself time some specific constant, Cn
-This is called and eigen equation, where gn(x) are eigen functions
and cn and eigen values
Ex) A Guitar String
The wave equation
Only a few suitable wave
forms, eigen functions, which
is determined by length
Y(x) = sin(3q)
l = 2L, L, 2L/3 ….
Wavefunction: Y(x)
depends on number of lobes
n =1, 2, 3, … and number of
nodes = n -1
Y(x) = sin(2q)
Y(x) n represents a series
of solutions, each with a
different energy E(n), which
are their corresponding eigen
values
Y(x) = sinq
x
q =px/2L
A Wave in Orbit
The circular path imposes a length limit on the wave function,
thereby allowing for only an whole number of nodes.
The Schrödinger Equation
H( Y(r)n ) = En Y(r)n
r = 3-D
coordinates of
electron
Y(r)n - is the n-th wavefunction corresponding to the electron
H
- is an operator acting on Y(r)n.
- This Hamiltonian operator used to calculate the total
energy , En , from Y(r)n
- It includes contributions from kinetic energy, and energy
from electron-electron and electron-nuclear iterations.
- The result ing Energy, E(n), depends on the quantum
number, n, and the original wavefunction Y(r)n
H( Y(n, l,m, s) ) = E(n) Y(n, l, m, s)
The wavefunction depends on four quantum numbers, each associated
with a different property of the electron.
n - Principle Quantum Number
Determines which shell the electron is in
and the energy of the electron, E(n)
l - Angular momentum Quantum Number
Subshells exist for each shell differing in the
angular momentum value.
m - Magnetic Quantum Number
n = 1, 2, 3, 4, …
E(n) = -Ry Z/n2
l = 0, 1 …n-1
L = h l/2p
m = -l …+l
Related to the orientation in space that of the
orbital.
s - Spin Quantum Number
Related to symmetry of wavefunction
s = 1/2, -1/2
Wavefunctions of H
Lets for the moment ignore
spin
l=0
n=1
m=0
States of l are labeled as:
l=0 S
l=2 D
l=1 P
l=3 F
Therefore this state is: 1s0 = 1s
n=2
n=2
l=0
2s
l=1
m=0
m= 1, 0, -1
2p1, 2p0, 2p-1
2px, 2py, 2pz
Wavefunctions of H
n=3
l=0
m=0
3s
n=3
l=1
m = 1,0, -1
3p1, 3p0, 3p-1
n=3
l=2
m= 2,1,0, -1,-2
3d2, 3d1, 3d0, 3d-1, 3d-2
3d(xy), 3d(xz), 3d(yz),
3d(x2-y2), 3dz2
Wavefunction of H
n=4
l=0
m=0
4s
n=4
l=1
m = 1,0, -1
4p1, 4p0, 4p-1
n=4
l=2
m = 2,1,0, -1,-2
4d2, 4d1, 4d0, 4d-1, 4d-2
n=4
l=3
m= 3,2,1,0, -1,-2,-3
4f3, 4f2, 4f1, 4f0, 4f-1, 4f-2, 4f-3
Heisenberg Uncertainty Principle
Measurement effects the state of the system.
There is a limit imposed on the degree of certainty to which you can
know the position (r) and momentum (p) of a particle
DpDr ≥ h/4p
Dp – error in p
Dr – error in r
p
r
r1
r?
Observation
p?
r2
Exercise
An electron is traveling between 0.11000 C and 0.11500 C.
What is the smallest error in the position you can expect? What
is the error in position if it were a proton?
Need error in momentum Dp?
We know that Dv = 0.00500 C
Therefore Dp = m Dv
For an electron
Dp = (9.109*10-31 kg)*(0.00500 * 2.998*108 m/s)
Dp = 1.36*10-24 kg m/s
Recall that DpDr ≥ h/4p
Dr ≥ (6.626*10-34 Js)/[(4*3.14159)*(1.36*10-24 kg m/s)]
Dr ≥ 3.88*10-11 m
For an proton
Dp = m Dv
Dp = (1.674 × 10-27 kg)*(0.00500 * 2.998*108 m/s)
Dp = 2.51*10-21 kg m/s
Dr ≥ h/(4pDp)
Dr ≥ (6.626*10-34 Js)/[(4*3.14159)*(2.51*10-21 kg m/s)]
Dr ≥ 2.10*10-14 m
Probability Distribution
A particle position and momentum cannot be known exactly
Therefore a particle is characterized by a probability distribution function
The probability distribution is determined by the wavefunction:
P α r2Y2(n,l,m,s;r)
Y(x)
P(x)
Hydrogen Orbitals
The hydrogen orbitals are determined from the wavefunctions
Ex) 1s a Y2( 1, 0, 0, r )
2s a Y2( 2, 0, 0; r )
The S orbitals with n > 0, and l = 0 - spherical, ie. identical in all directions
The probability distribution can be graphed as a function of the radius as
P(r) = r2 Y2
radial
probability
density plot
Notice the nodes in
the wavefunction
For 1s 0 nodes
For 2s 1 nodes
For 3s 2 nodes
Notice the shell
structure as n
increases
P Orbitals
In order to visualize the electron densities in 3D space associated with the p
orbitals, they are constructed from combinations of the hydrogen
wavefunctions with n > 1, and l = 1.
ie. Y(2,1,1), Y(2,1,0), Y(2,1,-1)
These are complex valued, and are recombined to make them real valued.
The resulting functions px and py are aligned along the x and y axis. The
remaining function Y(2,1,0) is aligned along z axis and is referreedd to as pz.
The px and py electron density functions do not correspond to m = 1 and -1,
as pz does to m =0.
P Orbitals
These dumbbell shaped orbital are referred to as the p (polar) orbitals, which
are labeled according to their orientation, 2px, 2py, 2pz
The orbitals are plotted as the boundary enclosing total of 90% probability
The p orbital has one nodal plane where the sign changes when crossed.
The number of nodes increases with n as n-1, ie. 1 for 2p
Px
Py
Pz
D Orbitals
The d orbitals are constructed from the hydrogen wavefunctions with n > 2,
and l = 2.
ie. Y(3,2,2), Y(3,2,1), Y(3,2,0), Y(3,2,-1), Y(3,2,-2)
Y(3,2,2) and Y(3,2,-2) a well as Y(3,2,1) and Y(3,2,-1), are
combined to make them real valued functions.
The four corresponding distribution functions are have four lobes in the xy,
xz and yz plane. (in between the axes)
D Orbitals
A fourth orbital exists in the xy plane aligned on the axes.
The remaining fifth orbital , dz2, resembles a Pz orbital with a donut like
shape in the xy plane (z2-x2 and z2-y2 are superimposed)
Sign changes when nodal plane (cone) is crossed
There are 2 nodes 3d
F Orbitals
Constructed from seven H
wavefunctions to make them
real valued
Composed of 8 lobes
There are 3 nodes for 4f
The Orbitals of the Hydrogen Atom
2 nodes
1 node
0 nodes
Radial
nodes
1 planar node
2 planar nodes
Concepts
Properties of waves (wavelength, frequency, amplitude, speed)
Electromagnetic spectrum, speed of light
Planck’s equation and Planck’s constant
Wave-particle duality (for light, electrons, etc.)
Atomic line spectra and relevant calculations
Ground vs. excited states
Heisenberg uncertainty principle
Bohr and Schrödinger models of the atom
Quantum numbers (n, l, ml)
Shells (n), subshells (s,p,d,f) and orbitals
Different kinds of atomic orbitals (s, p, d, f) and nodes