Electrons as Waves
Bohr model only works for one electron atom, 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
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
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
Wavefunctions
Standing waves
Ex) String on a guitar
Only a few wave forms
are suitable, which is
determined by length
Y(x) = sin(3q)
l = 2L, L, 2L/3 ….
Wavefunction: Y(x)
Y(x) = sin(2q)
depends on number
of lobes n =1, 2, 3, …
Therefore Y(n,x) is s
series of solutions,
each with a different
energy E(n)
Y(x) = sinq
x
q =px/2L
A Wave in Orbit
A circular path imposes a length on the wave form,
allowing for only an whole number of nodes.
A Little Math
F(y) - is function that depends on y
F(y) = cy - the function F acting on y to give back y
multiplied by some constant c
Ex)
F(y) = 3 y requires that c = 3
F(G(x)) - is the function, F, that acts on G(x),
where G(x) s a function of x.
F(G(x)) = cG(x) - The function F acting on G(x) gives back
G(x) multiplied by some number c
Ex) F(y) = 3 y, and y = G(x) = 4x2+2
F(G(x)) = 3 (4x2+2) = 12x2+2 = 3 G(x) where c = 3
The Schrödinger Equation
H( Y(n) ) = E(n) Y(n)
Y(n) - is the wavefunction
corresponding to the electron
r = 3-D
coordinates
of electron
H(Y(n))- means that H is a function that acts on Y(n)
- H is a function that calculates the total energy
using, Y(n)
- H contains electron-electron, electron-nuclear
iteration, and kinetic energy terms.
- The result is the Energy, E(n), which depends
on n, and the original wavefunction Y(n)
H( Y(n, l,m, s) ) = E(n) Y(n, l, m, s)
The wavefunction depends on four quantum number,
each associated, with a different property on 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
Related to the orientation in space that
of the orbital.
s - Spin Quantum Number
Related to symmetry of wavefunction
n = 1, 2, 3, 4, …
E(n) = -Ry Z/n2
l = 0, 1 …n-1
L = hl/2p
m = -l …+l
s = 1/2, -1/2
Wavefunctions of H
Lets for the moment ignore spin
n=1
l=0
m=0
States of m 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
Wavefunction 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 state of 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
Dp – error in p
DpDr ≥ h/4p
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:
Y(x)
P α r2Y2(n,l,m,s;r)
P(x)
Hydrogen Orbitals
The hydrogen orbital are determined from the wavefunctions
Ex) 1s a Y2( 1, 0, 0, r )
2s a Y2( 2, 0, 0; r )
S orbitals - are spherical, i.e. they are 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
The p orbitals are constructed from the hydrogen wavefunctions
with n > 1, and l = 1.
i.e. Y(2,1,1), Y(2,1,0), Y(2,1,-1)
Y(2,1,1) and Y(2,1,-1), are complex values, and are combined to
make them real valued.
The resulting functions are aligned along the x and y axis. The
remaining function Y(2,1,0) is aligned along z axis.
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
Note that when the nodal plane is crossed the orbital changes sign
The number of nodes increases with n as n-1, i.e 1 for 2p
Px
Py
Pz
D Orbitals
The d orbitals are constructed from the hydrogen wavefunctions
with n > 2, and l = 2.
i.e. 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
other fits between 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