Download Wave Mechanics

Wave Mechanics
• Better description of atoms is obtained using wave
mechanics.
Particles (Bohr)
Electrons
Waves (de Broglie)
•
In 1927, Schrödinger suggested that electrons can be described by
wave functions

Understanding Waves
• Traveling waves
i.e. waves in ocean
• Standing waves
i.e. a guitar string
Standing Waves
boundaries
1 node
Petrucci
Fig. 8-18
2 nodes
node:
A point which
undergoes no
displacement at
all
Standing Waves
Fig. 8-18
What are the possible values of l?
L
L = length
boundaries
l=2L
l=L
2L
l=
n
n=1,2,3,etc
1 node
2L
l= 
3
2 nodes
In General,
Permitted values of l are
said to be quantized!
Wave functions
• Erwin Schrödinger (1927)
– suggested electrons could be described by an
equation
– called wave function (y) - psi
• y is a standing wave within the boundary
of the system
• Simplest system - electron in a 1D box!
The Particle in a Box!!
The Particle in the Box
n ( x) =
2
 nπx 
sin

L  L 
L = length of box
x = displacement along x
axis
n = quantum number
Fig. 9-19, pg. 321
The Particle in a Box I
• particle must be inside a box of length L and the
particle moves in x direction
• similar picture to before…
Fig. 8-20, pg. 301
The Particle in a Box II
What mathematical
function to use for y?
y is a function of x
Functions for waves…
sine or cosine ???
sine
x=0; sin(x) = 0
Fig. 8-20, pg. 301
The Particle in a Box III
General solution
2  nx 
n (x) =
sin

L  L 
L = length of box
x = displacement along x axis
n = quantum number

Fig. 8-20, pg. 301
Particle in a Box IV
• What about energy?
Particle has kinetic energy:
1 2 m 2u2 p2
E K = mu =
=
2
2m
2m
Subatomic particles have wavelike properties:


h
h
l= =
p mu
Substituting for p:
de Broglie Eqn!
h2
EK =
2
2ml
Particle in a Box V
Recall from earlier that for standing waves…
In General,
2L
l=
n
Permitted values of l are
said to be quantized!
n=1,2,3,etc

Therefore,
h2
EK =
2
2ml
n 2h 2
EK =
2
8mL
n=1,2,3…
Energy is quantized!!!
Particle in a Box VI
n 2h 2
EK =
2
8mL
Note: lowest E
n=1,2,3…
0

Consequence:
the particle cannot be at rest
(consistent with Heisenberg principle)
Lowest E is when n=1
Called zero point energy
Particle in a Box VII
Where can we find the electron?
Heisenberg said “both its position and momentum cannot be known
exactly”
In 1926, Born proposed:
We should think about the
probability of finding an electron
in a particular position.
probability
 2
Max Born
Probability
2  nx 
n (x) =
sin

L  L 
2 2 nx 
 (x) = sin 

 L 
L
2
n

Fig. 8-20
Fig 9-20,
pg. 322
Summary
• Particle in 1D box
– Can be represented by wave function ()
with quantum numbers (n; n=1,2,3…)
– # of nodes = n-1
– Energy is quantized
– Probability of finding an electron in a
particular position is proportional to 2
IN AN ATOM THE ELECTRON CAN
MOVE IN 3-DIMENSIONS
Particle in 3D Box
n 2h 2
EK =
8mL2

1D box
2
2
2

n
n
h nx
y
EK =
 2  2  2z 
8m L x L y L z 
2
3D box
In 3-dimensions we need 3
quantum numbers!

Wave functions for Atoms
• Describing  is very complex… we will not go
into detail… we will look at the solutions.
• Atom is 3D; also +ve charge (protons) must
be considered (I.e. electrostatic attraction
of protons and electrons)
Schrödinger Equation:
ˆ
H = E
Atomic #
or
h 2   2   2   2   Ze2
 2  2  2  2  
 = E
8 me  x
y
z  r
Hydrogen Atom
Must solve…
        Ze
 2  2  2  2  
 = E
8 me  x
y
z  r
h
2
2
2
2
2
For the hydrogen atom, the solution to the Schrödinger
equation gives the wave functions for the electrons in
this atom.
These wavefunctions () are called orbitals.
• Wave functions are analyzed in terms of 3 variables required to
define a point with respect to the nucleus.
• Cartesian coordinates (x,y,z) or spherical polar coordinates (r,
q, f) can be used.
theta
Fig. 8-22 (pg. 303)
phi
In the spherical polar coordinate system, orbitals
can be expressed in terms of two functions:
r,q , f  = R(r )Y (q , f )
R(r) is the radial wave function
Y(q,f) is the angular wave function
Each orbital has three quantum numbers to define it.
The particular set of quantum numbers gives a certain functional form
to R(r) and q,f.
Quantum Numbers
Principal Quantum Number
n = 1,2,3,4…
Orbital Angular Momentum Quantum Number
l = 0,1,2,3,…n-1
Magnetic Quantum Number
ml = -l, -l + 1, -l +2,…0, 1, 2,…l - 1, l
Electron spin
• A fourth quantum number is used for electron
spin
ms = +1/2 or –1/2
• ms does not depend on any of the other quantum
numbers.
• An electron generates a magnetic field due to its
spin.
Example
For an orbital with n = 3 and ml = -1, what is (are)
the possible value(s) of l?
• All orbitals with the same value of n are in the
same principal electronic shell:
n = 1 (first principal shell)
n = 2 (second principal shell)
etc.
• All orbitals with the same values of n and l are in
the same subshell.
n=1
n=2
n=3
etc.
l=0
l = 0,1
l = 0,1,2
• Number of subshells = n
• Names given to subshells:
l=0
l=1
l=2
l=3
l =4
etc.
s
p
d
f
g
Number of orbitals in a subshell = the number of
allowed values of ml = 2l + 1
l=0
ml = 0
one s orbital
l=1
ml = -1,0,1
three p orbitals
l=2
ml = -2,-1,0,1,2
five d orbitals
•
•
To designate the s orbital in the first principal shell:
1s
To designate the p orbitals in the second principal shell:
2p
–
Fig 9-22
Pg. 326
Orbitals of the Hydrogen Atom I
• Orbitals are wave functions
• The square of the wave function, 2 gives the threedimensional probability distribution.
• We commonly draw orbitals as these probability
distributions.
• Orbitals have nodes and exhibit phase behavior just
like other waves.
• Recall, that (r,q,f = R(r)Yq,f)
Orbitals of the Hydrogen Atom II
For the 1s orbital
• Angular part: Yq,f) is the same for all s orbitals
1/ 2
•
Radial part: R(r)
 1 
Y ( s) =  
 4 
Z
R(1s) = 2 
 ao 
σ = 2Zr/na0
3/ 2
e
 / 2
The 1s orbital of Hydrogen
Fig. 9-23 (pg. 329)
Orbitals of the Hydrogen Atom II
For the 2s orbital
• Angular part: Yq,f) is the same for all s orbitals
1/ 2
•
Radial part: R(r)
 1 
Y ( s) =  
 4 
1 Z
 
R(2s) =
2 2  ao 
3/ 2
(2   )
 / 2
Number of
radial
nodes:
0
1
2
Fig.
9-23
Pg. 329
Orbitals of the Hydrogen Atom
Probabilities of finding an electron versus distance from the nucleus, we see
that s orbitals possess n−1 nodes, or regions where there is 0 probability of
finding an electron.
p Orbitals of Hydrogen
• The 2p orbital has no radial nodes at finite values of r
• The p orbitals vanish at r = 0
• The angular part of the wave function is a function of q and f
Number of radial nodes = n-l-1
Number of angular nodes = l
The three p Orbitals
Fig. 8-28, pg 311
d Orbitals
dxy
dxz
dyz
dx2-y2
dz2