Download WAVE MECHANICS (Schrödinger, 1926)

WAVE MECHANICS
(Schrödinger, 1926)
The currently accepted version of quantum
mechanics which takes into account the wave nature
of matter and the uncertainty principle.
* The state of an electron is described by a function
y, called the “wave function”.
* y can be obtained by solving Schrödinger’s
equation (a differential equation):
Hy=Ey
^
This equation can be solved exactly only for the H
atom
WAVE MECHANICS
* This equation has multiple solutions (“orbitals”),
each corresponding to a different energy level.
* Each orbital is characterized by three quantum
numbers:
n : principal quantum number
n=1,2,3,...
l : azimuthal quantum number
l= 0,1,…n-1
ml: magnetic quantum number
ml= -l,…,+l
WAVE MECHANICS
* The energy depends only on the principal quantum
number, as in the Bohr model:
En = -2.179 X 10-18J /n2
* The orbitals are named by giving the n value
followed by a letter symbol for l:
l= 0,1, 2, 3, 4, 5, ...
s p d f g h ...
* All orbitals with the same n are called a “shell”.
All orbitals with the same n and l are called a
“subshell”.
HYDROGEN ORBITALS
n
1
2
l
0
0
1
3
0
1
2
4
0
1
2
3
and so on...
subshell
1s
2s
2p
3s
3p
3d
4s
4p
4d
4f
ml
0
0
-1,0,+1
0
-1,0,+1
-2,-1,0,+1,+2
0
-1,0,+1
-2,-1,0,+1,+2
-3,-2,-1,0,+1,+2,+3
What is the physical meaning of the wave function?
BORN POSTULATE
The probability of finding an electron in a certain
region of space is proportional to y2, the square of
the value of the wavefunction at that region.
y can be positive or negative. y2 is always positive
y2 is called the “electron density”
E.g., the hydrogen ground state
y
1s
=
y21s =
1
1
3/2
ao
p
1
p
1
3
ao
e
-r/ao
e
(ao: first Bohr radius=0.529 Å)
-2r/ao
y21s
r
Higher s orbitals
All s orbitals are spherically symmetric
Balloon pictures of orbitals
The shape of the orbital is determined by the l
quantum number. Its orientation by ml.
Radial electron densities
The probability of finding an electron at a distance r from the
nucleus, regardless of direction
The radial electron density is proportional to r2y2
Dr
Surface = 4pr2
Volume of shell =
4pr2 Dr
r2y2
Radial electron densities
Maximum here corresponds to the first Bohr radius