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 , called the “wave function”.
* can be obtained by solving Schrödinger’s equation (a differential equation):
H = E
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 l subshell ml
1 0 1s 0
2 0 2s 0
1 2p -1,0,+1
3 0 3s 0
1 3p -1,0,+1
2 3d -2,-1,0,+1,+2
4 0 4s 0
1 4p -1,0,+1
2 4d -2,-1,0,+1,+2
3 4f -3,-2,-
1,0,+1,+2,+3 and so on...
BORN POSTULATE
The probability of finding an electron in a certain region of space is proportional to 2, the square of the value of the wavefunction at that region.
can be positive or negative. 2 is always positive
2 is called the “electron density”
What is the physical meaning of the wave function?
E.g., the hydrogen ground state
1 1 3/2
1s = e -r/ao (ao: first Bohr radius=0.529 Å)
ao
1 1 3
21s = e -2r/ao ao
21s
Higher s orbitals
Balloon pictures of orbitals
The shape of the orbital is determined by the l
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 r22
Surface = 4r2
r
Volume of shell = 4r2 r
Radial electron densities
r2 2
