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 numbern=1,2,3,...

l : azimuthal quantum numberl= 0,1,…n-1

ml: magnetic quantum numberml= -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 ml1 0 1s 02 0 2s 0

1 2p -1,0,+13 0 3s 0

1 3p -1,0,+12 3d -2,-1,0,+1,+2

4 0 4s 01 4p -1,0,+12 4d -2,-1,0,+1,+23 4f -3,-2,-

1,0,+1,+2,+3and 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 32

1s = e -2r/ao

ao

21s

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 densitiesThe 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

Maximum here corresponds to the first Bohr radius

r2 2