1. 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 ^

2. 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 m l : magnetic quantum number m l = -l,…,+l

3. WAVE MECHANICS * The energy depends only on the principal quantum number, as in the Bohr model: E n = -2.179 X 10 -18 J /n 2 * 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”.

4. HYDROGEN ORBITALS nl subshellm l 101s0 202s0 12p -1,0,+1 303s0 13p -1,0,+1 23d -2,-1,0,+1,+2 404s0 14p -1,0,+1 24d -2,-1,0,+1,+2 34f -3,-2,-1,0,+1,+2,+3 and so on...

5. 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?

6. E.g., the hydrogen ground state  1 1 3/2  1s = e -r/a o (a o : first Bohr radius=0.529 Å)  a o  1 1 3  2 1s = e -2r/a o  a o  2 1s r

7. Higher s orbitals All s orbitals are spherically symmetric

8. Balloon pictures of orbitals The shape of the orbital is determined by the l quantum number. Its orientation by m l.

9. 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 r 2  2 Surface = 4  r 2 rr Volume of shell = 4  r 2  r

10. Radial electron densities Maximum here corresponds to the first Bohr radius r22r22