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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