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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 , 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 ^

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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

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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”.

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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...

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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?

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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

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Higher s orbitals All s orbitals are spherically symmetric

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

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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 rr Volume of shell = 4 r 2 r

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Radial electron densities Maximum here corresponds to the first Bohr radius r22r22

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