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WAVE MECHANICS (Schrödinger, 1926)

* 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): H y = E y This equation can be solved exactly only for the H atom. ^. WAVE MECHANICS (Schrödinger, 1926).

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WAVE MECHANICS (Schrödinger, 1926)

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  1. * 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): H y = E y This equation can be solved exactly only for the H atom ^ WAVE MECHANICS(Schrödinger, 1926) The currently accepted version of quantum mechanics which takes into account the wave nature of matter and the uncertainty principle.

  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 ml: magnetic quantum number ml= -l,…,+l

  3. 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”.

  4. 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...

  5. 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”

  6. 1 1 3/2 y1s = e -r/ao (ao: first Bohr radius=0.529 Å) p ao 1 1 3 y21s = e -2r/ao p ao y21s r E.g., the hydrogen ground state

  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 ml.

  9. 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 r2y2 Dr Surface = 4pr2 Volume of shell = 4pr2 Dr

  10. Radial electron densities r2y2 Maximum here corresponds to the first Bohr radius

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