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Wave Mechanics
Erwin Schrِdinger (1887-1961)
The Schrِdinger Wave Equation (1926). The description of the behavior of electrons and other particles requires the use of wave concepts. Schrِdinger took this idea to the stage of a complete theory governed by the fundamental differential equation that bears his name.
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Wave mechanics is a series of postulates - its justification is that it works, i.e. provides predictions in agreement with experiment.
The Postulates:
The Schrِdinger Equation
H=E
E is the energy
H is the Hamiltonian operator, contains the kinetic and potential energies of all particles in the system.
The way in which expectation values can be obtained, e.g. the dipole moments of HF, HCl, HBr, and HI.
The probabilistic interpretation of
is called the wave function.
||2 V is the probability of finding the particle in the tiny volume V.
We can't say much more about this in Chem1211, because Schrِdinger's Equation is a differential equation.
Those of you who go on in math will get differential equations the year after calculus. So we put off a detailed discussion until you take physical chemistry, typically as juniors.
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When you get to P. Chem, we show you how to set up Schrِdinger's equation for the hydrogen atom and solve it exactly.
The exact wave functions for the H atom are called orbitals. An atomic orbital describes the distribution of an electron in an atom.
In solving Schrِdinger's equation, three quantum numbers appear in a natural, mathematically rigorous way.
They are:
n, the principal quantum number
n = 1,2,3,4,...
The Schrِdinger Equation is only soluble for positive integer values of n
l, the azimuthal quantum number, can be any non-negative integer < n l =
0, 1, 2, 3, .... (n - 1)
s, p, d, f,... spectroscopic designation
m, the magnetic quantum number, may be any integer between l and -l.
For this 3d orbital, the possible values of the magnetic quantum number are m = -2, -1, 0, +1, and +2.
For H, each orbital corresponds to a different probability distribution for the electron.
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The ground state 1s orbital
n=1, l=0, m=0
For s orbitals, the probability of finding the electron is the same in all directions but varies with distance from the nucleus.
We're most likely to find the electron relatively close to the nucleus.
Go to Overhead
No fixed orbits as in the Bohr or Sommerfeld theories. Instead the electron is in an electron probability cloud. We never know exactly where it is, but we do know where the electron will be 70% of the time, 80%, 90%, etc.
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In ordinary cartesian coordinates, we have three equivalent 2p orbitals
2px, 2py, 2pz
Each is cylindrically symmetrical with respect to rotation about x, y, or z.
On a blackboard, we draw as
2px
The idea is that the electron has some definite probability (e.g. 90%) of being within the contour.
The + means that the sign of the wavefunction is positive for positive values of x.
There is no probability of finding the e- on the yz plane, the nodal surface.
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Go to Figure 5-22
There is a fourth quantum number associated with the spin of the electron.
In addition to its planetary - like motion about the proton, the electron behaves in a certain sense as if it were spinning around its own axis.
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There are two spin quantum numbers: ms = ½ if the spin lines up with an applied magnetic field. ms = -½ if the spin opposes an applied magnetic field.
Many-Electron Atoms
Schrِdinger's Equation cannot be solved analytically for systems with two or more electrons.
However, even minicomputers can solve H-, He, Li+, Be+2, etc. to at least 12 significant figures. This is better than experimentalists can measure spectroscopically. Also, the quantum mechanics itself is probably incomplete if one wants more than 12 sig. figs.
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Li, Be+, B+2 - 3 electron systems may be solved (i.e. Schrِdinger's Equation) numerically to nine significant figures.
Be, B+, C+2 - 4 electrons: Schrِdinger's equation may be solved to about 6 significant figures, or about 0.1 kJ/mole for Be. About 6 sig. figs. for all other elements.
This is somewhat deceptive, since the total energies become large rapidly -- the absolute accuracy as a fraction of the total energy goes down as elements get larger. Fortunately the relative accuracy holds up pretty well....
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This Document Last updated
Sunday, July 11, 1999
Steven S. Wesolowski
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