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Unformatted text preview: PHYS 302: Unit 15 Quantum Mechanics Supplement – Athabasca University Unit 15 Resonant Cavities: Quantum Resonance
After completing this Unit 15 supplement, you should be able to
1. identify that the wave equation can be solved in cylindrical and spherical coordinate systems. 2. describe boundary conditions suited to cylindrical and spherical coordinate systems. 3. describe atomic energy levels arising from wave effects in atoms, and determine the energy of
EM-radiation arising from transitions between two atomic energy levels. In French (p. 246, Eq. 7-44), the wave equation governing a symmetric wave is
one solution as ( r , t ) C
r 2 2
r r 2
v t 2 . Eq. 7-45 gives sin 2 (vt kr ) . This makes sense, since the energy per unit area of such a wave would go to zero at infinity (i.e., it would spread out with distance to become imperceptible). Another boundary
condition would have to be supplied to determine C; this might be the wave on a surface of known radius.
The analog of a wave equation for a stationary state (similar to a normal mode) of energy E and that is spherically
h 8 2 m ( r 2 2 r ) (r ) e 0Z (r ) E (r )
2 2 2 The solution requires energies given by En nR2
, and the lowest of these states is (r ) 4Z3
a0 exp( rZ )
a0 52.9177×10−12 m.
The illustration on the next page shows the first few hydrogen atom orbitals (energy eigenfunctions) in crosssection, showing a colour-coded probability density for different n=1,2,3 and l=s, p, d, where the letters describe the
shape of the atomic orbital. Black=zero density, and white=highest density. The angular momentum quantum
number l is denoted in each column using the usual spectroscopic letter code (s means l=0; p: l=1; d: l=2). The main
quantum number n (=1,2,3,...) is marked to the right of each row. For all pictures the magnetic quantum number m
has been set to 0, and the cross-sectional plane is the x-z plane (z is the vertical axis). The probability density in
three-dimensional space is obtained by rotating the one shown here around the z-axis.
The energy involved in transitions between such states is readily calculated for various values of n. 1 PHYS 302: Unit 15 Quantum Mechanics Supplement – Athabasca University Note the striking similarity of this picture to the
diagrams of the normal modes of displacement of a
soap film membrane oscillating on a disk bound by a
wire frame. See French, A. P. Vibrations and Waves.
New York: W.W. Norton & Company, 1971, page
186, Fig. 6-13. See also Normal vibration modes of a
circular membrane (online). 2 ...
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