852HW7 - PHYS852 Quantum Mechanics II, Spring 2009 HOMEWORK...

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Unformatted text preview: PHYS852 Quantum Mechanics II, Spring 2009 HOMEWORK ASSIGNMENT 7: 1. Spontaneous emission: In this problem, we will use the Fermi Golden Rule to estimate the spontaneous emission rate of an atom. We consider an atom which is initially excited, and therefore has energy E e . It decays to the ground state, having E g = 0, by emitting a photon of energy planckover2pi1 . To use the FGR, = 2 planckover2pi1 | G ( E i ) | 2 n ( E i ) , we need the density of photon states, n ( planckover2pi1 ), as well as the atom-field coupling constant, G ( planckover2pi1 ). Clearly the decaying atom cannot tell whether it lives in an infinite space, or in a very large box. Hence the decay rate should be the same in either situation, and should be independent of L , the box length. We know from mathematics that any function defined on the interval [0 ,L ] which vanishes at the edge points, can be expanded in terms of sin( k n x ) where k n = n/L , with n being any positive integer. If the box is large enough, then the photon electric field must vanish at the walls. This fieldinteger....
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852HW7 - PHYS852 Quantum Mechanics II, Spring 2009 HOMEWORK...

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