Transition Probabilities
In order to develop the concept of transitions between excited states of atoms based completely on the formalism of quantum mechanics, time-dependent perturbation theory will be described and then applied to the problem of spontaneous emission of an excit
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Transition Probabilities
In order to develop the concept of transitions between excited states of atoms based completely on the formalism of quantum mechanics, time-dependent perturbation theory will be described and then applied to the problem of spontaneous emission of an excited atomic state. Every author faces a dilemma when trying to do this at an undergraduate level. The difficulty is that spontaneous emission can most properly be thought of as an interaction of the quantized radiation field with an atom. To do that interaction properly one should quantize the radiation field and apply second quantization to the atom. All of that is beyond the scope of this text, so what is to be done? Many authors choose to do a semiclassical treatment in which a classical radiation field interacts quantum mechanically with an atom. The level of detail varies considerably. This presentation will attempt to be complete, explicitly stating where a result from more advanced quantum mechanics is invoked, even as that result is explained and motivated.
4.1 Time-Dependent Perturbation Theory The time-dependent Schr¨odinger equation describes the time evolution of the state of a physical system and is given by Hψ(t) = i
d ψ(t) dt
(4.1)
At some time t0 (and for all earlier times), assume that the state of the system satisfies the time-independent equation Hψ(t0 ) = Eψ(t0 )
(4.2)
If H is time independent, (4.1) can be integrated to yield
R.L. Brooks, The Fundamentals of Atomic and Molecular Physics, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-1-4614-6678-9 4, © Springer Science+Business Media New York 2013
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4 Transition Probabilities
ψ(t) = e−iH(t−t0 )/ ψ(t0 )
(4.3)
This solution can be readily verified by differentiation. −iH −iH(t−t0 )/ d ψ(t) = e ψ(t0 ) dt −iH ψ(t) = d or i ψt = Hψ(t) dt
(4.1)
What does it mean to have an operator in the exponent as in (4.3)? It means just a series expansion as eAt = 1 + At +
A2 t2 + ··· 2
So when H is time independent, (4.3) gives ψ(t) = e−iE(t−t0 )/ ψ(t0 ) or ψ(t) = e−iEt/ ψ(0)
(4.4a) (4.4b)
Of course, since (4.2) is an eigenvalue equation the above is valid for each eigenvector k, or (4.5) ψk (t) = e−iEk t/ ψk (0) If the system were prepared in the k th eigenstate at time t = 0, the probability of finding it in the mth state at a later time is 2 | ψm (t) | ψk (0) |2 = e−iEm t/ ψm (0) | ψk (0) = δmk So when a Hamiltonian is time independent, a system prepared in a given state stays in that state. This is the justification for treating the time-independent problem by a separate formalism. Of course, an atom in an excited state decays spontaneously which does not occur for the Hamiltonian previously considered. Hence, the Hamiltonian previously considered must be incomplete. There must be some time dependence that has yet to be considered. Consider next the problem given by the equation (H0 + HI )ψ(t) = i
d ψ(t) dt
(4.6)
HI is a perturbation term in the Hamiltonian which will be time dependent, but only the time-independent solut
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