Continuous Spectrum due to a Constant Perturbation

We consider the transition in a continuous spectrum due to a constant perturbation. In this case, the perturbation is brought about by a quantum transition independent of time t. Hie transition probability of the states within the interval и -* i/ + dv.

If we do the change t = T -> oo , then

This is the transition probability per unit time.

It is necessary to differentiate the transition probability from the transition rate. The transition rate is a quantity with dimension per second. It has the units of frequency. Consider dW to relate the optical transition in an atom and to be of the order 10’sec‘. It is the mean transition frequency and its inverse is called the mean lifetime r. Suppose there exists a transition where at time moment t = 0 with N0 atoms. We find the number of atoms N{t) at time moment t Ф 0 from the equation:


where W is the full transition probability per unit time t:

Here, r is the mean lifetime relative to the given transition.

Harmonic Perturbation

For the case of v =f(t) being a periodic dependence then we have

Here, w is a constant angular frequency; и and w are operators that explicitly do not depend on time t as v should be Hermitian, i.e., v =v which is easily seen after the substitution in 11.100:

From where This follows that

Here we set to introduce the perturbation at t = 0 and cut it off at t = T. Thus, if we consider 11.100 and 11.95 then

For fixed t i.e., T, the transition probability dPm, is a function of only the variable w. For a resonance phenomenon we have for any of the two denominators in 11.104 the following frequencies


These two cases may correspond, respectively, to the absorption and emission of a quantum of radiation when electromagnetic interactions are involved.

Consider co > 0. Then covn = w{-w) gives the resonance conditions corresponding to a>m > 0 (vn < 0). For > 0, the system goes from the lower energy level £„ to the higher-level £v by the resonant absorption of an energy quantum ha> (Figure 11.9).

For a>v„ < 0, the resonant perturbation stimulates the passage of a system from the higher-level £,, to the lower level £„. This is accompanied by the induced emission of an energy quantum hw. Further, we

The resonance condition corresponding to a>,,„ > 0

FIGURE 11.9 The resonance condition corresponding to a>,,„ > 0 (a)v„ < 0). For a>,„ > 0 the system goes from the lower energy level £„ to the higher-level £v by the resonant absorption of an energy quantum feta.

consider the case for which a>r„ > 0 and to > 0 as the case for which to,,,, < 0 is considered analogously. Consider again 11.104. We do the denotation

The denominator of Vv~_,„ is zero for co = a>v and that of Vv%„ is zero for w = - a>v„. Thus, if we consider w close to v„then should be the dominant term compared to This follows that is the resonant term while V*„ is the anti-resonant term. It should be noted that V*_t becomes resonant for negative v„ close to to and as well close to -tov„.

It should be noted that we deal with a situation in which the initial state |n) is a discrete bound state and final state v) is one of a continuous set of dissociated states. Then

and we need consider only Thus 11.104 becomes:

We consider the moment T -> oo. It should be noted that

The resonance width Д a = can be obtained at the zeros on both sides of the resonant point and for

the transition probability 11.109 then

Then from here the time T is larger if the width is smaller. It should be noted that during the time interval [0, T] the perturbation performs numerous oscillations that are sinusoidal perturbations (Figure 11.10).

If T were small, then the perturbation would not have time to oscillate and this will be equivalent to a perturbation linear in time.

I I2


For T —>oo , the maximum ot the function -—^— tends to infinity and the width of the curve becomes

smaller and the curve increases in height. This follows that the curve tends to a Dirac ^-function and we have

The variation of the transition probability P, _ „(а>, Г) on the frequency w for fixed Г with the large

FIGURE11.10 The variation of the transition probability P, _ „(а>, Г) on the frequency w for fixed Г with the large

I I2t2


peak (resonant peak), centered at <»,„ = u>, with height -L— and width inversely proportional to T.



which is the transition rate. This formula cannot yield a numerical value since it contains a S -function. In order to compare 11.112 with the experiment it is necessary to do away with the S -function using integration.

Suppose that the energy £is the unique parameter, then

We now integrate 11.113 over the parameter £and this results in the transition rate for the continuous spectrum:

Here, />(£,.) is the density of state of the continuous spectrum and

From where it shows that if in the perturbation, there is absorption of light, then ha> is the quantum of light and 11.116 is the equation for the Einstein photo effect.


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