Quantum Resonances and Ensemble Average

We have argued in previous notes that the free particle wavefunction exp(-iEt+ipx) is a complex probability used to describe interactions. As a result, quantum calculations are statistical ones and only provide part of all possible information. If an interaction only involves space, one may use exp(ipx), as in a 2-slit calculation which uses an OR probability approach to deal with probabilistic interactions with both slits (if they are about hbar/p apart) by adding exp(ip dot r1) to exp(i p dot r2), where r1 and r2 are vectors from the center of each slit to the same point on a screen far away. The result   (exp(ip dot r1) + exp(i p dot r2) * (exp(-ip dot r1) + exp(-i p dot r2)) gives an ensemble result- it is a statistical calculation. A single photon or particle interferes and only goes to one point and one does not know what that point is (except that it does not go to minima points). Only by considering an ensemble can one give meaning to the idea that there is a certain probability for the single photon to move to one peak versus another. In short, one performs a statistical calculation and needs to interpret it as such, we argue.

   For a single particle bound state, one again has an ensemble average because an exp(ipx) interacts with V(x) in a statistical manner. One creates an ensemble average to find average kinetic energy:  {1/2m Sum over p a(p) pp/2m exp(ipx) } / { Sum over p a(p) exp(ipx)}. Again, this is a statistical result. For the overall bound state, exp(-i E average t+ i P x) where P=0 deals with ensemble averages and must be considered in a statistical sense. Also, this calculation involves the particle moving in and out of the bound region and so seems to be a clear large N ensemble calculation.Furthermore, there is information lost in this calculation which one should be aware of. For high energy levels, the particle should be moving first forwards, then backwards, and this is not shown. That does not mean that it does not occur. A statistical calculation does not need to show all information and one sometimes has to obtain this extra information outside of the statistical calculation.

   The reason we belabour the notion of a statistical calculation is that we believe that it also applies to the resonance decay factor  exp(- gt). In textbooks (1), one often sees exp(-iEt) with E-> E +  i g  and exp(-iEt) -> exp(-iEt -gt).  exp(-iEt) is a quantum factor which applies to a single particle and so we argue that it is a little confusing to make sense of exp(-gt) in a single particle context. In particular, one may also obtain exp(-gt) from dN/dt = -g N(t), which is a statistical equation applied to radioactive decay (among other things), i.e. a large N ensemble calculation which holds for a low rate of decay. Thus exp(-gt) from E+ig should be understood in terms of an ensemble calculation, we argue, which does not seem to be stressed (we suggest). 

  In particular, if one has some photons uniformly dispersed in an n1=1 index of refraction medium (1-dimension) region of length L surrounded by n2>>n1, with a very small probability for refraction, exp(-gt) would need to be understood as applying to an ensemble because exp(-gt) is continuous and photons refract (leave the resonance area) in a discrete manner. Thus, exp(-gt) is a statistical solution which removes quite a bit of information. In particular, if the probability to refract is very small, dN/dt= -1/ delta t1  = N(t) 2 Probability(refract). The solution to this is not exp(-gt). Rather at t=0, N(0) = N. Then one may find a delta t for which, on average (ensemble average) one photon has been lost. Then N(delta t) = N(0) -1 and one may find the delta t2 etc. This approach uses the quantum calculated Probability(refraction), but not exp(-gt) even though this example should be a resonance. As a result, we suggest that quantum calculations are statistical in nature and as such leave out information as they describe an ensemble. In some cases, as in the reflection-refraction case, one may obtain a more detailed description of what happens than given by the resonance factor exp(-gt) which one sees in textbooks and in the Gamow factor calculation etc.