N.24 A problem if the energy is given

Examining all shelf number combin­ations with the given energy and then picking out the combin­ation that has the most energy eigen­functions seems straight­forward enough, but it runs into a problem. The problem arises when it is required that the set of shelf numbers agrees with the given energy to mathematical precision. To see the problem, recall the simple model system of chapter 11.3 that had only three energy shelves. Now assume that the energy of the second shelf is not $\sqrt{9}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 3 as assumed there, (still arbitrary units), but slightly less at $\sqrt{8}$. The difference is small, and all figures of chapter 11.3 are essentially unchanged. However, if the average energy per particle is still assumed equal to 2.5, so that the total system energy equals the number of particles $I$ times that amount, then $I_2$ must be zero: it is impossible to take a non­zero multiple of an irrational number like $\sqrt{8}$ and end up with a rational number like $2.5I-I_1-4I_3$. What this means graphi­cally is that the oblique energy line in the equivalent of figure 11.5 does not hit any of the centers of the squares mathemati­cally exactly, except for the one at $I_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. So the conclusion would be that the system must have zero particles on the middle shelf.

Of course, physi­cally this is absolute non­sense; the energy of a large number of perturbed particles is not going to be certain to be 2.5 $I$ to mathematical precision. There will be some uncertainty in energy, and the correct shelf numbers are still those of the darkest square, even if its energy is 2.499,9...$I$ instead of 2.5 $I$ exactly. Here typical textbooks will pontificate about the accuracy of your system-energy measurement device. However, this book shudders to contemplate what happens physi­cally in your glass of ice water if you have three system-energy measurement devices, but your best one is in the shop, and you are uncertain whether to believe the unit you got for cheap at Wal-Mart or your backup unit with the sticking needle.

To avoid these conundrums, in this book it will simply be assumed that the right combin­ation of shelf occupation numbers is still the one at the maximum in figure 11.6, i.e. the maximum when the number of energy eigen­functions is mathemati­cally inter­polated by a continuous function. Sure, that may mean that the occupation numbers are no longer exact integers. But who is going to count 10$\POW9,{20}$ particles to check that it is exactly right? (And note that those other books end up doing the same thing anyway in the end, since the mathematics of an integer-valued function defined on a strip is so much more impossible than that of a continuous function defined on a line.)

If fractional particles bothers you, even among 10$\POW9,{20}$ of them, just fix things after the fact. After finding the fractional shelf numbers that have the biggest energy, select the whole shelf numbers nearest to it and then change the “given” energy to be 2.499,999,9...or whatever it turns out to be at those whole shelf numbers. Then you should have perfectly correct shelf numbers with the highest number of eigen­functions for the new given energy.