Physics 195aProblem set number 4 – Solution to Problem 18Due 2 PM, Thursday, October 31, 2002Notes about course:• Homework should be turned in to the TA’s mail slot on the first floorof East Bridge.• Collaboration policy: OK to work together in small groups, and to helpwith each other’s understanding. Best to first give problems a good tryby yourself. Don’t just copy someone else’s work – whatever you turnin should be what you think you understand.• There is a web page for this course, which should be referred to for themost up-to-date information. The URL:http://www.hep.caltech.edu/˜fcp/ph195/• TA: Anura Abeyesinghe, [email protected]• If you think a problem is completely trivial (and hence a waste of yourtime), you don’t have to do it. Just write “trivial” where your solutionwould go, and you will get credit for it. Of course, this means you arevolunteering to help the rest of the class understand it, if they don’tfind it so simple. . .READING: Finish reading the “Density Matrix Formalism” course note.PROBLEMS:18. Let us try another example of the discussion we have been having inclass concerning the use of the uncertainty relation on “localized” wavefunctions. Consider the three dimensional generalization. Hence, letP (a) be the proability to find the particle, of mass m in a sphere ofradius a centered at the origin.(a) Recall that in the one dimensional case, if the probability of findingthe particle in the interval (−a, a)wasα, then a simple lowe rbound on the kinetic energy was obtained as:T ≥18mα2a2. (1)7Make a simple, but rigorous, generalization of this result to thethree dimensional case. Don’t worry about finding the “best”bound; even a “conservative” bound may be good enough to an-swer some questions of interest.Solution: The limit on T in Eqn. 1 corresponds to a limit on themomentum ofp2≥14α2a2. (2)In three dimensions the kinetic energy isT =12mp2x+ p2y+ p2z. (3)If the probability to find the particle within radius a is P (a), thenin each dimension we certainly have that the probability to be inthe interval (−a, a)isatleastP (a). Thus, we can apply Eqn. 2in each dimension, and hence, in sum:T ≥38mP (a)2a2. (4)(b) We know that an atomic size is of order 10−10m. Suppose thatwe have an electron which is known to be in a sphere of radius10−10m with 50% probability. What lower b ound can you put onits kinetic energy? Is the result consistent with expectation; e.g.,with what you know about the kinetic energy of the electron inhydrogen?Solution:T ≥38122(200 MeV-fm)20.5MeV10−20m2≥ 0.8eV. (5)In the ground state, the expectation value of the kinetic energyof the electron in hydrogen is 13.6 eV, consistent with our bound,although our bound is not especially good.(c) In ancient times, before the neutron was discovered, it was sup-posed that the nucleus contained both electrons and protons. Acomfortable nuclear size is 5 × 10−15m. Find a lower bound on8the kinetic energy of an electron if the probability to be withinthis radius is 90%. If there is a problem with the validity of yourbound, see if you can fix it.Solution:T ≥38(0.9)2(200 MeV-fm)20.5MeV25× 10−30m2≥ 4GeV. (6)The electron is relativistic, inconsistent with our assumption incomputing the kinetic energy with a non-relativistic equation. Ourderivation that p2≥1/4a2in the one-dimensional case shouldstill be valid. The relativistic kinetic energy is T = E −m ≈|p|,where the approximation is in the limit E  m. Let’s presumewe can estimate T with p2.ThenT ≥√32a≥200 MeV-fm5fm=40MeV. (7)Let’s compare this with an estimated order of magnitude for t heelectrostatic potential energy of an electron and a proton sepa-rated by 5 fm:|V | =e2a≈200 MeV-fm100 × 5fm=0.4MeV (8)This is much smaller than our limit on the kinetic energy, pre-senting a theoretical problem with binding and with expectationsfrom the virial theorem.(d) Now find a lower bound for a proton in the nucleus, if it has aprobability of 90% to be within a region of radius 5 × 10−15m.Solution:T ≥38(0.9)2(200 MeV-fm)2900 MeV25 ×10−30m2≥ 0.5MeV. (9)19. Some more thoughts about time reversal: Exercise 13 of the “Ideas ofQuantum Mechanics” course note.920. The von Neumann mixing theorem: Exercise 3 of the “Density MatrixFormalism” course note.21. Operators in product spaces: Exercise 4 of the “Density Matrix For-malism” course note.22. Entropy in a two-state system: Exercise 5 of the “Density Matrix For-malism” course note.23. Measuring the density matrix: Exercise 6 of the “Density Matrix For-malism” course