Physics 195aProblem set number 6Due 2 PM, Thursday, November 14, 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: Read the “The Simple Harmonic Oscillator: Creation and De-struction Operators” course note.PROBLEMS:30. K0system in density matrix formalism: Exercise 2 of the K0coursenote.31. [Worth two problems] “Regeneration”: Exercise 3 of the K0coursenote.32. Qualitative features of wave functions: Exercise 1 of the HarmonicOscillator course note.1433. One of the failings of classical mechanics is that matter should be“unstable”. Let us investigate this in the following system: Considera system consisting of N particles with masses mkand charges qk,k =1, 2,...,N, where we suppose some of the charges are positive andsome negative. The Hamiltonian of this multiparticle system is:H =Nk=1p2k2mk+N≥j>k≥1qkqj|xk− xj|,where pk= |pk| is the magnitude of the momentum of the particlelabelled “k”.(a) Assume we have solved the equations of the motion, with solutionsxk= sk(t). Show that for any ω>0 we can select a number c>0such that xk= csk(ωt) is also a solution of the equations of motion.Remember, we are dealing with the classical equations of motionhere.(b) Find scaling laws relating the total energy, total momentum, to-tal angular momentum, position of an individual particle, andmomentum of an individual particle for the original sk(t)solutionand the scaled csk(ωt) solution. The only parameter in your scal-ing laws should be ω. Make sure that any time dependence isclearly stated.(c) Hence, draw the final conclusion that there does not exist anystable “ground state” of lowest energy. As an aside, what Kepler’slaw follows from your analysis?(d) We assert that quantum mechanics does not suffer from this dis-ease, but this must be proven. You have seen (or, if not, see thefollowing problem) the analysis for the hydrogen atom in quan-tum mechanics, and know that it has a ground state of finiteenergy. However, it might happen for larger systems that sta-bility is lost in quantum mechanics – there are typically severalnegative terms in the potential function which could win over thepostive kinetic energy terms. We wish to prove that this is, in fact,not the case. The Hamiltonian is as above, but now pk= −i∂k(∂k=(∂∂xk,∂∂yk,∂∂zk)).Find a rigorous lower bound on the expectation value of H.Itdoesn’t have to be very “good”– any lower bound will settle this15question of principle. You may take it as given that the lowerbound exists for the hydrogen atom, since we have already demon-strated this. You may also find it convenient to consider center-of-mass and relative coordinates between particle pairs.34. The one-electron atom (review?): Continuing from problem 29, nowconsider the case of the one-electron atom, with an electron under theinfluence of a Coulomb field due to the nucleus of charge Ze:V (r)=−Ze2r, (10)(a) Without knowing the details of the potential, we may evaluatethe form of the radial wave function (Rn(r)=un(r)/r ,whereψnm(x)=Rn(r)Ym(θ, φ)) for small r,aslongasthepotentialdepends on r more slowly than 1/r2. Here, n is a quantum numberfor the radial motion. Likewise, we find the asymptotic form ofthe wave function for large r, as long as the potential approacheszero as r becomes large. Find the allowable forms for the radialwave functions in these two limits.(b) Find the bound state eigenvalues and eigenfunctions of the one-electron atom. [Hint: it is convenient to express the wave function,or rather un, with its asymptotic dependence explicit, so thatmay be “divided out” in solving the rest of the problem.] Youmay express your answer in terms of the Associated LaguerrePolynomials:L2+1n+(x)=n−−1k=0(−)k+1(n + )!(n −  − 1 − k)!(2 +1+k)!k!xk.