Physics 195aProblem set number 5 – Solutions to Problems 24 and 29Due 2 PM, Thursday, November 7, 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 K0: An Interesting Example of a ‘Two-State’System” course note.PROBLEMS:24. Suppose we have a system with total angular momentum 1. Pick abasis corresponding to the three eigenvectors of the z-component ofangular momentum, Jz, with eigenvalues +1, 0, −1, respectively. Weare given an ensemble described by density matrix:ρ =14211110101.11(a) Is ρ a permissible density matrix? Give your reasoning. For theremainder of this problem, assume that it is permissible. Does itdescribe a pure or mixed state? Give your reasoning.Solution: Clearly ρ is hermitian. It is also trace one. This isalmost sufficient for ρ to be a valid density matrix. We can seethis by noting that, given a hermitian matrix, we can make atransformation of basis to one in which ρ is diagonal. Such atransformation preserves the trace. In this diagonal basis, ρ is ofthe form:ρ = a|e1e1| + b|e2e2| + c|e3e3|,where a, b, c are real numbers such that a + b + c =1. Thisisclearly in the form of a density operator. Another way of arguingthis is to consider the n-term dyad representation for a hermitianmatrix.However, we must also have that ρ is positive, in the sense thata, b, c cannot be negative. Otherwise, we would interpret someprobabilities as negative. There are various ways to check this.For example, we can check that the expectation value of ρ withrespect to any state is not negative. Thus, let an arbitrary statebe: |ψ =(α, β, γ). Thenψ|ρ|ψ =2|α|2+ |β|2+ |γ|2+2(α∗β)+2(α∗γ). (10)This quantity can never be negative, by virtue of the relation:|x|2+ |y|2+2(x∗y)=|x + y|2≥ 0. (11)Therefore ρ is a valid density operator.To determine whether ρ is a pure or mixed state, we consider:Tr(ρ2)=116(6+2+2)=58.This is not equal to one, so ρ is a mixed state. Alternatively, onecan show explicitly that ρ2= ρ.(b) Given the ensemble described by ρ, what is the average value ofJz?12Solution: We are working in a diagonal basis for Jz:Jz=10 0 00 000−1.The average value of Jzis:Jz =Tr(ρJz)=14(2 + 0 −1) =14.(c) What is the spread (standard deviation) in measured values of Jz?Answer: We’ll need the average value of J2zfor this:J2z =Tr(ρJ2z)=14(2+0+1)=34.Then:∆Jz=J2z−Jz2=√114.25. Coherent states with density matrices: Exercise 7 of the “Density Ma-trix Formalism” course note.26. Density matrix for a spin 1/2 system in a magnetic field: Exercise 8 ofthe “Density Matrix Formalism” course note.27. Entropy for a system of spin 1/2 particles in a magnetic field: Exercise9 of the “Density Matrix Formalism” course note.28. Hamiltonian in the particle-antiparticle basis: Exercise 1 of the K0course note.29. Review of Schr¨odinger equation in three dimensions: Central potentialproblem. There are some areas of elementary quantum mechanics thatI want to make sure don’t fall through the cracks in your education,in particular, the central force problem and the specific case of theone-electron atom.Suppose we have two particles, of masses m1and m2, described byposition coordinates x1and x2. Assume that they interact with eachother via a potential V (x1, x2).13(a) Write down the Hamiltonian for this system. Show that it may betransformed to a description in terms of center-of-mass and rela-tive coordinates. Show that the problem then reduces to two prob-lems: one for the center-of-mass motion, and one for the relativemotion, if the potential can be separated into a term dependingonly on the position of the center-of-mass, plus a term dependingonly on the relative locations of the particles. Now assume thatthe potential does not depend on the center-of-mass position, andsolve for the center-of-mass motion. Is your solution sensible?Solution: Let pi= −i∇idenote the momentum of particle i,with magnitude pi. The Hamiltonian is:H =p212m1+p222m2+ V (x1, x2)=−∇212m1−∇222m2+ V (x1, x2) (12)Define total mass and center-of-mass position and momentumvariables:M = m1+ m2(13)X =m1x1+ m2x2M(14)P = M˙X = p1+ p2. (15)Define the reduced mass and relative position and momentumvariables:m =m1m2m1+ m2(16)x = x1− x2(17)p = m˙x =m2p1− m1p2M. (18)We may solve for x1and x2in terms of X and x:x1= X +m2Mx (19)x2= X −m1Mx (20)(21)14Then the Hamiltonian can be written in terms of center-of-massand relative coordinates according to (letting P = |P| and p =|p|):H =P22M+p22m+ V (X +m2Mx, X −m1Mx) (22)Now let UT(X, x)=V (X +m2Mx, X −m1Mx), and assume that it isof the form:UT(X, x)=UCM(X)+U(x). (23)Let ∇CMbe the gradient operator with respect to X,and∇ be thegradient with respect to x. Then we may write the Schr¨odingerequation as:−∇2CM2M−∇22m+ UCM(X)+U(x)ψ(X, x)=Eψ(X, x). (24)We expand ψ in a series of terms of the form Φ(X)φ(x), and applythe technique of separation of variables to obtain:1Φ−∇2CM2M+ UCM(X)Φ=ECM(25)1φ−∇22M+ U(x)φ = E − ECM(26)Now we assume UCM(X) = 0. We may solve for the center-of-massmotion:−∇2CM2MΦ(X)=ECMΦ(X) (27)The solution isΦ(X)=AeiP·x+ Be−iP·x, (28)with ECM=P22M. This is simply the motion of a free particle ofmass M.(b) Suppose V is a function of the separation between the two particlesonly, V = V (|x|), where x ≡ x1−x2. Solve the Schr¨odinger equa-tion for the angular dependence and show that the Schr¨odingerequation may be reduced to an equivalent one dimensional prob-lem. Give the “effective