Due At 9:00 AM, Monday March 13, 2006Physics 402 Take Home ExamDue At 9:00 AM, Monday March 13, 2006This exam is open notes and open book. You may also use Mathematica or other symbolic manipulation programs. If you use Mathematica or a similar program you must include the output to get credit. Do not seek outside help. (I trust you.) If you cannot do a section of the exam do not panic. The exam is written in such a way as you can often do a later section of a problem while missing earlier parts. To aid you in this, I will often ask you to show that something is true rather than asking for the answer. To get credit you must show how you obtained your answer from the basic physical and mathematical principles. You may use formulae that we derived in class or in the book as a starting point. Since you have considerable time on this exam, I fully expect your answers to be clear. If you have questions you may e-mail me ([email protected]) or call me at the office (301) 405-6117 or at home (301) 654-7702 (Before 10:00 p.m.) 1) At time t=0 a hydrogen atom is in the state 1,2,30,0,21,1,20,0,1222121)0(ψψψψψii−+−= where the state mln,,ψ is an energy eigenstate characterized by the usual quantum numbers n, l, and m and (0) in the ket indicates the time. You may neglect spin in this problem.a) If the z component of the angular momentum, zL of the state )0(ψis measured what is the probability the measured value will be +? What is the probability that it will be 0? What is the probability that it will be −? Briefly explain your reasoning.b) If the energy of the state )0(ψis measured what is the probability the measured value will be 0E (where 0E is the ground state energy) ? What is the probability that it will be 20E? What is the probability that it will be 40E? Briefly explain your reasoning.c) Find the expectation value of the energy, the z component of the angular momentum and the square of the angular momentum in this state. That is find )0(ˆ)0(ψψH, )0(ˆ)0(ψψzL and )0(ˆ)0(2ψψL.d) What is the expectation value of the x-component of the angular momentum, )0(ˆ)0(ψψxL? e) Find the state as a function of time, i.e. find )(tψ2) Consider a system of two particles one of which is spin one and the other is spin 1/2. They interact via a rotational invariant (scalar) Hamiltonian. In this problem the spatial degrees of freedom are irrelevant---you may assume that each particle is fixed in some spatial wave function. Since the Hamiltonian is a scalar it commutes with all three components of the total angular momentum which in this case is the total spin of the system.a) Show that there are a total of 6 possible spin states for the two particle system. (Hint: this is as easy it sounds!)b) On general grounds involving the nature of angular momentum , the eigenvalues of the Hamiltonian must have degeneracies. In particular there should be one set of two degenerate states and one set of four degenerate states. Explain why. c) It turns out that the most general Hamiltonian for such a system can be written as 221ˆˆ1ˆˆssbaH⋅+=where a, b are constants with the dimension of energy and 1ˆ is the identity operator 1ˆs is the spin operator for the spin one particle and 2ˆsis the spin operator for the spin ½ operator. Find the energies for the set of two degenerate levels and the set of four degenerate levels in terms of a, b. Hint: You may find it helpful to think about the total spin of the system.3) The matrix elements of xˆyˆzˆ between energy eigenstates of the hydrogen atom play an important role in the calculation of the rate at which photons are emitted (for reasons that are beyond the scope of this course). Here we will work in the simplified limit where spin can be neglected and the proton will be assumed to be so heavy that it sits without moving in the center of the atom. As usual the states are labeled by three quantum numbers n,l m . The associated wave functions in spherical coordinates are ),(),,(φθφθψmlnnlmYrur=. We will consider matrix elements between the states mln and the ground state is 001.a) As a first step derive expressions for x , y, and z in terms of r and spherical harmonics. That is derive ),(34012/1φθπYrz= and analogous expressions for x and y. You may use the expression for spherical harmonics in the book.b) Use part a) show that ( )( ) ( )( )∫∫−Ω=−∞),(),(),()()(61001ˆ1111*1*02/1φθφθφθYYYdrurrudrxmlnmln( )( ) ( )( )∫∫+Ω=−∞),(),(),()()(61001ˆ1111*1*02/1φθφθφθYYYdrurrudriymlnmln( )( )( )∫∫Ω=∞),(),()()(31001ˆ01*1*02/1φθφθYYdrurrudrzmlnmlnwhere as usual Ω is the solid angle i.e. where ∫∫ ∫≡Ωππθθφ020)sin(dddc) Show that 0001ˆ=xmln unless l=1 and m=1 or m= -10001ˆ=ymln unless l=1 and m=1 or m= -10001ˆ=zmln unless l=1 and m=0These results are examples of selection rules. These give conditions which matrix elements must have to be nonzero. They play a central role in the absorption and emission of photons. The results of part c) imply for example that the dominant method by which photons are emitted (so-called electric dipole radiation) can only land an atom in the ground state if the state emitting the photon is in a p-wave state (i.e. an l=1 state).4) A spin ½ particle interacts with an external potential. All of the energy levels are non-degenerate. The ground state energy is denotedgE, the first excite state isgEE21=, the second excited state is gEE42= , the third excited state is gEE73=, the third excited state is gEE124=. The analogous states are denoted g, 1, 2 , 3 and the 4. Note that these states represent the full state (i.e. both the spin and space degrees of freedom)a) From the information provided above one can deduce the Hamiltonian of the system does not commute with the total angular momentum. Explain why. What does this tell you about whether the Hamiltonian is rotationally invariant? Now suppose that we have two of these spin ½ particles (they are identical) both of which are in this potential. Moreover these particles are non-interacting in the sense that the Hamiltonian has no interaction between them. b) How many energy eigenstates are there in this two particle system with an energy less than gE12? Explicitly construct these