Final Examination. Tuesday December 15, :30 am 12:30 pm. particles that are in the same spin state 1 2, + 1 2

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1 Department of Physics Quantum Mechanics I, Physics 57 Temple University Instructor: Z.-E. Meziani Final Examination Tuesday December 5, 5 :3 am :3 pm Problem. pts) Consider a system of three non interacting, identical spin particles that are in the same spin state, + > and confined to move in a one dimensional infinite potential well of length a: V x) for < x < a and V x) for other values of x. a) Determine the energy and the wave function of the ground state. The energy levels and wave functions for a single particle in an infinite well potential are given by E n π nπx ) ma n, ψ n x) a sin ) a where m is the mass of the particle, a the width of the infinite well and n an integer. Since the three particles we are considering are fermions spin /) and all in the spin state, + > they cannot be in the same energy state due to Pauli exclusion s principle. Each energy state of the infinite well is occupied by one particle at most as such states with n, n and n 3 will be occupied by one particle. Therefore, the total energy of the three particle system is written as: E gs E + E + E 3 E + E + 9E E 7h π ma ) The total wave function of the three Fermions should be antisymmetric. The spin part is defined to be symmetric therefore we need to anti-symmetrize the spatial part, which is done through the slater determinant. ψ gs x, x, x 3 ) 3! ψ x ) ψ x ) ψ x 3 ) ψ x ) ψ x ) ψ x 3 ) ψ 3 x ) ψ 3 x ) ψ 3 x 3 ) χ,+ b) Determine the energy and the wave function of the first excited state. s )χ,+ s )χ,+ s 3 ) 3) The first excited state is obtained easily by raising the energy level of the particle in n 3 to the next energy state thus n. Therefore, the total energy of the three particle system is written as: E fes E + E + E E + E + 6E E h π ma )

2 Again the total wave function of the three fermions should be antisymmetric. The spin part is defined to be symmetric therefore we need to anti-symmetrize the spatial part. ψ gs x, x, x 3 ) ψ x ) ψ x ) ψ x 3 ) 3! ψ x ) ψ x ) ψ x 3 ) ψ x ) ψ x ) ψ x 3 ) χ,+ s )χ,+ s )χ,+ s 3 ) 5) Problem #. pts) A hydrogen atom, initially i.e., for t < ) is in its ground state, is placed starting at time t in a time dependent electric field pointing along the z axis Et) E τ ˆk/τ + t ) 6) where τ is a constant having the dimension of time and ˆk is the unit vector along the z direction. a) Calculate the probability that the atom will be found in the p state at t + You might make use of the following result: e iωt τ + t dt π τ e ωτ 7) a-s) The potential resulting from the interaction of the electric field and the electron is written as: V t) e r Et) 8) The probability of transition from the initial state to a final state is given in time dependent perturbation theory by P if i t ψ f ˆV t ) ψ i e iω fit dt 9) The probability of transition from s ground state) to the p state is then given by P s p ˆV t ) e iω fit dt where t is a dummy variable of integration and we set the upper limit of time integration to t +. ˆV t ) e r Et ) ee τ τ + t z ee τ τ r cos θ + t ) We need to express both the initial state and the final state in the { r } representation, in order to evaluate the transition matrix element as: r cos θ ψpr cos θψ s d r ψpr cos θψ s r dr sin θdθdφ ) ) given that r ψ s R r)y θ, φ) πa 3 e r/a 3) r ψ p R r)y θ, φ) 8πa 3 r a e r/a cos θ

3 We evaluate the matrix element as a triple integral of separate variable r, cos θ and φ: r cos θ 3 a π r 3 Rr)R r)dr π sin θ cos θdθ dφ ) r e 3r/a dr 5) 8 a 3 5 where we have used the result of the integration of sin θ cos θdθ 3 6) through the change of variable x cos θ and dx sin θdθ. Regrouping all the terms by inserting the matrix element results back into equation ) we find P s p 5 e E τ a 3 5 e E τ a 3 3 π e E a 3 e ω fiτ e iω fit τ + t dt e iω fit τ + t dt 7) 8) Where ω fi 3/)E s and E s 3.6eV. Problem 3. pts) Consider two spin / s S and S, coupled by an interaction of the form at) S S ; at) is a function of time which approaches zero when t approaches infinity, and takes on non-negligible values on the order of a ) only inside an interval, whose width is of the order of τ, about t. a) At t, the system is in the state +, an eigenstate of S z and S z with the eigenvalues +/ and /). Calculate, without approximations, the state of the system at t +. Show that the probability P+ +) of finding, at t +, the system in the state + depends only on the integral at)dt. a-s) First, we write the state of the system at t in the basis of Ŝ and Ŝz, namely { S, M } where, and, are kets of the S, M basis. ψ) +,, +, ] 9) 3

4 W t) S, M at) S S S, M ) at) S S S) S, M ) at) SS + ) 3 ) S, M ) 3) So S, M is an eigenket of W t). We find the time evolution by Schrödinger equation: i d ψ dt H ψ ) i S,M ċ S,M t) S, M W t)c S,M S, M 5) iċ S,M t) at)c S,M t) cs,m + ) c S,M ) dc S,M c S,M infty SS + ) 3 ] at)dt SS + ) 3 ln c S,M + ) ln c S,M ) i { i c S,M + ) c S,M ) exp SS + ) 3 SS + ) 3 6) ] ) i 7) ] at)dt 8) infty ] } at)dt 9), +,, ] 3) with ψ+ ) exp i P+ +) + ψ+ ) 3) ) + at)dt exp i3 ) + at)dt,, 3) Now we can evaluate the probability P+ +) exp i P+ +) at)dt ) exp i3 at)dt ) 33) which only depends on at)dt b) Calculate P + +) by using first-order time dependent perturbation theory. Discuss he validity conditions for such an approximation by comparing the results obtained with those of the preceding question. b-s) In st -order time-dependent perturbation theory, we have P+ +) + e iωfit W fi t)dt 3)

5 where ω fi and W fi t) and W fi t) + W t) + ] at),, ) W fi t) S S S,, ) ) 35) at), S S S ),, S S S), ] 36) at) 3 ] 3 at) 37) P+ +) at)dt 38) + thus P+ +) at)dt) 39) + For P+ +) at)dt), we can expand the result from part a) in a Taylor series and neglect all but the lowest-order terms: + ) ] P+ +) at)dt) + O at)dt ) under this condition, st order time-dependent perturbation theory is valid. c) Now assume that the two spins are also interacting in a static magnetic field B parallel to Oz. The corresponding Zeeman Hamiltonian can be written H B γ S z + γ S z ) ) where γ and γ are the gyromagnetic ratios of the two spins, assumed to be different. Assume that at) a e t τ. Calculate P+ +) by first-order time-dependent perturbation theory. With fixed a and τ, discuss the variation of P+ +) with respect to B. c-s) Again, but now with ω fi P+ +) + e iωfit W fi t)dt ω fi E + E + H + B γ H + B γ ) 3) ) + γ + ) ) γ + 5) As in b), ω fi B γ γ ) B γ + γ ) B γ γ ) 6) W fi t) at) 7) 5

6 P+ +) + e ib γ γ )t at)dt a + e ib γ γ )t t τ dt a B exp γ γ ) ) πτ dt /τ 8) 9) 5) P+ +) τ πa B exp γ γ ) τ ) 5) We find a gaussian function describing the probability as a function of the magnetic field strength. Note that this expression for P+ +) in the limit B, agrees with part b). Figure : Probability as a function of the magnetic field strength 6