Indiana University PHYSICS P30 Final Exam 5 December 004 This is a closed-book examination. You may not refer to lecture notes, textbooks, or any other course materials. You may use a calculator, but solely for the purpose of arithmetic computation. A list of potentially useful formulas, definitions, etc is given after the problems. The exam consists of five problems (of which the first problem is a series of short-answer questions), for a total of 75 points, plus a short Extra Credit problem (#6). You must show all work: clearly state and justify any arguments, assumptions, approximations etc., as well as the use of any formulas. Unless otherwise indicated, evaluate all integrals and derivatives, and perform any arithmetic calculations if a numerical answer is requested.
[] (0 points) Short-answer questions. You only need to do four of these, as I will only count the best four. (a) [5 points] Give an example of an experiment which indicates that the energy of a photon is equal to hν. State clearly what is measured and how the conclusion is inferred. (b) [5 points] Cite an experiment which demonstrates the de Broglie relation λ = h/p; indicate how the conclusion is related to the observation made. (c) [5 points] What is the spectroscopic notation for Oxygen (Z=8)? (d) [5 points] What is a p-type semiconductor? Describe how it differs from an intrinsic semiconductor: in particular, what are the mobile charge-carriers, and where do they come from? (e) [5 points] An electron and a positron have the same mass energy mc = 0.5 MeV but opposite electric charge. They can and sometimes do spontaneously emerge from the vacuum. About how long could the pair last before they must pop back into the vacuum and disappear from existence? [] (5 points) An electron in a hydrogen level is in the f orbital. Recall that the sequence for the angular momentum quantum number in spectroscopic notation is (s, p, d, f, g,...). (a) [4 points] What is the value of the orbital quantum number l? (b) [4 points] What is the magnitude of the orbital angular momentum L? (Make sure you report this in the correct units!) (c) [3 points] What are the allowed values of the magnetic quantum number m? (d) [4 points] What is the minimum principal quantum number n that is allowed for this state, and what is the associated energy?
[3] (5 points) A particle is constrained to lie on a ring of radius R lying in the x-y plane. It has a properly-normalized wave-function of the form: Ψ(φ, t) = π cos (bφ) e iωt, where b and ω are real constants and φ is the angle representing the location of the particle around the ring relative to the x-axis. (a) [4 points] Is Ψ(φ, t) an eigenfunction of energy? If so, what is the energy eigenvalue? (b) [3 points] Is Ψ(φ, t) an eigenfunction of the square of the z-component of angular momentum L z? (Recall L z = i h / φ.) If so, what is the eigenvalue? (c) [4 points] What is the expectation value for the square of the z-component of angular momentum? Explain the relationship between this answer and that in part (b), if there is any. (d) [4 points] the wave-function must satisfy? What are the possible allowed values for b, given the conditions that 3
[4] (0 points) Benjamin Sisko is the arbitrator in a dispute between the Ferengi and the Cardassians. The Ferengi are claiming that Gul Dukat, a Cardassian, destroyed some precious metals belonging to them. They present as evidence two photographs taken from a Ferengi airship of rest length L = 000 m, moving past the scene of the purported events with a velocity of v = 0.8 c. The first photograph, taken from the rear of the ship x = 0 at time t = 0 (according to a timestamp on the photo) shows Dukat firing his laser weapon. The second photo, taken from the front of the ship a time t = L /c shows the explosion of the precious metals. (The Ferengi ship was fortuitously positioned so that the two events occurred just next to these locations on the ship at just these times.) The Ferengi state that the time delay is consistent with the travel time of the laser pulse over the distance from Dukat to the metals. The lawyer for Dukat objects, saying that these measurements are not valid because they were made from a moving frame of reference. He claims that if the measurements had been made in the rest frame of Dukat and the metals they would have shown that the shot fired by Dukat could not have been the same shot that destroyed the metals. Being an expert on relativistic matters, Sisko quickly rules in favor of one of the sides. Which one? Explain your answer. [Note: Be sure to account for the space-time distortion due to the proximity of the Bajoran wormhole. ] just kidding... 4
[5] (5 points) There are only a few potential energy wells for which the energy eigenvalues and associated wave functions can be determined exactly. An interesting onedimensional example is the well: [ V (x) = V 0 cosh(x/a) where V 0 and a are positive real constants, and the hyperbolic cosine and and sine functions are defined as: cosh(s) = es + e s ; and sinh(s) = es e s, obeying the useful identity sinh (s) = cosh (s). (a) [3 points] Sketch the potential energy as a function of x/a. [You may want to first sketch the function cosh(x/a) to get started.] (b) [3 points] Write down the time-independent Schrödinger Equation describing the energy eigenfucntions for this potential well. Specify the energy eigenvalue as E = E since we will be interested here only in bound state solutions, and specify the wave-function as ψ(x). (c) [5 points] Try a solution of the form ψ(x) = [/ cosh(x/a)] p, where p is some positive number, yet to be determined. Determine the value of p in terms of V 0. To do this it will be helpful to employ the above identity to express the dependence on x in all terms in the Schrödinger Equation solely as a dependence on powers of cosh(x/a). Determine the energy eigenvalue E. (d) [4 points] Is this wave-function the lowest-energy state (i.e., ground state) of the system? Explain why or why not. ] [6] Extra Credit (4 points) In Quantum Electrodynamics, Compton scattering (e+γ e+γ) can be considered as a combination of () the absorpton of an incoming photon by an incoming electron, resulting in a scattered outgoing electron, and () the emission of a photon by an incoming electron, resulting in a scattered outgoing electron. There are two Feynman diagrams that describe Compton Scattering, corresponding to different ways in which the two reactions are combined. Draw these diagrams. (Note that the intermediate virtual particle in these cases is an electron.) 5
Possibly useful formulas, etc Lorentz Transformations: where β v/c, and γ / β. x = γ x βγ ct p x = γ p x βγ E/c y = y p y = p y z = z p z = p z ct = γ ct βγ x, E = γ E βγ cp x, (For the inverse L.T. s, swap the primes and flip the sign of terms containing β.) Relativistic Addition of velocities: (As for the L.T. s, v is the velocity of F relative to F, assumed to be parallel to x, x dir ns) u x = u x v u u x v/c x = u x + v + u x v/c u y = u y γ ( u x v/c ) u y = u y γ ( + u x v/c ) Relativistic Momentum and Energy, Newton nd Law, etc.: p = mvγ E = mc γ F = dp/dt E = m c 4 + p c (Einstein energy-mass-momentum relation) E 0 = mc (rest energy) T = mc (γ ) (kinetic energy) β = pc/e, γ = E/mc, βγ = pc/mc (params for boosting into rest frame) Relativistic Doppler Shift & Stellar Aberration: ν = ν γ ( + β cos θ ) tan θ = Again for inverse relations, swap primes and replace β by β. Lorentz-Invariant Space-Time Interval & Mass sin θ γ (cos θ + β) ( S) = c ( t) ( x) ( y) ( z) M c 4 = E c p x c p y c p z 6
More Possibly useful formulas, etc Planck s Constant, Einstein photon energy & debroglie wavelength h = h/(π) =.05 0 34 J s = 6.58 0 MeV s; E γ = hν; λ = h/p. Compton Scattering, Compton wavelength hc = 97 ev nm = 97 MeV fm; λ λ f λ i = λ C ( cos θ), where λ C h/(m e c). Rydberg-Ritz, Bohr Radius, etc. for Hydrogen: ( = (E i E f )/hc = R y ) Rydberg-Ritz Formula λ fi n f n i ( = e ), energy levels of Bohr atom 8πε 0 hc r i r f where r n = n a 0, with a 0 = λ C /πα where α e /(4πε 0 hc) = /37 and λ C h/(m e c). QM Operators (position representation): Expectation Values: x = x, p = i h x, p = h x, Ê = i h t, Ĥ = p /m + V ( x) = h m x + V ( x) Q = x= Schrödinger Equation (Time-Dependent): Ψ(x, t) i h t Schrödinger Equation (Time-Independent): E ψ(x) = Wave Packets: Ψ(x, t) = = Ψ (x, t) QΨ(x, t) dx h Ψ(x, t) Ĥ Ψ(x, t) = + V (x) Ψ(x, t) m x h ψ(x) Ĥ ψ(x) = + V (x) ψ(x), where Ψ(x, t) = ψ(x) e iet/ h m x p= A(p) e i(px Et)/ h dp, Heisenberg Uncertainty Relations: δx δp h/ where δe δt h/, where δq ( q q ) / A (p)a(p) dp = π h 7
Possibly useful formulas, etc, cont d Normaliz n cond n for Schrödinger Eqn w/ Central Potential: π r=0 θ=0 π φ=0 R nl(r)r nl (r)y lm(θ, φ)y lm (θ, φ) r sin θ dr dθ dφ = S.E. Solutions to the -D infinite well potential (V = 0 from x = 0 to x = L): ψ n (x) = ( ) nπx L sin L. E n = h π n, where m = mass of particle here. ml Schrödinger Equation for Hydrogen atom after separation of variables: (ignoring spin-orbit, spin-spin, interactions) d Φ m (φ) dφ + m Φ m (φ) = 0 d P lm (θ) + [ ] dp lm (θ) + l(l + ) m dθ tan θ dθ sin P lm (θ) = 0 θ h d [ ] (rr nl (r)) e + m e dr 4πε 0 r + h l(l + ) rr m e r nl (r) = E n rr nl (r) Angular momentum operators L z = i h φ L = h [ θ + tan θ θ + sin θ ] φ Zeeman Effect: E = µ B (m l + m s )B where µ B = e h m e Atomic Electric Dipole Transition Rate: R α ψf( r ) ( e r ) ψ i ( r ) dv 8