PHYSICS 250 May 4, Final Exam - Solutions

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1 Name: PHYSICS 250 May 4, 999 Final Exam - Solutions Instructions: Work all problems. You may use a calculator and two pages of notes you may have prepared. There are problems of varying length and difficulty. If you have trouble with one problem, go onto another problem. Hope you enjoyed the class. Have a nice summer!

2 . (0 pts.) The frequencies of spectral lines in light from a distant galaxy are found to be two thirds as great as those of the same lines in light from nearby stars. Find the recession speed of the distant galaxy. f β f ' + β β β β ( 2 ) β 9 9β 4 + 4β β 5 β 5 ν (0 pts.) How much time does a meter stick moving at v 0.c relative to an observer take to pass the observer? The meter stick is parallel to its direction of motion. ν 0.c γ β 2 L t m m.05 x c m/sec nsec. (0 pts.) The helium-4 nucleus is composed of two neutrons and two protons. Its mass is M 728 Mev/c 2. How much energy is required to break up a helium-4 nucleus into its constituent neutrons and protons? M He 728 MeV/c 2 m p MeV/c 2 m n MeV/c 2 E 2( 98.79) + 2( 99.57) MeV 2

3 0 4. (5 pts.) The sun s mass is M 2 0 kg, its radius is R m, and its surface temperature is T K. How long will it take for the sun to lose % of its mass by radiation? R σt W m 2 K 4 ( K ) W m 2 P W m 2 4πR watts joules/sec E sun MC kg C joules 8 t yr sec.45 0 years 5. (20 pts.) Calculate the de Broglie wavelength of (a) A neutron with kinetic energy K 0 MeV. K λ p 2 2m p 2mK MeV 7 MeV/c h hc 2400eV Ȧ p pc 7 MeV Ȧ (b) An electron with kinetic energy K MeV. E 2 p 2 c m 0c 2 4 pc.5 m 0c (.5) 2 ( 0.5) 2.42 MeV λ hc 2400eV Ȧ pc Ȧ

4 6. (20 pts.) A particle is confined to the one-dimensional finite potential well of depth V 0 as shown in the figure below. V 0 E 2 E (a) (b) (c) (d) (a) Which of the wavefunctions shown in figures (a)-(d) is an acceptable wavefunction for a partivcle with energy E < V 0? c (b) Which of the wavefunctions shown in figures (a)-(d) is an acceptable wavefunction for particle with energy E 2 > V 0? a 4

5 7. (5 pts.) Three wavefunctions for a particle confined by a one-dimensional potential are shown below. Circle the wavefunction with the highest kinetic energy. (a) (b) (c) 8. (20 pts.) Helium- is a spin-/2 Fermion with M.72 0 MeV/c 2. A noninteracting gas 2 of these atoms has number density N V 6 0 cm. Calculate (a) its Fermi wavenumber, k F k F π --- N ( π cm ) V cm m E F (b) its Fermi energy. h 2 2 k E F F m ( evs) 2 c 2 ( m) ev ev J 5

6 9. (5 pts.) An electron in a hydrogen atom is in a p state with m l. Calculate the probability that the particle lies in the cone 2π < θ< π. ψ LM ( θφ, ) ψ ( θφ, ) e iφ sinθ 2π 2 [ ψ Lm ( θφ, )] π sin 2 θ P( 2π < θ< π) π dφ 8π sin θ sinθ dθ 2π π π sin θ dθ 8π π 2π -- ( cos 2 θ) ( d cosθ) 4 2π -- x 2 x ( ) dx 4 x [ ] % 6

7 0. (5 pts.) A beam of electrons with energy E.0eV is incident an a barrier 6.00eV high and 0.200nm wide. Estimate the fraction of electrons that is transmitted through the barrier. V 0 6eV 2mV ( k 0 E) h 2 2mc 2 ( V 0 E) h 2 c 2 E ev T 6 E V E k e 2 L 0 V 0 k 2 2π mc 2 ( V hc 0 E) 2π eV 240eVnm eV.45 nm 240eVnm T nm 0.2nm e e

8 . (20 pts.) This problem concerns the Li 2+ ion ( Z ). (a) Draw an energy level diagram for its electron states showing all states with principal quantum number n. Indicate the energies, quantum numbers n and l, and degeneracies of the states. Draw arrows showing which transitions are permitted. Ignore electron spin. l 0 l l 2.6eV n 5 Z 2 E n n 2 0.6eV 22.4eV n 2 n E 9.6 E eV 0.6eV E.6 (b) What wavelength photons appear in the emission spectrum from these states? λ 2 hc 240eVnm nm 729Ȧ E E E 2 E hc λ λ hc 240eVnm nm 4Ȧ E E E λ 2 hc nm 5Ȧ E E 2 E 8

9 2. (20 pts.)the Manganese atom has a total spin of 5/2 and a total orbital angular momentum of zero. (a) A beam of these atoms is passed through a Stern-Gerlach apparatus with average magnetic field at its center of.2t. How many beams of Mn atoms should be detected at the output of the apparatus? spin states and 6 beams (b) What is the wavelength of electromagnetic radiation that will flip Mn spins at the center of the apparatus? (Recall the selection rule for spin systems with zero orbital angular momentum: ±.) m s E gµ B Bm s E gµ B B m s gµ B B hc hc E λ λ E nm λ nm 8.92mm (c) If N of these atoms are in equilibrium at temperature T 4K in the center of the apparatus, find the ratio of the number of atoms in the highest energy magnetic state to the lowest energy magnetic state. E max gµ B B E min gµ B B PE max ( ) e E max E min PE ( min ) ( ) k B T e 5gµ BB k B T exp e

10 . (20 pts.) Six non-interacting electrons are placed in a 2D square well potential with sides of length 0.nm. Calculate (a) the ground state energy h E nx n y mL 2 ( n x + n y ) ( n x + n y )ε ε ( hc) mc 2 L 2 ( 240) ( 0.) 2 4.8eV E 22 8ε E G 2E + 4E 2 2 2ε + 4 5ε E 2 E 2 5ε 24ε eV E 2ε E G 00eV (b) the energy and degeneracy of the first excited state. You may find it helpful to show occupancy of energy levels in an energy level diagram. + 4 states of form Degeneracy 8 E' 2E + E 2 + E ε + 5ε + 8ε 27ε E' ev 0

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