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King Fahd University of Petroleum and Minerals Department of Physics PHYS 212-MODERN PHYSICS Term 071 Second Major Exam Solution Sunday 02 December 2007 05:30-06:20 pm Name: ID No.: 1 1. Calculate the de Broglie wavelength for a proton moving at 106 m/s. We have h h 6.626 10 34 J .s 3.96 10 13 m 396 fm p mv 1.673 10 27 kg 10 6 m.s 1 2. A proton has a kinetic energy of 1.0 MeV. If its momentum is measured with an uncertainty of 5%, what is the minimum uncertainty in its position? We have p2 K p 2 Km 2m We also have px x 2 2p Moreover, p 0.05 p 0.05 p 0.05 2 Km p Therefore, x 2 0.05 2Km c 0.1 2Kmc 2 197.3eV .nm 0.1 2 10 6 938.3 10 6 eV 2 46 fm 3. A free electron has a wavefunction x A sin 5 1010 x where x is measured in meters. Find (a) the electron’s de Broglie wavelength, (b) the electron’s momentum, and (c) the electron’s energy in eV. We have (a) k 5 1010 m 1 (b) p h 2 2 2 3.14 0.1256nm k 5 1010 m 1 k 1.055 10 34 J .s 5 1010 m 1 5.275 10 24 kgm / s 2 hc 1240.8 eV 2 nm 2 p2 95.5eV 2m 2 2 mc 2 2 0.12562 nm 2 0.511 10 6 eV 2 (c) K 2 4. A proton is confined to moving in a one-dimensional box of width 0.200 nm. (a) Find the lowest possible energy of the proton. (b) What is the lowest possible energy of an electron confined to the same box? (c) How do you account for the large difference in your results for (a) and (b)? We have 2 2 E1 2mL2 2 c 2 2mc 2 L2 (a) For a proton E1 2 197.32 eV 2 nm 2 2 938.3 10 eV 0.2 nm 2 6 2 5.11 10 3 eV (b) For an electron E1 2 197.32 eV 2 nm 2 2 0.511 10 6 eV 0.2 nm 2 2 9.38eV (c) According to Heisenberg’s uncertainty principle, for the same x , and therefore the same p , the particle with the smaller mass will have associated with it a larger uncertainty on its speed. It will be moving faster and therefore will have more kinetic energy than the particle with a larger mass. 5. Find the points of maximum and minimum probability density for the nth state of a particle in a one-dimensional box. Check your results for the n = 2 state. The probability density for the nth state is x 2 2 nx sin 2 L L It is maximum when nx 3 5 L 3L 5 L , , ,..., x , , ,... , L 2 2 2 2n 2n 2n It is minimum when nx L 2L 0, ,2 , x 0, , ,... L n n 3 Of course 0 x L in all cases. For the n = 2 state we get Maxima at x L 3L , 4 4 L Minima at x 0, , L 2 6. Write down the time-dependent Schrodinger’s equation for a particle of mass m moving in a region where the potential energy is U(x). 2 2 x, t U x x, t i x, t 2 2m x t 7. If x, t is the wavefunction describing a particle at position x at time t. Write down an expression for the probability of finding the particle in the interval from x a to x b at time t. 2 b P x, t dx a 8. What is a stationary state? It is a state for which the probability density is independent of time. 9. When do we say that an observable has sharp values? When the state function of the system is an eigenfunction of the operator associated with that observable. 10. Discuss the minimum energy of a quantum system using Heisenberg’s uncertainty principle. According to Heisenberg’s uncertainty principle, the minimum energy of a quantum state cannot be zero because the particle is constantly moving since there is a minimum uncertainty on its momentum. 4