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Unformatted text preview: Physics 2D Spring 2010 Final Exam Friday June 11 11:30am to 2:30pm [Note: This is a closed-book exam; total score of this exam is 100 points.] Some possibly useful equations and relations: relativistic energy-momentum relation: E 2 = p 2 c 2 + ( m c 2 ) 2 where m is the rest mass Einsteins photon theory: E = hf deBroglies relation for matter wave: = h/p or f = E/h k = 2 / , = 2 f , c = f = /k , h = h/ (2 ) uncertainty principle: x p h 2 time-indepdent Schr odinger equation:- h 2 2 m 2 + U ( ~ r ) ( ~ r ) = E ( ~ r ) trignometric relations: sin( ) = 0 for = 0 , , 2 , 3 ,... cos( ) = 0 for = / 2 , 3 / 2 , 5 / 2 ,... sin(- ) =- sin( ), cos(- ) = cos( ) d d sin( ) =- cos( ), d d cos( ) =- sin( ) 1 1. Particle decay. [10 points] A particle of mass M , initially at rest in the lab frame, decays into a particle of smaller mass m which moves with speed u in one direction, and a (massless) photon which moves with speed c in the opposite direction. Find u in unit of c for m = M/ 2. What fraction of the initial energy is carried away by the photon? [Hint: Start with the conservation of total energy and momentum. You will need to make use of the energy-momentum relation for a massless particle.] 2. One-dimensional square well. [10 points] A particle of mass m is trapped in a square potential well of width a , i.e., with U ( x ) = 0 for < x < a , and U ( x ) = U > 0 elsewhere. (a) Use Heisenbergs uncertainty principle to estimate the zero-point energy of the particle in an infinite potential well, i.e., with U ....
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