ps11 - Homework (11) for Physics 316, Modern Physics I

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Unformatted text preview: Homework (11) for Physics 316, Modern Physics I (Hoffstaetter/Drasco/Thibault) Due Date: Friday, 04/22/05 - 9:55 in 132 Rockefeller Hall Exercise 1: Consider the two dimensional wave equation for the vibration of a rectangular membrane, 1 2z 2z 2z + 2 = 2 2 . (1) x2 y v t The membrane has a boundary at x = 0, y = 0 and x = Lx , y = Ly at which it does not oscillate, i.e. z = 0. a) Show that z = A sin(kx x) sin(ky y) cos(t) (2) can be a solution for specific kx , ky , and . b) Show for Lx = Ly that there can be different solutions z(x, y, t) for the same frequency . This is called degeneracy. Exercise 2: Show that the spherical finite well potential, V (r) = V0 H(r - R) with H(x) = 2 2 h if(x> 0, 1, 0), does not have a spherically symmetric bound state if V0 < 8mR2 . Exercise 3: Derive a formula for the curl in spherical coordinates in the form f (r, , ) = A(r, , )er + B(r, , )e + C(r, , )e , using the arguments with which 2 (3) (r, , ) was derived in class. Exercise 4: Calculate the probability that the electron in the ground state of hydrogen, which is spherically symmetric, is in the classically forbidden region, where E < V (r). Does this probability increase or decrease for the two spherical symmetric states of higher energy? If the electron is in the ground state, what is the probability of the electron being within the radius of the nucleus, assuming it has a radius of 1fm? Exercise 5: Since the electron has charge -e, an electron probability density corresponds to a charge density (x) = -e . Assume the electron is in the ground state to a) find the electric charge Q(r) inside a sphere with radius r around the Hydrogen nucleus, which has charge +e. b) find the corresponding potential V (r) and the electric field E(r). (from [1,511]) Exercise 6: Read [1, Sec. 11] [1] An Introduction to Quantum Physics, French and Taylor, Norton, W. W. & Company, Inc. (1990) ...
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