Homework5 - value of the angular momentum L 7 Prove that...

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PHYS343 Spring 2007 Homework #5 Due Friday March 30,2007 COB 1. For the ground state of hydrogen, calculate the probability of finding the electron in the interval o o a 4 r a < < . 2. (a) For the ground state of the hydrogen atom, determine the radial distance for which the probability of finding the electron less than this distance is 90%. Give a numerical answer to two significant figures. (b) Repeat the calculation for a 99% probability. 3. Estimate the probability that the electron will be found inside the proton for (a) the 2s state and (b) the 2p state. Assume that a proton has a radius of approximately 1 fm. 4. (a) What are the possible values of L and Lz for the 3p state? (b) For the 3d state? 5. A hydrogen atom is in an excited state, n=5. (a) What are the possible values of the quantum numbers and m ? (b) What are the possible values of the orbital angular momentum L? 6. (a) Prove that the function ( 29 θ cos e r C ψ δ 2 r - = is a solution of the hydrogen atom, where 2 2 mke δ H = . (b) Determine the energy of the state. (c) What is the
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Unformatted text preview: value of the angular momentum L? 7. Prove that the operator for the x component of orbital angular momentum is given by ( 29 ( 29 ( 29 ( 29 ∂ ∂ + ∂ ∂-= ϕ θ sin cos θ cos θ sin i L x H . 8. Consider a model of an electron as a tiny uniform sphere of size 10-18 m corresponding to the experimental limit on possible electron structure. Suppose that the electron intrinsic angular momentum is due to the spinning motion of a sphere with an angular frequency ω . Calculate the value of ω . Explain why this is a “bad” model of electron spin. 9. If you modeled the nucleus of hydrogen as a 1-D particle in a box with the distance between the walls of approximately 1 fm, what would be the energies of the first four states? 10. Assuming an electron is trapped in a 2-D square box whose sides are 1 nanometer. What are the energy values of the four lowest states?...
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