# hwsol10 - Physics 31 Spring 2008 Solution to HW#10 Problem...

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Physics 31 Spring, 2008 Solution to HW #10 Problem A A particle of mass m moves in a three dimen- sional box of unequal sides L 1 , L 2 ,and L 3 . Find the energies of the six lowest states if L 1 = L , L 2 =2 L ,and L 3 =3 L . Which of these states are degenerate (have the same energy)? From the solution given in class, we know the energy levels are E n 1 ,n 2 ,n 3 = π 2 ¯ h 2 2 mL 2 · n 2 1 + ³ n 2 2 ´ 2 + ³ n 3 3 ´ 2 ¸ Here is a table of the quantum numbers that give the six lowest energies: n 1 n 2 n 3 E n 1 ,n 2 ,n 3 ÷ π 2 ¯ h 2 2 mL 2 1 1 1 1.361 1 1 2 1.694 1 2 1 2.111 1 1 3 2.250 1 2 2 2.444 1 2 3 3.000 No levels are degenerate. Problem B Radium ( 226 88 Ra) decays by α emission to radon ( 222 86 Rn). The energy of the emitted α particle is E = 4 . 78 MeV. Estimate the number of times per second that the alpha particle strikes the nuclear boundary. ( Hint: You will need to estimate the nuclear radius; just use a number of the right order of magnitude.) Estimate the half life of
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