solm(1) - Physics 742 Graduate Quantum Mechanics 2 Homework...

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Physics 742 – Graduate Quantum Mechanics 2 Homework Set M 1. [5] Tritium is 3 H an isotope of hydrogen which radioactively decays to 3 He. Assume an isolated 3 H has its electron in the ground state when it suddenly radioactively decays to 3 He, but the nucleus stays in the same place (no recoil). What is the probability that the atoms remains in the ground state? What is the probability that it goes into each of the n = 2 states 2 lm ? The probability is just () 2 100 PI F n lm →= , but the eigenstate of the initial Hamiltonian are not the same as the eigenstate of the final Hamiltonian. The angular integrals will vanish unless l = m = 0, and in this case the angular integral yields one, so we have () () 2 00 0 1 0 0 100 n nlm r R r R r dr δδ = The prime doesn’t denote derivative, but rather that the radial wave function must be evaluated for Helium, which has the same wave function except 1 2 aa . We’ll let Maple finish it for us, using the online wave functions for hydrogen. > for n from 1 to 6 do integrate(r^2*radial(1,0)* subs(a=a/2,radial(n,0)),r=0. .infinity)^2; end do; Just for fun, I worked out the first six solutions, with the probabilities listed below. The other 2 lm states, of course, have probability zero.
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This note was uploaded on 02/15/2011 for the course PHYS 742 taught by Professor Unknow during the Spring '04 term at Harvard.

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solm(1) - Physics 742 Graduate Quantum Mechanics 2 Homework...

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