A tritium atom (³H, with one proton and two neutrons in its nucleus) decays into a He (helium-3) atom, whose nucleus contains two protons and one neutron. The transformation can be considered as instantaneous. If before the decay the ³H atom was in its ground state, what is the probability that the newly-created ³He atom will be in its ground state? The radial wavefunctions of the ground states of hydrogen-like atoms are given by (34) where ao 0.05 nm is the Bohr radius and Z the atomic number. R₁,0 (r) = 2e-Zr/ao Problem 2 A particle of mass m moves (non-relativistically) in the three-dimensional potential V = ½k (x² + y² + z² + εxy), where x, y, and z are the three spacial coordinates, k is the spring constant, and e is a small real number; w= √k/m. (a) For the raising and lowering operators ât = (P) and â = √(+P), where and pe are the position and linear-momentum operator, respectively, show the following three relationships: [â, â¹] = 1 â \n) = at (n) mw √nn - 1) √n + 1/n+1). (b) Determine the energy of the ground state up to and including 2nd order perturbation theory. (c) Determine the effect of the perturbation on the first excited state of the system. Is the degeneracy of the system lifted?

A tritium atom (³H, with one proton and two neutrons in its nucleus) decays into a He (helium-3) atom, whose nucleus contains two protons and one neutron. The transformation can be considered as instantaneous. If before the decay the ³H atom was in its ground state, what is the probability that the newly-created ³He atom will be in its ground state? The radial wavefunctions of the ground states of hydrogen-like atoms are given by (34) where ao 0.05 nm is the Bohr radius and Z the atomic number. R₁,0 (r) = 2e-Zr/ao Problem 2 A particle of mass m moves (non-relativistically) in the three-dimensional potential V = ½k (x² + y² + z² + εxy), where x, y, and z are the three spacial coordinates, k is the spring constant, and e is a small real number; w= √k/m. (a) For the raising and lowering operators ât = (P) and â = √(+P), where and pe are the position and linear-momentum operator, respectively, show the following three relationships: [â, â¹] = 1 â \n) = at (n) mw √nn - 1) √n + 1/n+1). (b) Determine the energy of the ground state up to and including 2nd order perturbation theory. (c) Determine the effect of the perturbation on the first excited state of the system. Is the degeneracy of the system lifted?

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