Homework 4

Problem 1:

Consider a deuterium atom (composed of a nucleus of spin 1 and an electron).  The electronic angular momentum is J = L + S, where L is the orbital angular momentum of the electron and S is its spin.  The total angular momentum of the atom is F = J + I, where I is the nuclear spin.  The eigenvalues of J2 and F2 are J(J+1)h2 and F(F+1)h2 respectively.

(a)  What are the possible values of the quantum numbers J and F for a deuterium atom in the 1s ground state?

(b)  What are the possible values of the quantum numbers J and F for a deuterium atom in the 2p excited state state?

Problem 2:

The muon is a heavy electron which is identical to the electron except that it has a rest mass mm=207me , where me is the electron mass.  The muon is unstable and it can decay into the final state  are massless particles of spin ½ , which are called the electron antineutrino and muon neutrino respectively.  They may be considered non-interacting with ordinary matter in what follows.  Because the muon has the same charge as the electron, it can form an atom with the proton, which we denote by Hm. Consider the ground state of Hm.

(a)  What is the ground state energy E0m of Hm.

(b)  Suppose in the ground state of Hm the muon decays at time t = 0 into  such that the 3-momentum Find the probability that the electron is in the ground state of ordinary hydrogen at t>0.

Problem 3:

The following wavefunctions refer to the hydrogen atom.

(a)  ( eigenfunction of H, L2, Lz, Sz )

(b)  ( eigenfunction of H. L2, J2, Jz )

For each eigenfunction calculate 

Problem 4:

The three matrices, Mx, My, and Mz, each with 256 rows and columns, obey the commutation rules Image4257.gif (1104 bytes) The eigenvalues of Mz are ±2h (each once), (each eight times), (each 28 times), (each 56 times), and 0 (70 times).  State the 256 eigenvalues of the matrix M2=Mx2+My2+Mz2.

Problem 5:

A particle of spin ½ is in a D-state of orbital angular momentum.  What are its possible states of total angular momentum?  Suppose the single particle Hamiltonian is  What are the values of energy for each of the different states of total angular momentum in terms of the constants A, B, and C?