- A hydrogen atom is in an eigenstate (ψ) ofJ^2 ,L^2 , and ofJz such thatJ^2 ψ =^154 ̄h^2 ψ,
L^2 ψ= 6 ̄h^2 ψ,Jzψ=−^12 ̄hψ, and of course the electron’s spin is^12. Determine the quantum
numbers of this state as well as you can. If a measurement ofLzis made, what are the possible
outcomes and what are the probabilities of each. - A hydrogen atom is in the stateψ=R 32 Y 21 χ−. If a measurement ofJ^2 and ofJzis made,
what are the possible outcomes of this measurement and what are the probabilities for each
outcome? If a measurement of the energy of the state is made, what are the possible energies
and the probabilities of each? You may ignore the nuclear spin in this problem. - Two identical spin 1 particles are bound together into a state withorbital angular momentum
l. What are the allowed states of total spin (s) forl= 2, forl= 1, and forl= 0? List all
the allowed states giving, for each state, the values of the quantum numbers for total angular
momentum (j), orbital angular momentum (l) and spin angular momentum (s) iflis 2 or less.
You need not list all the differentmjvalues. - List all the allowed states of total spin and total z-component of spin for 2 identical spin 1
particles. Whatℓvalues are allowed for each of these states? Explicitly write down the(2s+1)
states for the highestsin terms ofχ(1)+,χ(2)+,χ(1) 0 ,χ(2) 0 ,χ(1)−, andχ(2)−. - Two different spin^12 particles have a Hamiltonian given byH=E 0 + ̄hA 2 S~ 1 ·S~ 2 +B ̄h(S 1 z+S 2 z).
Find the allowed energies and the energy eigenstates in terms of thefour basis states|+ +〉,
|+−〉,|−+〉, and|−−〉. - A spin 1 particle is in anℓ= 2 state. Find the allowed values of the total angular momentum
quantum number,j. Write out the|j,mj〉states for the largest allowedjvalue, in terms of
the|ml,ms〉basis. (That is give one state for everymjvalue.) If the particle is prepared in
the state|ml= 0,ms= 0〉, what is the probability to measureJ^2 = 12 ̄h^2? - Two different spin 21 particles have a Hamiltonian given byH=E 0 +AS~ 1 ·S~ 2 +B(S 1 z+S 2 z).
Find the allowed energies and the energy eigenstates in terms of thefour product states
χ(1)+χ(2)+,χ(1)+χ(2)−,χ(1)−χ(2)+, andχ(1)−χ(2)−.
frankie
(Frankie)
#1