## Transcribed Text

Problem I
Consider two spin-1/2 particles. Let s(1) and s(2 describe their respective spin operators, while
S
the operator of the total spin of the two particles. If we use notation (1,0) to
represent the eigenstate of the total spin withs 1, m, =0, compute S (1,0), where S_ is the
lowering ladder operator of the total spin in two different ways:
1). use the generic formula defining action of this operator on an arbitrary angular momentum
(e.g. Eq. 4.136 in Griffiths)
2. Present the state vector (1,0) as a linear combination of eigenstates of the spin operators for
individual particles and compute the same quantity by presenting S as a sum =
Make sure that the both results agree.
Problem 2
Consider operator J - L+S, where S is operator of spin with s 1/2 and L is a generic
operator of the angular momentum. Eigenvectors jm; of the operator J can be presented as
linear combinations of the eigenvectors sm,) of operator S1 and eigenvectors (mm) of operator
L:
(1)
where A and B are constants.
a) Show that jm;) given by Eq. 1 is the eigenvector of the operator J. = L. + S. for
arbitrary A and B with eigenvalue m; if m, = m;-1/2;m_=m; +1/2
b) Since jm;) is an eigenvector of J2. we know that If Eq.
(1) is correct we must have
(s+L)
which is only possible for particular values of coefficients A and B. Find for which coefficients Eq. 2 is
true. To this end compute its left-hand side by presenting
(s+L)
and using representation of x andy-components of these operators in terms of ladder operators L.,S,
Problem 3
Consider two electrons in a Helium atom neglecting direct Coulomb interaction between them.
(Thus each electron can be thought as in a pure hydrogen-like potential with all the same energy
levels and wavefunctions).
a) Find the lowest possible energy of the two electrons assuming that they are in a singlet
spin state. Write down the respective orbital wavefunction.
b) The same for electrons in a triplet spin state.
Problem 4
Consider two non-interacting particles, each of mass m in the infinite square well, one in state
W/ (ground state), while the other in the state W, Write down the correct two-particle wave
function Y (4,02 and calculate expectation value ((x,-5)>) for this state assuming that
a) The particles are distinguishable
b) The particles are identical spinless bosons
c) The particles are identical spin 1/2 fermions in a singlet spin state
d) The particles are identical spin 1/2 fermions in a triplet spin state
Problem 5
A particle is initially (for / <0) is in its first excited state in an infinite potential well located
between points x 0 and x-a, is a subject, starting at time 1-0, to a time-dependent
perturbation
V(D)V,
where the perturbation strength Vo is positive and small, and T is a characteristic time scale of
the perturbation. Calculate the probability that the particle is found in its second excited state at
/ 8 Do the same for the third excited state.

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