Consider two spin-1/2 particles. Let s(1) and s(2 describe their respective spin operators, while
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.
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
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
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
and using representation of x andy-components of these operators in terms of ladder operators L.,S,
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.
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
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
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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