 # Quantum Mechanics Problems

## 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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