Sample Test Problems

  1. A particle of mass $m$ in 3 dimensions is in a potential $V(x,y,z)={1\over 2}k(x^2+2y^2+3z^2)$. Find the energy eigenstates in terms of 3 quantum numbers. What is the energy of the ground state and first excited state?
  2. * N identical fermions are bound (at low temperature) in a potential $V(r)={1\over 2}m\omega^2r^2$. Use separation in Cartesian coordinates to find the energy eigenvalues in terms of a set of three quantum numbers (which correspond to 3 mutually commuting operators). Find the Fermi energy of the system. If you are having trouble finding the number of states with energy less than $E_F$, you may assume that it is $\alpha (E_F/\hbar\omega)^3$.
  3. A particle of mass m is in the potential $V({\bf r})={1\over 2}m\omega^2 (x^2 + y^2)$. Find operators that commute with the Hamiltonian and use them to simplify the Schrödinger equation. Solve this problem in the simplest way possible to find the eigen-energies in terms of a set of "quantum numbers" that describe the system.
  4. A particle is in a cubic box. That is, the potential is zero inside a cube of side L and infinite outside the cube. Find the 3 lowest allowed energies. Find the number of states (level of degeneracy) at each of these 3 energies.
  5. A particle of mass m is bound in the 3 dimensional potential $V({\bf r}) = kr^2$.
    a)
    Find the energy levels for this particle.
    b)
    Determine the number of degenerate states for the first three energy levels.
  6. A particle of mass $m$ is in a cubic box. That is, the potential is zero inside a cube of side $L$ and infinite outside.
    a)
    Find the three lowest allowed energies.
    b)
    Find the number of states (level of degeneracy) at each of these three energies.
    c)
    Find the Fermi Energy $E_F$ for $N$ particles in the box. (N is large.)
  7. A particle is confined in a rectangular box of length $L$, width $W$, and ``tallness'' $T$. Find the energy eigenvalues in terms of a set of three quantum numbers (which correspond to 3 mutually commuting operators). What are the energies of the three lowest energy states if $L=2a$, $W=1a$, and $T=0.5a$.
  8. A particle of mass m is bound in the 3 dimensional potential $V({\bf r}) = kr^2$.
  9. a) Find the energy levels for this particle.
  10. b) Determine the number of degenerate states for the first three energy levels.
  11. In 3 dimensions, a particle of mass $m$ is bound in a potential $V({\bf r})={-e^2\over \sqrt{x^2+z^2}}$.
    a)
    The definite energy states will, of course, be eigenfunctions of $H$. What other operators can they be eigenfunctions of?
    b)
    Simplify the three dimensional Schr$\ddot {\rm o}$dinger equation by using these operators.

Jim Branson 2013-04-22