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    Vibrational frequency, Earth mass, harmonic function, hydrogen atom degeneracy

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    1. a) In the infrared spectrum of H79Br, there is an intense line at 2630 cm¡1. Calculated the force constant
    of H79Br and the period of vibration of H79Br.
    b) The force constant of 79Br79Br is 240 N ¢m¡1. Calculated the fundamental vibrational frequency
    and zero-point energy of 79Br79Br
    c) Calculate the average z component of angular momentum, hLzi, for an electron in a ring of constant
    potential when it has the wavefunction:
    i) y(f ) = r1
    p sin3f
    ii) y(f ) = r 1
    2p e¡3if :

    2. The Earth (mass »=6£1024 kg) rotates about the sun (at an average distance of r »=1:5£1011 m) once
    per year (one year = 10p million seconds).
    a) What is the Earth's angular momentum with respect to the sun in units of ¯h?
    b) What is the smallest principle quantum number n that allows the electron in a hydrogen atom to
    have this value of angular momentum?
    c) How large is such an atom? Use hˆri for the H atom in the 1s state to express this size.

    3. The ve l = 2 spherical harmonic functions are:
    Y2;0(q ;f ) =
    p ¡2cos2q ¡sin2q ¢
    Y2;§1(q ;f ) = ¨
    2p cosq sinq e§if
    Y2;§2(q ;f ) =
    2p sin2q e§2if
    a) Prove that these functions are eigenfunctions of ˆL2 and ˆLz with appropriate eigenvalues.
    b) Write Y2;0(q ;f ) in terms of xr
    , yr
    , and z
    r .
    c) Use superposition to relate Y2;1 and Y2;¡1 to real functions of xr
    , yr
    , and z
    r . Hint: Remember that
    e§if = cosf §i sinf
    Chem 120A, Spring 2006 1
    d) Show that the smplitude,
    y(q ;f ) =
    [Y1;0(q ;f )+Y2;0(q ;f )]
    is an eigenfunction of ˆLz but not ˆL2. Y1;0(q ;f ) =q 3
    4p cosq .

    4. In class you were shown that not only are the 2l +1 m states of each l degenerate for the hydrogen
    atom, but there is an additional degeneracy for each n from the fact that 0 · l · n¡1.
    a) Given that under parity
    what is the rst n for which a superposition of two different l states of even parity can be constructed?
    What are the values of l?
    b) The same question as in a), but for the case of odd parity.

    5. This is a subtle question. When (in physical space, a.k.a. position representation) an eigenfunction of
    some Hamiltonian, ˆH , is ys(~r) with a corresponding eigenvalue Es (i.e. ˆHys(~r) = Esys(~r)), then we
    know that
    Es = Z y¤s (~r) ˆHys(~r)d3r;
    where here we assume that ys(~r) is normalized. Now, the amazing thing about the hydrogen atom
    (or any tow-body coulomb interaction) is that the 2s and 2p states are degenerate. Suppose instead of
    having a coulomb interaction that the relative interaction in atomic units is actually
    V0(r) = ¡e¡gr
    r ;
    where g << 1.
    a) Show that the new relative Hamiltonian, &#136;H 0, may be written as,
    0 = &#136;H +µ¡e¡gr
    r +
    r ¶;
    where &#136;H is the hydrogen atom Hamiltonian.
    b) Assume that g is so small that the eigenfunctions of the hydrogen atom Hamiltonian &#136;H are also
    eigenfunctions of &#136;H0. Show that no matter how small g is, the energies of the 2s and 2pz are no longer
    degenerate. Are the three 2p states still degenerate?
    Extra credit: Suppose instead of a regular Coulomb interaction that the potential was given by ¡1
    r1+e ,
    with e << 1. Show that the 2s and 2pz states are not degenerate using the ideas advanced in the
    previous part of this problem.

    6. Starting with the relationship [ &#136; x; &#136; p] = i¯h and the denition~L =~r£~p, show that:
    £&#136;Lx;&#136;Ly¤ = i¯h&#136;Lz
    £&#136;Lz;&#136;L2¤ = 0
    The following commutator relations may be useful: [A;BC]=B[A;C]+[A;B]C and [AB;C]=A[B;C]+

    7. Consider a U atom stripped of all but one electron: U91+.
    a) What is the ion's ionization energy in both eV and in atomic units?
    b) What is h&#136;ri for the 1s state of this ion in both Angstroms and atomic units?
    c) For some principle quantum number n¤, U91+ will have h&#136;rin¤s ¸ h&#136;ri1s for H. What is the smallest
    value that n¤ can be?
    d) What does the plot of h&#136;ri1s look like for all the one electron species from H to U91+?

    8. Consider two possible spin states:
    Y1® = ¯¯

    Y2® = r1
    0® +ir3

    a) What is the expectation value of Sz for each of these states? (Expectation value = hSzi=­Y¯¯ Sz¯¯
    b) What is hSxi for each of these states?
    c) What is hSyi for each of these states?

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