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Expectation value of: energy, angular momentum

  1. Dec 3, 2011 #1
    Hi all!

    1. The problem statement, all variables and given/known data

    If we consider the hydrogen atom as a spinless particle. Let this system in the state
    [itex] \Psi ( \vec{r} )= \frac{1}{6} [4 \Psi_{100} ( \vec{r} )+ 3 \Psi_{211}- \Psi_{210} ( \vec{r} ) + \sqrt{10}\Psi_{21-1} ( \vec{r} )] [/itex]

    Calculate:

    1) Expectation value of energy when measured from this state.
    2) Expectation value of z-component orbital angular momentum
    3) Expectation value of x-component orbital angular momentum


    2. Relevant equations

    [itex]\langle \vec{r} | nlm \rangle =\Psi_{nlm} ( \vec{r} ) = R_{nl} (r) Y_{lm} (\Omega) [/itex]

    [itex]E_n = -\frac { \alpha^2}{2 n^2} \mu c^2 [/itex]


    3. The attempt at a solution

    1) For the expectation value for the energy , [itex] \langle H \rangle = \langle \Psi ( \vec{r} ) | H | \Psi ( \vec{r} ) \rangle = \frac {1}{36} [ 16 \langle \Psi_{100} | H | \Psi_{100} \rangle + 9 \langle \Psi_{211} | H | \Psi_{211} \rangle + \langle \Psi_{210} | H | \Psi_{210} \rangle + 10 \langle \Psi_{21-1} | H | \Psi_{21-1} \rangle ] =?? [/itex]


    In this point I should be able to put the eigen-energy [itex]E_n = -\frac { \alpha^2}{2 n^2} \mu c^2 [/itex] but I don't know how I can do that.


    2)
    I did the same as before..

    [itex] \langle L_z \rangle = \langle \Psi ( \vec{r} ) | L_z | \Psi ( \vec{r} ) \rangle = \frac {1}{36} [ 16 \langle \Psi_{100} | L_z | \Psi_{100} \rangle + 9 \langle \Psi_{211} | L_z | \Psi_{211} \rangle + \langle \Psi_{210} | L_z | \Psi_{210} \rangle + 10 \langle \Psi_{21-1} | L_z | \Psi_{21-1} \rangle ] [/itex] but I have no idea what's the next step-

    3) the same problem as before.

    Do you know what I'm doing wrong?

    Thanks in advance!
     
  2. jcsd
  3. Dec 3, 2011 #2

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    Well, you know that

    [tex] H|nlm\rangle =E_n |nlm\rangle [/tex]

    for the discrete portion of the spectrum and also that [itex] |nlm\rangle [/itex] has unit norm. Use this for point 1)

    For point 2), use that

    [tex] L_z |nlm\rangle = m |nlm\rangle [/tex]

    Also for point 3), express L_x in terms of L+- whose action you know on [itex]|nlm\rangle [/itex] from the general theory of angular momentum.
     
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