1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Gross Structure of Hydrogen

  1. Feb 17, 2015 #1
    1. The problem statement, all variables and given/known data
    Show that for hydrogen the matrix element <2 0 0|z|2 1 0> = -3a0 where a0 is the Bohr Radius.

    On account of the non-zero value of this matrix element, when an electric field is applied to a hydrogen
    atom in its first excited state, the atom's energy is linear in the field strength.

    2. Relevant equations

    Energy of electron: -ħ2/2a02μn2

    3. The attempt at a solution
    <2 0 0| and |2 1 0> are bra and ket states of Hydrogen |n l m> where n is the principle quantum number, l is the orbital number and m is the magnetic number. I think I'm just struggling to work out what the operator z does (does it just point out the z coordinate of the electron?) Any advice on how I can approach this, specifically what matrix is being referred to, would be great.
  2. jcsd
  3. Feb 17, 2015 #2


    User Avatar
    Homework Helper

    the "z" between the bra and the ket is the z function , which is anti-symmetric along the z coordinate.
    It is needed so that the L=1 ket state, after multiplied by z, has non-zero overlap with the (symmetric) L-0 bra state.
    (so that, any operator that is non-symmetric in z (I wonder what that might be?) might initiate a transition).
  4. Feb 18, 2015 #3
    I'm not really sure what operator would be non-symmetric in z since the hydrogen atom is spherically symmetrical?
  5. Feb 18, 2015 #4
    And I'm still not really sure what matrix is being referred to in the question
  6. Feb 18, 2015 #5


    User Avatar

    Staff: Mentor

    Read the second part of the question.

    When you have a basis of states ##|\phi_i\rangle##, you can construct a matrix representation of any operator ##\hat{A}##, where the elements are
    A_ij = \langle \phi_i | \hat{A} | \phi_j \rangle
    This is why these bracket "sandwiches" are often referred to as matrix elements. Note that the wave function can then be written as a vector.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted