H-Atom p.13

Projects for you!

The below is a list of project you should be able to complete. For some of the projects Mathematica may be of use. Here's a file containing the Mathematica I used to make these pages.

  1. On page 3 the differential equation and recursion relation for the polynomial coefficients of H() are given. Consider the following example of the H(x) function for a 10g H-atom wavefunction.

    H(x)=240240 - 120120 x + 21840 x2 - 1820 x3 + 70 x4 - x5

    1. What is l? nr? ?
    2. Show that this H satisfies the H differential equation.
    3. Show that the coefficients satisfy the recursion relation.
    4. This H is not quite a Laguerre polynomial...it is off by an overall factor. What is that overall factor?

  2. At the end of page 3 there is a table showing that at least for n=1-5 the total degeneracy is n2. Prove this result in general.

  3. On page 4 the method of normalizing the H-atom wavefunction is described: it requires use of the Laguerre recursion relation. Follow similar steps to find the expectation value of r and r2...i.e., show: For circular, large n orbits, find an expression for a classical orbit radius consistent with the above two expressions. For classical s orbits we expect r is less than twice the same energy circular orbit. Explain this, and show that these expressions are consistent with this result.

  4. With additional work one can show: You may want to look up (e.g., in Szegö) how to write Ln2l+1 in terms of a sum of Lk2l. However a faster proof is via the Feynman-Hellmann theorem.

  5. By the previous problem, ns have larger average size than say np orbitals. Where is the probability density largest for s orbitals? Explain how this can be consistent with with the both the classical orbit and the result of a larger average radius.

  6. Calculate the wavelength and energy of the following transitions. Report what "type" of light is involved (e.g., , X-ray, UV, visible, IR, microwave, radio).
    1. H-atom: 2 1
    2. H-atom: 3 2
    3. H-atom: 4 3 ; 3
    4. H-atom: 5 4 ; 4
    5. H-atom: 11 10
    6. H-atom: 101 100
    7. C-atom: 2 1
    8. U-atom: 2 1

  7. For what value of n would an H-atom ns orbital have a <r>
    1. the size of a "bucky ball" C60: r=5Å
    2. the size of a big bacteria cell: r=0.5µm
    3. the size of a pin head: r=0.4mm

  8. The radius of a nucleus is approximately 1.2 (2Z)1/3 fm. For what value of Z would <r> for 1s be "only" 10 times the size of the nucleus. 100×?

  9. We can approximate the nucleus as a uniform ball of charge. Find the electrostatic potential as a function of r inside such a ball of charge. When we solved the H-atom problem we used an exact 1/r, whereas we now know the potential deviates from 1/r inside the nucleus. Using the difference between the 1/r and the actual potential as a perturbation, find the energy shift of 1s states as a function of Z.

  10. [Gasiorowicz] An electron is in the ground state of tritium (a radioactive isotope of hydrogen whose nucleus has 2 neutrons and 1 proton, i.e., H3). A nuclear reaction instantaneously changes the nucleus to He3 (a rare isotope of helium whose nucleus has 1 neutron and 2 protons). Calculate the probability that the electron remains in the 1s state.

  11. An electron is in the following superposition state:

    =N(|100> - 2|200> + i|210> + |211>)

    where |210> means the wavefunction has n=2, l=1, m=0.

    1. Find the normalization N.
    2. Find the expectation value of energy.
    3. Find the expectation value of r (use Mathematica).

  12. On page 5 probability densities for a variety of wavefunctions with nr=2 are displayed. Why are the d wavefunctions nearly a factor of two larger than the p wavefunctions? Describe how changing nr would change these pictures.

    There are four "x-y" graphs on this page. The first two (with x-axis labels z & x) are very similar and the next two (with x-axis labels x & z) are very similar. Why are the pairs similar? why are they different? Describe how changing nr would change these graphs.

  13. Review the material on WKB applied to 3-d problems. Applied to the H-atom, there is no need to rederive the l(l+1)(l+½)2 relation, simply change the potential from r2 to 1/r. Now find the WKB energies for the H-atom.

  14. On page 6 we used perturbation theory to find the energy shift from electron-electron repulsion. The states 2s & 2p (m=-1,0,1) are degenerate, yet we did not use degenerate perturbation theory. Why could we get away with this error?

  15. Continue the "l-tilting" calculations of page 6 for the 3s, 3p, and 3d cases.

  16. In excited two-electron atoms (e.g., 1s2p) one expects that the 1s shields the 2p electron, so that a better approximation to the 2p wavefunction would use a central charge of Z-1 rather than Z. We can do this if we make "unperturbed" Hamiltonian a sum: an H-atom with Z for electron 1 and an H-atom with Z-1 for electron 2. Since the total Hamiltonian must add up to the total energy, the perturbation must include a bit of the nuclear electrostatic potential for electron 2 and the electron-electron repulsion. The result is called Heisenberg's method and it is described starting in §24 of Quantum Mechanics of One- and Two- Electron Atoms by Bethe and Salpeter. Re-invent this method, note any odd problems you find along the way (there are several!) and find the energy of the 1s2p state in He.

  17. The ground state energy of two electron atoms is an interesting problem for the Rayleigh-Ritz (variational) method. The basic idea is to calculate the expectation value of the Hamiltonian using a trial wavefunction, i.e., a function with some variable parameters in it. We are guaranteed that the expectation value of the Hamiltonian always lies above the actual ground state energy. Using the variable parameters one can get the best (lowest) such upper bound. Using our two-electron Hamiltonian:

    and the trial wavefunction: = exp[-a(r1+r2)]

    Here is the process of finding the expectation value as a function of a and then finding the lowest such upper bound:

    Justify each step in the above derivation. Some things to consider: How did the derivative terms come about, e.g., what happened to the angular part of the Laplacians. In line 1 we have three terms in square brackets "[]", on the third line its just two...what happened? The term that starts as the third square bracket term become an infinite sum and then just the first term in that sum...what happened? What are the Pl and what is the argument: 12? Explain how each "dV" was eventually replaced and why. Evaluate each integral with Mathematica and verify the results.

  18. Let's apply the above formula to three two electrons atoms: H-, He, and Li+.

    energy in
    System less
    one electron
    energy of
    that ion (eV)
    energy (eV)
    energy (eV)

    For comparison, ignoring the interaction, each 1s electron would give a binding energy of ½, so for 1s2 we would have an energy of -1. For large Z we approach that limit.

    The variational calculation is fairly accurate, its estimated energies are about 1½ eV too high. (It is, after all, just an upper limit on the energy.) Unfortunately that 1½ eV difference is critical for the existence of the ion: H-. By our calculation H- is less strongly bound than H and so should spontaneously dissociate. But it turns out that H- is stable and is critical for understanding the opacity of the atmosphere of the Sun. Furthermore, H- is difficult to make in the lab so calculation was the basis for understanding this ion. In fact, discrepancies between early calculations and observational evidence on the Sun's radiation pointed out the inaccuracies of the then-existing wave functions for H-.

    Thus improved calculations of H- properties were very much on the agenda of physics in the 1950s. Our variational calculation allows the electrons expand away from the nucleus (as one would expect, given the partial shielding by the other electron) [i.e., a has been reduced from the hydrogen value of 1]. One expects an additional effect: the electrons should avoid each other. It turns out that we can to a bit better if we allow the electrons to have different "orbital radii"...i.e., to use a trial wavefunction like:

    exp[-ar1-br2] + exp[-br1-ar2]

    Use this wavefunction and show that the H- ionization energy is positive (i.e., H- is more stable than H+e-)

  19. Better yet is to use a trial wavefunction in which the electron positions are (anti)correlated (so if one electron is at r1 the other electron is unlikely to be found near r1). This requires a more complex trial wavefunction, for example letting the wavefunction depend on the angle between r1 and r2. Three parameter trial wavefunction like this are able to predict binding energies to a fraction of an eV (and get the ionization energy of H- accurate to about 5%). See Quantum Mechanics of One- and Two- Electron Atoms by Bethe and Salpeter for details.

    We consider here a trial wavefunction:

    (exp[-ar1-br2] + exp[-br1-ar2]) (1+c|r2-r1|)

    It is a mess to find the energy expectation value, but here is the result in Mathematica form (so you can cut and paste):

    e=(-2*a^11*b^2 + a^12*b^2 - 16*a^10*b^3 + 8*a^11*b^3 - 56*a^9*b^4 + 
         29*a^10*b^4 - 204*a^8*b^5 + 64*a^9*b^5 - 426*a^7*b^6 + 226*a^8*b^6 - 
         426*a^6*b^7 + 368*a^7*b^7 - 204*a^5*b^8 + 226*a^6*b^8 - 56*a^4*b^9 + 
         64*a^5*b^9 - 16*a^3*b^10 + 29*a^4*b^10 - 2*a^2*b^11 + 8*a^3*b^11 + 
         a^2*b^12 - 6*a^11*b*c + 3*a^12*b*c - 48*a^10*b^2*c + 24*a^11*b^2*c - 
         170*a^9*b^3*c + 85*a^10*b^3*c - 360*a^8*b^4*c + 176*a^9*b^4*c - 
         1232*a^7*b^5*c + 165*a^8*b^5*c - 2000*a^6*b^6*c + 1147*a^7*b^6*c - 
         1232*a^5*b^7*c + 1147*a^6*b^7*c - 360*a^4*b^8*c + 165*a^5*b^8*c - 
         170*a^3*b^9*c + 176*a^4*b^9*c - 48*a^2*b^10*c + 85*a^3*b^10*c - 
         6*a*b^11*c + 24*a^2*b^11*c + 3*a*b^12*c - 6*a^11*c^2 + 3*a^12*c^2 - 
         48*a^10*b*c^2 + 24*a^11*b*c^2 - 171*a^9*b^2*c^2 + 86*a^10*b^2*c^2 - 
         364*a^8*b^3*c^2 + 184*a^9*b^3*c^2 - 536*a^7*b^4*c^2 + 269*a^8*b^4*c^2 - 
         2363*a^6*b^5*c^2 + 48*a^7*b^5*c^2 - 2363*a^5*b^6*c^2 + 
         2868*a^6*b^6*c^2 - 536*a^4*b^7*c^2 + 48*a^5*b^7*c^2 - 364*a^3*b^8*c^2 + 
         269*a^4*b^8*c^2 - 171*a^2*b^9*c^2 + 184*a^3*b^9*c^2 - 48*a*b^10*c^2 + 
         86*a^2*b^10*c^2 - 6*b^11*c^2 + 24*a*b^11*c^2 + 3*b^12*c^2)/
       (2*(a^10*b^2 + 8*a^9*b^3 + 28*a^8*b^4 + 120*a^7*b^5 + 198*a^6*b^6 + 
           120*a^5*b^7 + 28*a^4*b^8 + 8*a^3*b^9 + a^2*b^10 + 3*a^10*b*c + 
           24*a^9*b^2*c + 86*a^8*b^3*c + 184*a^7*b^4*c + 823*a^6*b^5*c + 
           823*a^5*b^6*c + 184*a^4*b^7*c + 86*a^3*b^8*c + 24*a^2*b^9*c + 
           3*a*b^10*c + 3*a^10*c^2 + 24*a^9*b*c^2 + 87*a^8*b^2*c^2 + 
           192*a^7*b^3*c^2 + 294*a^6*b^4*c^2 + 1872*a^5*b^5*c^2 + 
           294*a^4*b^6*c^2 + 192*a^3*b^7*c^2 + 87*a^2*b^8*c^2 + 24*a*b^9*c^2 + 

    Find the minimum of this function. You should find a reasonable value for the ionization potential of H- near 0.7 eV. A 24 parameter trial wavefunction (published in 1956) gave an ionization potential of: 0.7545 eV.

  20. On page 9 there are two "oddities" in the transition metals and 8 "oddities" at the end of the page. For each oddity report what the electron configuration "should" have been. Can you find any "rules" which would "explain" some of the oddities?

  21. On page 10 I outline the approach for calculating the Stark shifts for the n=3 orbitals. Do this calculation; remember to check your resulting eigenenergies using the given parabolic solution.

  22. How close would an ion have to get to a n=2 H-atom to perturb the energy level enough so that the light produced in the transition 21 is shifted by 1 Å. (Approximate the coulomb field of the ion: assume the ion's E field at the atom's nucleus is uniform throughout the atom.)

    Using the parabolic solution to the Stark Effect, find how close an ion could come to a n=100 H-atom without the calculated shift being large enough to ionize the atom.

  23. Carry out a crystal field calculation like that on page 11 but assume tetrahedral symmetry rather than octahedral. I.e., assume the charges are at: (1,1,1), (1,-1,-1), (-1,1,-1), and (-1,-1,1).

  24. Carry out a crystal field calculation like that on page 11 but assume square-planar symmetry rather than octahedral. I.e., assume the charges are at: (1,0,0), (-1,0,0), (0,1,0), and (0,-1,0). (This problem is more difficult than the previous problem.)

  25. Plot out the sp2 state Tri a.

  26. Calculate the dipole, i.e., <r> for a sp1, sp2, and sp3 state. (Use symmetry! You should know which direction <r> points.) Which wavefunction reaches out the furthest?