Homework 5 Due Mon Nov 7, 2005
Levine 6.1, 6.16, 6.17, 6.47, 7.27, 7.29 **Levine 9.2 and 9.4**
1) A Dirac notation equivalent of our integrals for finding Fourier coefficients is the
“projection operator” Pn = |n> <n| (it may look strange but you know how to “use” it). Pn
projects out the component of any arbitrary ket that lies along the basis ket |n> . (i.e. if
ψ= a|1> + b|2> + c|3>) then Pnψ = |1>(<1| a|1> + <1|b|2> + <1|c|3>)= a|1>. Combine
the definition of completeness in Levine 7.39 with Levine 7.71 to show that
1|| =><
∑
n
nn
This is the more common statement of completeness for an orthonormal basis set.
Note: this is really a notational problem—just translate Levine’s equations into Dirac
notation, and using the definitions you are familiar with show that this new operator must
return f when operating on any function f.
2) What is the most probable distance (in meters) of finding a 1s electron from a Uranium
nucleus that has been ionized of all but 1 electron? What is the potential energy of an
electron this far away from a uranium nucleus (in Joules)? The virial theorem (an
important result from chapter 14 that we haven’t had time to cover) tells us that 2 * <KE>
of this electron will equal -<U> (or that <KE>=-<E>), the average PE of this electron.
Use a classical estimate to calculate how fast such an electron is moving. Comment.
3) Following Levine, use a computer to make linear combinations of Y(1,-1), Y(1,1),
Y(1,0) to generate the familiar px, py and pz orbitals. Make 3D plots of the angular
components to verify that you indeed get the angular factors from px and py. Why can we
take linear combinations of these eigenfunctions and create new eigenfunctions of H,
when we know that in general, linear combinations of eigenfunctions are not also
eigenfunctions.
** 4) See next page:
