1) Solving the Hamiltonian 1.1) A common problem that we encounter in quantum mechanics is matrix diagonalization. This arises when the Schrodinger equation H W) = E |1,b) is solved by expressing W) as a linear combination

of N orthonormal functions Mm): N

lib) : Zen Man) ((¢kl ¢n) : 5km): where the coefﬁcient cn may be complex. Show that, by expressing W) in this basis, the

Schrodinger equation takes the form of a linear system of N equations: N

2%.k((¢klﬂ|¢n> was...) = [Us 6 {1....,N}. where the index k distinguishes among the equations in the system. 1.2) Let N = 2. Explicitly write out the two equations that make up the system of equations referenced

in the previous question. 1.3) Write out the determinantal equation that must be satisﬁed in order for a nontrivial solution to

exist for this system of equations. MathChapter E in McQuarrie provides a discussion on this

topic. 1.4) The Hamiltonian operator (and any other operator whose eigenvalues correspond to physical

observables) is Hermitian. A Hermitian operator has the property (¢k| H Mn) : (gbn| H |qbk)*. In

matrix form, this means that the element at position 1743,11. [denoted Hm) is the complex conjugate

of HT“, (and vice-versa). If all elements are real, then H3,” : Hﬂk. Assume that all terms in the

determinant you wrote in the previous question are real, and solve it for E. 1.5) Let H11 : 3/2, H12 : —\/§/2, and H22 : 5/2. Solve for the two values of E and their respective

(normalized!) eigenfunctions. 1.6) Write out the determinantal equation for the case in which the eigenfunctions are normalized, but

not orthogonal. That is, 1, kzn “ka $11) : {Slang kﬁé n 1.7) In class, we derived the energy levels and wavefunctions for H2+ under the Born-Oppenheimer

approximation. We let the wavefunction be RD) 2 c1 |13A) + c2 |133), where the basis functions

represent 13 hydrogen orbitals centered on each atom. Then, we redeﬁned our Hamiltonian as

H : Hg + V, where H0 is a hydrogen atom Hamiltonian, and ‘7 contains the potential from

another nucleus. Expressed this way, the Hamiltonian in matrix form is: 3,, + J Emma) + K

1E1135“?) + K E13 + J l where E13 is the energy of a 13 orbital of an H atom, J is the Coulomb integral (13A| V |13A},

K is the exchange integral (1314”? |1sB), and EU?) is the overlap integral (13A| 133), which

depends on the internuclear separation R (constant under the B0 approximation). The H atom

basis functions are normalized, but they are not orthogonal. Using the strategy from the previous

question, solve for the energies and normalized eigenfunctions.