Chemistry 356 017: Problem set No. 6; Reading: Mathchapters F and G, MQ - Ch. 7-8, Lecture notes on hydrogen atom. The H atom involves spherical coordinates and angular momentum, which leads to the shapes of s, p, d,... atomic orbitals. It is important to be familiar with integration in spherical coordinates. This will allow us to calculate properties of atomic wavefunctions. There is no need however to memorie the spherical form of the operators. I will always give them on an exam. In the lecture notes you find a fairly extensive treatment of angular momentum theory. This is important and will show up at various other places in the course. I expect you to be able to manipulate operator equations as discussed in the notes. The angular momentum operators in spherical coordinates do not need to be memoried. They will always be given. However I do l l il expect you to know the form for the ml functions Y ( θ, ϕ) sin θe ϕ, and that you can generate the other functions in the multiplet from this form (possibly in Cartesian coordinates). The variational principle is a very important concept in quantum mechanics. Unfortunately, actual calculations are often tedious, as is clear from the book. Linear variations using an expansion Ψ c k Φ k k l will be used frequently, in particular in a x form, which explains much of chemical bonding. We will usually assume that the expansion functions Φ k are orthonormal, as this simplifies the math. First-order perturbation theory amounts to calculating an expectation value over wave function that is determined by a 'related' eroth-order problem.
Problems: 1. MQ - 7.10. MS 7.18 3. MS 7.45 4. In this excercise you are asked to prove a number of relations in connection with angular momentum theory. They are mentioned in the lecture notes. You should try to use the general strategies that are mentioned in the notes, but avoid using the notes extensively, as very similar problems are discussed there. Prove the following: a. L!, L! i" L! (starting from the classical definition!!! L r p) b. L!, Ly y x 0 c.! L L! L! L L! " d. L!, L! " L! e.! Lx 0, and hence L! for any functionψ( θ, ϕ ), using the hermiticity of L! x. L 5. Derive explicitly the unnormalied forms of the d-functions in spherical coordinates by acting sequentially with L! on Y ( θ, ϕ ). Also show that! L Y ( θ, ϕ ) 0. Combine the functions Y m and Y m and find real cartesian d-functions that are eigenfunctions of L!. Are they unique? 6. Express! L x in terms of! L and! L and use this to show that! Lx 0 for any function Y m l ( θ, ϕ ). 7. MQ G-, G-4 (they go together). 8. MQ 8-1 9. See below. 10. See below.
9. Variational principle for excited states! (final exam 010) Consider the following trial wavefunction (not normalied): 3 ( x A x) x( x A)( x A), A x A Ψ ( x) 0, x< A or x> A for the Harmonic oscillator Hamiltonian ˆ 1 d 1 H h ω[ x ] (using dimensionless dx coordinates). Here A is the variational parameter, to be determined. First sketch the wave function for the value A1 (use the second form of the wave function for your convenience). Clearly show if the function is even or odd with respect to reflection in x0. Calculate the energy E as a function of A for a normalied wave function. If you have troubles with the integrals you can use EA A A ( ) h ω(5 /6). Next optimie the parameter A, using the variational principle, and calculate the optimal energy. You will 3 find (if everything is done correctly) that your energy is greater than E1 h ω. Please reason carefully that the trial wave function is orthogonal to the exact ground state (by symmetry, for any value of A), and explain that therefore the trial energy EA ( ) E. 1 You have found an approximation for the first excited state!
10. Prior final exam question on angular momentum. Relevant for final exam. The operator J! L! S! is an angular momentum operator with components Jˆ Lˆ Sˆ, Jˆ Lˆ Sˆ, Jˆ Lˆ Sˆ that satisfy the general commutation relations for x x x y y y angular momentum operators (listed on the next page for your convenience). In addition ˆ ˆ ˆ ˆ ˆ L,,, ˆ i S j L S j S L j 0 i, j x, y,,, You are asked to prove a number of identities below making use of (4.1) and the general relations. You are advised to use the results listed in previous parts of the question to prove the subsequent parts of the question. If you are stuck in the middle, you can go on with the next part, simply using the information from the previous part of the question (without proof). 4.1 Demonstrate the following: a.!,! (!,! J L J L J!, L! 0 soyoucan usealso 0) x y b. [ J, L ˆ ] 0 (hence it follows [! J, L! ] 0. No proof required) c.!!! J L S ( L! S! L! S! L! S! ) x x y y d. LS ˆ ˆ LS ˆ ˆ ( LS ˆ ˆ LS ˆ ˆ ) e. x x y y J L Sˆ L Sˆ L Sˆ L Sˆ (use the information from c. and d.) ˆ ˆ ˆ ˆ ˆ f. S!, J! " L! S! " L! S! (use form of Ĵ from e.) General relations on angular momentum theory for your perusal: A general vector-operator K! with components K!, K!, K!,!!! x y K Kx K y K is said to be an angular momentum operator if it satisfies the commutation relations K!, K! i" K! ; K!, K! i" K! ; K!, K! i" K! (A.1) x y x y y x In such a case we can define K! K! ik! ; K! K! ik! (A.) x y x y
and write!!!! K K! K K " K K! K! K! " K! (A.3) K!, K! " K! ; K!, K! " K! (A.4) We also know that!,!!,!!,!! K K K K K K K, K! K!, K! x y 0 (A.5) For your information: This question (in addition to some further parts) can be used to show that ˆ ˆ ˆ ˆ L, S, J, J all commute, and hence they have common eigenfunctions. This is the reason we can label atomic eigenfunctions by quantum numbers L, S, J, M J. However, S ˆ and L ˆ do not commute with Ĵ (see part f), and M S and M L are not good quantum numbers. Indeed we do not use them in the atomic term symbols. We only use them to find the possible L and S (highest M S and M L ).