Quantum assignment
- 1. Department of Chemistry Indian Institute of Technology Madras, Chennai CY101: Part A II Semester Assignment 1 Quantum Chemistry – Elementary quantum mechanical models 1) Consider an electron with a kinetic energy 2 eV. Calculate its momentum, velocity and the de Broglie wavelength. Is it reasonable to assume that the mass of the electron is independent of its speed? 2) Assume that an electron moves in a box of length 200 pm. Estimate the kinetic energy of electron using a particle in a box model. State your approximations. What is the de Broglie wavelength of the electron? 3) What are the eigenfunctions of a free particle kinetic energy operator? Obtain the momentum eigenvalues and eigenfunctions for the free particle. Are these wave functions “acceptable” in quantum mechanics? If not, how do you make them “acceptable”? 4) Verify that the eigenfunctions Ψn (x) = 2 L sin nπx L are eigenfunctions of the Hamiltonian for the particle in a box but not its momentum. 5) The wave function for a particle in a one dimensional box is given as ψ (x) = 2 L c1 sin πx L + c2 sin 2πx L ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ . Calculate the expectation value for the energy and momentum of this particle. If the energy of the particle is found to be five times the smallest eigenvalue for the particle in a box, obtain a value for c1 and c2 Are they unique? 6) You have used a procedure for separation of variables to derive the eigenfunctions of a particle in a 2-D or a 3-D box. Obtain by a similar procedure the solution of the time dependent Schrodinger equation
- 2. i ∂ψ ∂t = ˆHψ where ˆH is independent of time and ( )n xΦ are eigenfunctions of ˆH , i.e. ˆHΦn (x) = En Φn (x). 7) What are the energy eigenfunctions and eigenvalues for a particle in a 1-D box whose boundaries are 2 L x = ± (Box length is L and the mass of the particle is m)? 8) For the one dimensional box, obtain an expression for the maxima of the probability densities 2 ( )nP xψ= for all n . 9) Consider the wave functions 1 ( ) ikx k x e a ψ = in the interval , 2 2 a a⎛ ⎞ −⎜ ⎟ ⎝ ⎠ . Are these wave functions normalized and orthogonal in the limit a → ∞ ? Show your answer explicitly. 10) Calculate the expectation values x , 2 x , p , 2 p for a free particle in a 1-D box, with 2 sinn n x L L π ψ = . Calculate the product Δx Δp = x2 − x 2 p2 − p 2 where xΔ and pΔ are uncertainties in the particle position and momentum. Show that this verifies the Heisenberg’s uncertainty principle. 11) For the free particle in a two dimensional box, calculate the expectation values for the following quantities: 2 2 ,xy x and y −i x ∂ ∂y − y ∂ ∂x What is your interpretation of the operator −i x ∂ ∂y − y ∂ ∂x ? 12) An arbitrary normalized wave function for a particle in a one dimensional box is given by ψ (x) = Cx2 (L − x)2 . a) Obtain an expression for C.
- 3. b) Express ( )xψ as 1 2 ( ) sinn n n x x C L L π ψ ∞ = = ∑ . Obtain an expression for the coefficient Cn. c) What is the value of the sum 2 1 n n C ∞ = ∑ ? d) What is the average value for the energy of the particle in this state? 13) Consider the rectangular wave function ψ (x) = 1 L , 0 < x < L ψ (x) = 0, otherwise Express 1 2 ( ) sinn n n x x C L L π ψ ∞ = = ∑ and obtain an expression for Cn and 2 1 n n C ∞ = ∑ . Is this an acceptable wave function? 14) A linear one dimensional chain of conjugated polyene contains 12 C-C links (-C=C-C=C-)3 with an average bond distance of 1.33 Angstrom. Ignore the end effect and assume that the pi electrons are free to move along the bonds. Determine the energy difference between the highest occupied energy level and the lowest unoccupied energy level. Assume that at most two electrons can occupy each energy level. 15) Calculate the commutator ˆ ˆ ˆ ˆ ˆˆ, x y zx y z p p p⎡ ⎤+ + + +⎣ ⎦. 16) If ˆA , ˆB and ˆC are three operators, show that the commutator ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ, , ,A BC B A C A B C⎡ ⎤ ⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦ ⎣ ⎦ and ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ, , ,AB C A B C A C B⎡ ⎤ ⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦ ⎣ ⎦ . Use these to calculate the commutator 2 2 ˆ ˆ, xx p⎡ ⎤⎣ ⎦. 17)For the free particle in a cubical box of side L, calculate the energy E for the first eight states and identify the degeneracy of each of them. 18) Calculate the ground state energy of the electron in the hydrogen atom using the wave functionψ (r) = ce−r/a0 . The Hamiltonian is
- 4. H = − 2 2m ∂2 ∂x2 + ∂2 ∂y2 + ∂2 ∂z2 ⎛ ⎝⎜ ⎞ ⎠⎟ − e2 4πε0r . 19) Assuming a simple 3d cubic box model, calculate the longest wave length that an electron can have in a box of size 118 pm. Calculate the average kinetic energy and determine what fraction this constitutes of the total energy calculated using the exact wave function in the previous problem? 20) Examine carefully all the fifteen angular wave functions (l=1, 2 and 3) and the animations given in your web site. Identify all angular nodes.