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4. Which of the hydrogen-like wavefunctions, 2p,2p, and 2p, is an eigenfunction of (a) A, (b)...
6. 2p,, 2py, and 2p, orbitals form an orthonormal basis set for a hydrogen state with n 1. These orbitals can be made by linear combinations of 2po, 2p, and 2p., as follows: 2 and 1- Compute the magnitude of total rotational angular momentum and the average z-component of the total angular momentum of 2p., 2p,, and 2p, states. (6 pts) Hint: use the following properties of H states and 1 1 for 2po. 2p, and 2p., and ml 0,...
1. (50 points) Consider the particle in a one-dimensional box (0 s x S L). Assume a term is added to the Hamiltonian of the form: πχ V(x)g sin Sketch the potential and the expected eigenfunction (small g). In the limit of small g, find the second order correction to the ground state energy 2. (50 points) For a diatomic molecule rotating in free space, the Hamiltonian may be written: 12 21 Where L is the total angular momentum operator,...
SMA #8: Bohr and Schrödinger Models of Hydrogen Here we investigate the relationship between the Schrödinger and Bohr models of hydrogen-like atoms, following our work in class on both 9 1. Using the appropriate Schrödinger wavefunctions, compute the most probable electron-proton radii (i.e., distances) for 1s, 2p, and 3d states. Do these agree with the corresponding Bohr radii? Hint #1: Remember to maximize the "radial distribution function" P(r) = [rR(r)], i.e., to include the radial Jacobian factor (r2) in your...
Consider a wave function for a hydrogen-like atom: 81 V πα3 a) Find the corresponding values of the quantum num bers n, 1, and m. (b) By measuring the angular momentum, what is the probability of finding 1-0? (c) Construct ψ(r, θ, φ) and another wave function with the same values of n and (azimuthal) quantum number, m+1 (d) Calculate the most probable value of r for an electron in the state corresponding to ψ(r, θ, φ) 1, but with...
question 7 9 10
), where n, a, and are constant, is an eigenfunction of p. 7. (a) p. =- what is p. ? (b) sin( i ax what is the eigenvalue? (107) (9) = v ydt for a normalized wavefunction. Please find (1) for(a) v. - and (b) 42p. 4/2008 re s in sind. (hint : integrate over all space: sin Odrdodø (sin? xdx = [l-c952de, 5 xede = (203) 3 2 10. A particle of mass m is...
(a) Show that (@) = sin e- is an eigenfunction of both Î, and Î", where = -1 1 a 1 22 sin + sin 020 sin0 and derive the corresponding eigenvalues. You may use the identity 1 a 1 sin sin 2 sin sin 0 80 sino 31 (sin 00 (5 marks) (6) Consider the function $(,0,4)= A - 1/200 sin 6e-ip, 20 where A is a constant and an is the Bohr radius. This is a hydrogen atom...
I. The state vector φ(t) at time t can be decomposed on the {I +), l-)} basis: Write down the system of coupled differential equations which the components c+ (t) and c() satisfy 2. Let |φ(1-0)) be decomposed on the {lx+), lx-) basis. Show that c+(1)-(Ηφ(t)) is written as sin with Ω 2VA2+B2. Here f.Ω is the energy difference of the two levels. Show that c+(1) (as well as c (t)) satisfies the differential equation c) 0. We assume that...
Problem 4 Consider the hydrogenic wavefunctions Un,1,mi = Rn,l(r)Yı,mı (0,0) for an elec- tron in a hydrogen atom (Z = 1). The electron is in the 2s state. a) Determine the location of the radial node in terms of ao. b) Calculate the most probable radius of an electron in a 2s state, and com- pare this radius with the most probable radius of the ls state (ao). c) Using the formula provided in class, determine the mean radius for...
1. (a) (10 pt) Which of the following wavefunctions CAN be a valid total wavefunction for the excited electronic state of He: 2p13d1 with the total electron spin in the triplet state (25+1 = 3), where N is the normalization constant. Explain your choice. (i) NW2p(1)3d(2) [0(1)(2)-6(1)a(2)] (ii) N [42p(1)93a(2) - za142p(2)] [a(1)B(2)+B(1)a(2)] (iii) N (42p(1)43a(2) + Y3d(1)420(2)) [a(1)$(2)-B(1)a(2)] (iv) N (42p(1)^3a(2) - 43a(1)#2p(2)] a(1)a(2) (b) (10 pt) Determine the normalization constant N in the following wavefunctigo: N [Uzo(1)3d(2) W3d(1)42p(2))...