1. (3 pts) What is the degeneracy of J=0 and J=1 for a linear,
symmetric, and spherical rotor? For
each rotor, give the complete set of quantum numbers for each
state. (Each state should have a
unique set of quantum numbers.)
1. (3 pts) What is the degeneracy of J=0 and J=1 for a linear, symmetric, and...
(5 pts) Give an example of a relation on a set that is a) both symmetric and antisymmetric. b) neither symmetric nor antisymmetric. (3 pts each) For each of the following find an indexed collection {An}nen of distinct sets (no two sets are equal) such that (a) n =1 An = {0} (b) Um_1 An = [0, 1] (c) n =1 An = {-1,0,1} (5 pts each) Give example of an explicit function f in each of the following category...
Orbitals and Quantum Numbers Each atomic orbital is specified by a unique set of n, l and ml quantum numbers: 1a. What quantum number/s do the two spherical orbitals have in common? What quantum number/s would be different? Are these orbitals s, p or d? 1b. Write down a possible set (n, l, ml) of quantum numbers for each spherical orbital. 1c. Consider the dumb-bell shaped orbitals. What quantum number/s do these three orbitals have in common? What quantum number/s...
Suppose A is a symmetric 3 by 3 matrix with eigenvalues 0, 1, 2 (a) What properties 4. can be guaranteed for the corresponding unit eigenvectors u, v, w? In terms of u, v, w describe the nullspace, left nullspace, (b) row space, and column space of A (c) Find a vector x that satisfies Ax v +w. Is x unique? Under what conditions on b does Ax = b have a solution? (d) (e) If u, v, w are...
3 Angular Momentum and Spherical Harmonics For a quantum mechanical system that is able to rotate in 3D, one can always define a set of angular momentum operators J. Jy, J., often collectively written as a vector J. They must satisfy the commutation relations (, ] = ihſ, , Îu] = ihſ, J., ſu] = ihỈy. (1) In a more condensed notation, we may write [1,1]] = Žiheikh, i, j= 1,2,3 k=1 Here we've used the Levi-Civita symbol, defined as...
5. (8 pts) Suppose B is a 4 x 5 matrix, and the associated linear transformation T(E) Bd, is onto. (a) (3) Find dim Nul(B) (b) (2) Does Ba 5 have a unique solution for every B (c) (3) Give a geometric interpretation to the solution set of Bt- 0 5. (8 pts) Suppose B is a 4 x 5 matrix, and the associated linear transformation T(E) Bd, is onto. (a) (3) Find dim Nul(B) (b) (2) Does Ba 5...
3. The spacing between the J = 0 and J = 1 lines (corresponding to l = 0,1 = 1 quantum numbers) of the rotational spectrum of HI is 13.2 cm-1, which corresponds to B = AE0–1/2 = 6.6 cm-1. The principal isotope of Iodine has a mass of 127 amu. From this information, determine: (a) What is the moment of inertia of HI? (b) What is the HI bond length? (c) What is AE0_1 for deuterium iodide?
N N (6 pts.) The goal of this problem is to help you understand better the quantum number rules and to prove to you that every electron in an atom has a unique set of 4 quantum numbers.) First, write out the full electron configuration for scandium (Sc). Then, write out the complete set of quantum numbers for each electron in Sc listing them in order of filling. 2. Example: The complete electron configu ration of oxygen (O) is 1s22s'2p....
Question 15 1 pts What is the maximum number of electrons in an atom that can have the following set of quantum numbers? n = 4,1 = 3, m = -2, ms = +1/2 06 00 O 10 0 Next > < Previous
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
3. (12 pts) Determine whether the following binary relation is: (1) reflexive, (2) symmetric, (3) antisymmetric, (4) transitive. a) The relation Ron Z where aRb means a = b. Circle your answers. (4 pts) Ris Reflexive? Symmetric? Antisymmetric? Transitive? Yes or No Yes or No Yes or No Yes or No b) The relation R on the set of all people where aRb means that a is taller than b. Circle your answers. (4 pts) Ris Reflexive? Symmetric? Antisymmetric? Transitive?...