OO T-Mobile F 6:45 PM Back DOC-20170421-WA0011 School of Arts and Sciences Depar

Question

OO T-Mobile F 6:45 PM Back DOC-20170421-WA0011 School of Arts and Sciences Department of Chemistry Quantum Mechanics (CCHE 542) (Spring 2017) Assignment 4 Read this instruction: If you take the photo from your phone you should save those pictures and place them inthe word file and submit the assignment as one word file (or pdf file). DO NOT SUBMIT multiple image files. (1) write all the steps to prove [lg, l] ihly (12 points) (2) The gradient of a function f(x,y,z), denoted grad f or is (2 points) 0x. oz Find V.Vf

Explanation / Answer

1. Prove that [Lz,Lx] = ihLy

Lx = Y*Pz Z*Py, Ly = Z*Px X*Pz, and Lz = X*Py Y*Px,

so that [Lx,Ly] = [Y*Pz Z*Py,Z*Px X*Pz]

= [Y*Pz,Z*Px] [Y*Pz,X*Pz] [Z*Py,Z*Px] + [Z*Py,X*Pz]

= Y*Px[Pz,Z] 0 0 + Py*X[Z,Pz ]

= ihY*Px + ihX*Py

= ihLz

For any right-handed triplet of unit-vectors, the choice which to call vector ex, is arbitrary, but once the choice is made, the labels of the other two are fixed due to handedness. Thus cyclic permutations of x,y,z are all equivalent to different labeling choices. This means that with different choices, our result would become [Ly,Lz] = ihLx

or [Lz,Lx] = ihLy

2.

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