ECE495N-F08-HW_4 - ECE 495N Fall08 ME118 MWF 1130A 1220P...

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9/10/08 ECE 495N, Fall’08 ME118, MWF 1130A – 1220P HW#4: Due Wednesday Oct.15 in class. Note: Problems 4 and 5 are carried over from HW#3 Problem 1: Consider the (2x2) matrix A = cos θ sin e i ϕ sin + cos Show that the following V 1 cos( /2) /2 sin( /2) + and 2 sin( /2) cos( /2) + are eigenvectors of [A]. What are the corresponding eigenvalues? Are they orthogonal (that is, is 1 + 2 = 0) ? Note: the superscript ‘+’ denotes Hermitian conjugate. Problem 2: Define a (2x2) matrix [ ] [{ 1 } { 2 }] Now calculate the matrix [B] = [V + ] [A] [V]. (A and [ ] [{ 1 } { 2 }] are given in Problem 1). Assuming ,0 2 π θϕ = , use MATLAB to find [V] and check that [B] = [V + ] [A] [V]. (HINT: use “[V, B]=eig(A)” command) Problem 3: Using the 2s and the three 2p levels as basis functions write down the Hamiltonian matrix for a hydrogen atom in an electric field F directed along the x-axis. Find the eigenvalues and eigenvectors. You may find Section 4.4.2 (page 99) of the text useful.
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9/10/08
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This note was uploaded on 12/30/2010 for the course ECE 495N taught by Professor S.datta during the Spring '08 term at Indiana University-Purdue University Fort Wayne.

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ECE495N-F08-HW_4 - ECE 495N Fall08 ME118 MWF 1130A 1220P...

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