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# hw6 - a = i 1 2 2 i 3 b = i 1 2 3(a Construct a and b in...

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Assignment #6 PHYS-446 due in class on 14 November 2008 1. (Griffiths, problem 3.4) (a) Show that the sum of two hermitian operators is hermitian. (b) suppose Q is hermitian, and is a complex number. Under what condition (on ) is Q hermitian? (c) When is the product of two hermitian operators hermitian? (d) Show that the position operator ( x = x ) and the hamiltonian operator ( H = −ℏ 2 / 2m d 2 / dx 2 V x ) are hermitian. 2. Given that the position operator x and momentum operator p are both Hermitian. Show that the Hamiltonian (or energy) operator H describing a one-dimensional harmonic oscillator is also Hermitian. H = p 2 2m 1 2 k x 2 where m 0 , k 0 3. Consider a three-dimensional vector space spanned by an orthonormal basis 1 , 2 , 3 . Kets a and b are given by
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Unformatted text preview: a = i 1 2 2 i 3 b = i 1 2 3 (a) Construct a and b in terms of the dual basis 1 , 2 , 3 (b) Find a b and b a , and show that b a = a b (c) Find all nine matrix elements of the operator A = a b in this basis. (d) Show whether the operator A is Hermitian or not. 4. Consider a three-dimensional ket space where operator B is defined as B = [ b ib ib ] (a) Find the eigenvalues and eigenvectors of operator B . (b) Find the matrix representation of operator B with respect to the vector basis given by its eigenvectors....
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