7—Operators and Matrices 2057.26 The set of Hermite polynomials starts out as
H 0 = 1, H 1 = 2x, H 2 = 4x^2 − 2 , H 3 = 8x^3 − 12 x, H 4 = 16x^4 − 48 x^2 + 12,(a) For the vector space of polynomials inxof degree≤ 3 choose a basis of Hermite polynomials and compute
the matrix of components of the differentiation operator,d/dx.
(b) Compute the components of the operatord^2 /dx^2 and show the relation between this matrix and the preceding
one.
7.27 On the vector space of functions ofx, define the translation operator
Taf=g means g(x) =f(x−a)This picks up a function and moves it byato the right.
(a) Pick a simple example functionfand test this definition graphically to verify that it does what I said.
(b) On the space of polynomials of degree≤ 3 and using a basis of your choice, find the components of this
operator.
(c) Square the resulting matrix and verify that the result is as it should be.
(d) What is the inverse of the matrix? (You should be able to guess the answer and then verify it. Or you can
work out the inverse the traditional way.)
7.28 The force by a magnetic field on a moving charge isF~ =q~v×B~. The operation~v×B~ defines a linear
operator on~v, stated asf(~v) =~v×B~. What are the components of this operator expressed in terms of the
three components of the vectorB~? What are the eigenvectors and eigenvalues of this operator? You may pick
your basis at will for the part of the problem in which you find the eigenvectors.
7.29 In section7.7you have an operatorMexpressed in two different bases. What is its determinant computed
in each basis?
7.30 In a given basis, an operator has the values
A(~e 1 ) =~e 1 + 3~e 2 and A(~e 2 ) = 2~e 1 + 4~e 4Draw a picture of what this does.
Find the eigenvalues and eigenvectors ofAand see how this corresponds to the picture you just drew.