7—Operators and Matrices 175This 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 cubic polynomials 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.)
(e) What if the parameteraishuge? Interpret some of the components of this first matrix and show
why they are clearly correct. (If they are.)
(f) What is the determinant of this operator?
(g) What are the eigenvectors and eigenvalues of this operator?
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? For this last part, pick the basis in which you want to do the computations. If you’re not
careful about this choice, you are asking for a lot of algebra. Ans: eigenvalues: 0 ,±iB
7.29 In section7.8you 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 4
(a) Draw a picture of what this does. (b) Find the eigenvalues and eigenvectors and determinant ofA
and see how this corresponds to the picture you just drew.
7.31 The characteristic polynomial of a matrixM isdet(M−λI). Iis the identity matrix andλis
the variable in the polynomial. Write the characteristic polynomial for the general 2 × 2 matrix. Then
in place ofλin this polynomial, put the matrixMitself. The constant term will have to include the
factorIas usual. For this 2 × 2 case verify the Cayley-Hamilton Theorem, that the matrix satisfies its
own characteristic equation, making this polynomial inMthe zero matrix.
7.32 (a) For the magnetic field operator defined in problem7.28, placezˆ=~e 3 along the direction of
B~. Then take~e 1 = (xˆ−iˆy)/√ 2 ,~e 2 = (xˆ+iˆy)/√ 2 and find the components of the linear operator
representing the magnetic field. (b) A charged particle is placed in this field and the equations of
motion arem~a=F~=q~v×B~. Translate this into the operator language with a matrix like that of
problem7.28, and writeF~=m~ain this language and this basis. Ans: (part)mr ̈ 1 =−iqBr 1 , where
r 1 = (x+iy)/
√
2. m ̈r 2 = +iqBr 2 , wherer 2 = (x−iy)/
√
2.
7.33 For the operator in problem7.27part (b), what are the eigenvectors and eigenvalues?
7.34 Anilpotentoperator was defined in section7.14. For the operator defined in problem7.8, show
that it is nilpotent. How does this translate into the successive powers of its matrix components?
7.35 A cube of uniform mass density has sideaand massm. Evaluate its moment of inertia about an
axis along a longest diagonal of the cube. Note: If you find yourself entangled in a calculation having