Mathematical Tools for Physics

7—Operators and Matrices 206

7.31 The characteristic polynomial of a matrixMisdet(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 matrixM itself. The constant term will have to include the factorIfor this to make sense.

For this 2 × 2 case verify the Cayley-Hamilton Theorem, that the matrix satisfies its own characteristic equation,

making this polynomial the zero matrix.

7.32 For the magnetic field operator defined in problem 28 , placeˆz=~e 3 along the direction ofB~. Then take
~e 1 = (ˆx−iyˆ)/

√

2 ,~e 2 = (ˆx+iˆy)/

√

2 and find the components of the linear operator representing the magnetic

field. 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 problem 28 , and writeF~=m~ain this language and this

basis.

Ans: (part)m ̈r 1 =−iqBr 1 , wherer 1 = (x+iy)/

√

2. m ̈r 2 = +iqBr 1 , wherer 2 = (x−iy)/

√

2.

7.33 For the operator in problem 27 part (b), what are the eigenvectors and eigenvalues?

7.34 Anilpotentoperator is one such that if you take enough successive powers of the operator (a finite number)

you get the zero operator. For the operator defined in problem 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 multiple integrals with

hopeless limits of integration, toss it out and start over. You may even find problem 18 useful. Ans:ma^2 / 6

7.36 Show that the set of all 2 × 2 matrices forms a vector space. Produce a basis for it, and so what is its

dimension?

7.37 In the vector space of the preceding problem, a similarity transformation is an operator.f(M) =S−^1 MS.

ForS, use the rotation matrix of Eq. ( 11 ) and compute the components of this operatorf. The obvious choice

of basis would be matrices with a single non-zero element 1. Instead, try the basisI,σx,σy,σz.

7.38 What are the eigenvectors and eigenvalues of the operator in the preceding problem? Now you’ll be happy

that I suggested the basis that I did.