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PH 205: Mathematical methods of physics

Problem Set 3

1. In this problem you will construct bases for the dual of the vector space of real valued functions of the form

f(x) =n

anxn,

with x in some domain (x1, x2). Recall that given a set of basis vectors{vi} of a vector space, you can obtaina basis for the dual space by constructing linear functionals {j}with the property that

j(vi) =ij .

First consider the set of Legendre polynomials whose elements form an orthogonal basis set for the above vectorspace with x1= 1 and x2 = 1.

(a) Use the orthogonality property of the Legendre polynomials to obtain the basis vectors for the dual space.Do the same with the Hermite polynomials and x1 = 0 and x2 = .

Now, consider the set of monomials {1, x , x2, x3, . . . }. This set can also be thought of as a basis for the given

vector space. However, as you have shown in problem set # 2, there is no way to orthogonalize the monomialsusing an inner product in the form of an integral with a kernel. Thus, a construction of a set of basis vectorsfor the dual space analogous o the ones staring with the Legendre and Hermite polynomials does not exist formonomials.

(c) Neverthelss, show by explicit construction that it is still possible to obtain linear functionals {j} withj(vi) =ij that form a basis for the dual space where vi is a monomial. (Hint: Use derivatives.)

(d) What would be the analogous construction for the basis set of Legendre polynomials?

2. The rank-nullity theorem has a neat application in dimensional analysis. Recall that dimensional analysisallows you to construct secondary physical quantities in terms of more fundamental dimensionful ones. Thefundamental quantities are typically mass, length, time, charge and temperature and the secondary ones canbe velocity, momentum, force, energy, electric field, magnetic field, entropy, specific heat etc. For concretenessassume that there are p fundamental quantities f1,f2, . . . f p and you want to construct n secondary quantitiesq1, q2 . . . q n using them. Dimensional analysis tells you that each qi is of the form

qi =

pj=1

(fj)aji ,

where the aji s are numbers (typically integers) and form a matrix A. A secondary quantity qi is dimensionlessifaji = 0j.

(a) Assume that the fs are mass, length and time and the qs are velocity, energy, angular momentum andtorque. Work out the matrix A.

A set of secondary quantities q1, q2, . . . q m for some m are said to be independent if it is not possible to form

any dimensionless quantity of the form

m

j=1(qj)bj using them. The fundamental quantities f are independent

by definition.

(b) Show that there is a one to one correspondence between Ker(A) and all possible dimensionless combinationsthat can be formed out ofq1, q2 . . . q n. Verify this for (a).

(c) Show that the rank ofA (call it r) is equal to the maximum number of independent quantities from amongq1, q2 . . . q n. Again, verify this for (a).

(d) Now, consider a relation p(q1, q2 . . . q n) = C, where C is a dimensionful quantity and the function pinvolves a product of each of the qs raised to some power. Show that

p(q1, q2. . . q n) =h(q1, q2, . . . q r, d1, d2, . . . dk),

where theds are dimensionless quantities. Each argument is raised to some power in the function h. Whatis the value ofk (think rank-nullity)? Check this result for (a) by picking some function p.

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(e) Argue that the the idependent quantities q1, q2, . . . q r can be elminated so that the relation p(q1, q2. . . q n) =Ccan be cast in the form g (d1, d2 . . . dk) =D, where g is also a function where each argumentis raised to some power and D is a dimensionless quantity. Thus, what you have shown is that anydimensionful equation involving the quantities qis equivalent to a relation among a smaller number (k) ofdimensionless quantities. This is a consequence of the rank-nullity theorem.

3. The Pauli spin matrices are used to represent the operators for the different components of the spin of anelectron. They are defined as

x =

0 11 0

, y =

0 ii 0

and z =

1 00 1

.

The operator for the spin along the dierction of the unit vector n can then be written as

n = n.,

where= xx+y y+z z.

(a) What are the eigenvalues and eigenvectors ofn?

(b) Calculate the commutator [n, m] for unit vectors n and m.

(c) Calculate (n)N

, where N is a non-negative integer. Use the result to argue that the operators sin(n)

and cos(n), where is a complex number are of the form A+Bn, where A and B are constantsthat depend on . What are the values of these constants?

4. Odds and ends.

(a) For a square matrix A, show that

log(det A) = Trlog A

(b) Let Xbe a column vector of unit norm. Obtain all the eigenvalues of the matrix XX.

(c) Prove that the diagonalizing matrices for Hermitian (real symmetric) matrices are unitary (orthogonal).

(d) Given a matrix A, the matrix B = AAis clearly Hermitian. Is the converse true, i.e. for every Hermitianmatrix B , does there exist a matrix A such that B= AA?

(e) Prove that the product of two Hermitian matrices is Hermitian iff they commute.

(f) Prove that [f(x), p] = idfdx

. here x and p are the position and the momentum operators and f(x) is afunction that has a Taylor expansion in x. For this problem use only the fact that [x, p] =iand not thatp=

iddx

. (Hint: Calculate [xn, p] for positive integersn.)

(g) An orthogonal n n matrix has a trace equal to t. What is the value of its determinant?

5. The value of the Gaussian integral

exp(x2) dx =

,

whereis a positive real number.

(a) Warm up: Calculate

x1=

x2=

exp(x21x2

2) dx1dx2,

where, >0.

(b) Now consider the integral

I=

x1=

x2=

exp(XTM X) dx1dx2,

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where X=

x1x2

and XT is the transpose ofX. Mis a real symmetric matrix. Prove that the integral

exists only ifMhas only positive eigenvalues and its value is detM

. To prove this you will need to one

of the results from problem 3 and the fact that when you transform co-ordinates from X =

x1x2

to

Y =

y1y2

, through the transformationY =OX, where O is a matrix, the area element transforms as

dx1dx2 = dy1dy2

| det O|.

The factor | det O| is called the Jacobian of the transformation from X to Y. (Hint: Convert the integralto the form of (a).)

(c) How does the above result generalize to the case when X is an N-dimensional column vector of reals andthe matrix M is an NN real symmteric matrix?