PH 205: Mathematical methods of physics

Problem Set 2

1. In a very strange city, people always have to pass through point O when they go from any point P1 to any otherpoint P2. They have a well defined notion of distance from any point P to O given by a function dO(P ), whichhas the property that dO(P ) ≥ 0 ∀P , with the equality holding only for P = O. The distance between twopoints P1 and P2 is defined by the function d(P1, P2)

d(P1, P2) =

{0 if P1 = P2

dO(P1) + dO(P2) if P1 = P2

(a) Does the distance d(P1, P2) fulfill the requirement of a metric?

Now, think of the city as a two dimensional vector space, whose elements are simply the position vectors ofall points in the city from the origin O. Further, dO(P ) =

√X2 + Y 2, where X and Y are the usual x and y

coordinates of P .

(b) Does dO(P ) satisfy the requirements of a norm on the two dimensional space?

(c) Is the metric translationally invariant, i.e. is d(P1 + p, P2 + p) = d(P1, P2) ∀p? Here addition P + p meansthat if P has coordinates X and Y and p has coordinates x and y, then P + p has coordinates X + x andY + y.

2. Recall that the p norm of a vector v in Rn with components vi, where i goes from 1 to n is defined as

∥v∥p =

(n∑

i=1

|vi|p)1/p

,

where p is a positive integer. Verify that ∥ ∥p does indeed satisfy the requirements of a norm for p = 1, 2 and∞. For other values of p, the fact that ∥ ∥p is a norm follows from Minkowski’s inequality. You can look up theproof of this inequality in many standard references (including Wikipedia). Prove that ∥ ∥p can be obtainedfrom an inner product only for p = 2.

3. The most general inner product for the monomials {xn}, where n is a non-negative integer is defined as

⟨xp, xq⟩ =∫ b

a

K(x)xpxqdx,

where a and b are appropriate limits and K(x) is an appropriate Kernel or weighting function. With the abovedefinition of the inner product, we can use the Gram-Schmidt orthogonalization procedure to produce orthogonalpolynomial sequences.

(a) Can the monomials {xn} be orthogonalized for any choice of the Kernel?

In class, it was shown that for a = −1, b = 1 and K(x) = 1, one obtains the Legendre polynomials. In thisproblem you will obtain other such sequences.

(b) With a = −∞, b = ∞ and K(x) = e−x2

, obtain the first 5 polynomials in the sequence. Verify that theseare the first 5 Hermite polynomials by looking them up.

(c) With a = 0, b = ∞ and K(x) = e−x, obtain the first 5 polynomials in the sequence. Verify that these arethe first 5 Laguerre polynomials by looking them up.

(d) Verify that the Hermite and Laguerre polynomials you have calculated above and the Legendre polynomialsyou learnt about in class have recurrence relations of the form

xPn(x) = AnPn+1(x) +BnPn−1(x),+CnPn(x)

where Pn(x) is the polynomial of degree n and An and Bn are coefficients that in general depend on n.

2

(e) Prove that the above recurrence relation has to hold as a consequence of the orthogonalization process.(Hint: Use the fact Pn is orthogonal to all Pm for m < n and that ⟨xPm, Pn⟩ = ⟨Pm, xPn⟩, ∀m,n, whilenoting that xPn is a polynomial of degree n+ 1.)

4. The special theory of relativity combines time with space to form what is called Minkowski space. A vector v inthis space is represented as (t, r), where t is the time coordinate and r is the usual radius vector for the spatialcoordinates. The “inner product” for this space between two vectors v1 and v2 with components (t1, r1) and(t2, r2) is

⟨v1, v2⟩ = c2t1t2 − r1.r2,

where c is a constant equal to the speed of light in vacuum and . represents the usual scalar product betweentwo spatial vectors. From the above definition of the inner product it is clear that ⟨v, v⟩ need not be positivedefinite. The following terminology is used to describe the different possible vectors v in Minkowski space:

(a) ⟨v, v⟩ > 0: time-like

(b) ⟨v, v⟩ < 0: space-like

(c) ⟨v, v⟩ = 0: light-like (Show with an example that light-like vectors other than v = 0 exist)

The norm of v in each case is defined as, ∥v∥ =√|⟨v, v⟩|. Now, let us first consider Minkowski space with only

one spatial dimension, i.e. with r = xx.

(a) Given that v = 0 and ⟨u, v⟩ = 0, prove that

i. u is space-like if v is time-like

ii. u is time-like if v is space-like

iii. u is light-like if v is light-like

(b) Prove the following statements relating to the Cauchy-Schwarz inequality

iv. |⟨u, v⟩| ≥ ∥u∥∥v∥ if u and v are both time-like

v. |⟨u, v⟩| ≥ ∥u∥∥v∥ if u and v are both space-like

vi. |⟨u, v⟩| ≥ ∥u∥∥v∥ if even one is time-like

vii. |⟨u, v⟩| can be greater than, equal to or less than ∥u∥∥v∥ if one of out of u and v is time-like and theother space-like.

(c) Now, consider Minkowski space with three spatial dimensions, i.e r = xx+ yy+ zz. Which of the relationsi-vii hold for this space?

5. In class, you saw how a sequence of Gaussians can be thought to “converge” to a Dirac delta function. Thedelta function can similarly also be thought of as a limiting case of sequence of Lorentzians. The sequence isdefined as

fn(x) =1

π

n

n2x2 + 1.

(a) Convince yourself that this notion makes sense by sketching the sequence of functions and observingpictorially that it converges to the delta function.

In reality, this sequence of functions (as also the sequence of Gaussians) does not converge to the delta functionin the standard mathematical sense of “bunching together and converging” (the Cauchy convergence criterion).In this problem you will see this for yourself. Consider the usual inner product on the domain (−∞,∞),

⟨f, g⟩ =∫ ∞

−∞f(x)g(x)dx,

for real valued functions f(x) and g(x). This inner product allows us to define a norm ∥f∥ and a metric d(f, g)in the usual way.

(b) Calculate d(fn, fm).

(c) If the functions bunch together, you can eventually get them to be as close to one another as you like.More formally, given any number ϵ, you can find an m such that for all n > m, d(fn, fm) < ϵ. Show thatthis does not happen for the sequence of Lorentzians, i.e. d(fn, fm) cannot be made smaller than ϵ ∀n > mfor any m.