PH 205: Mathematical Methods of Physics: Problem Set 2

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 P
1
to any other
point P
2
. They have a well dened notion of distance from any point P to O given by a function d
O
(P), which
has the property that d
O
(P) 0 P, with the equality holding only for P = O. The distance between two
points P
1
and P
2
is dened by the function d(P
1
, P
2
)
d(P
1
, P
2
) =
{
0 if P
1
= P
2
d
O
(P
1
) + d
O
(P
2
) if P
1
= P
2
(a) Does the distance d(P
1
, P
2
) fulll the requirement of a metric?
Now, think of the city as a two dimensional vector space, whose elements are simply the position vectors of
all points in the city from the origin O. Further, d
O
(P) =
X
2
+ Y
2
, where X and Y are the usual x and y
coordinates of P.
(b) Does d
O
(P) satisfy the requirements of a norm on the two dimensional space?
(c) Is the metric translationally invariant, i.e. is d(P
1
+p, P
2
+p) = d(P
1
, P
2
) p? Here addition P +p means
that if P has coordinates X and Y and p has coordinates x and y, then P + p has coordinates X + x and
Y + y.
2. Recall that the p norm of a vector v in R
n
with components v
i
, where i goes from 1 to n is dened as
v
p
=
(
n
i=1
|v
i
|
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 Minkowskis inequality. You can look up the
proof of this inequality in many standard references (including Wikipedia). Prove that
p
can be obtained
from an inner product only for p = 2.
3. The most general inner product for the monomials {x
n
}, where n is a non-negative integer is dened as
x
p
, x
q
=
b
a
K(x)x
p
x
q
dx,
where a and b are appropriate limits and K(x) is an appropriate Kernel or weighting function. With the above
denition of the inner product, we can use the Gram-Schmidt orthogonalization procedure to produce orthogonal
polynomial sequences.
(a) Can the monomials {x
n
} 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 this
problem you will obtain other such sequences.
(b) With a = , b = and K(x) = e
x
2
, obtain the rst 5 polynomials in the sequence. Verify that these
are the rst 5 Hermite polynomials by looking them up.
(c) With a = 0, b = and K(x) = e
x
, obtain the rst 5 polynomials in the sequence. Verify that these are
the rst 5 Laguerre polynomials by looking them up.
(d) Verify that the Hermite and Laguerre polynomials you have calculated above and the Legendre polynomials
you learnt about in class have recurrence relations of the form
xP
n
(x) = A
n
P
n+1
(x) + B
n
P
n1
(x), +C
n
P
n
(x)
where P
n
(x) is the polynomial of degree n and A
n
and B
n
are coecients 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 P
n
is orthogonal to all P
m
for m < n and that xP
m
, P
n
= P
m
, xP
n
, m, n, while
noting that xP
n
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 in
this space is represented as (t, r), where t is the time coordinate and r is the usual radius vector for the spatial
coordinates. The inner product for this space between two vectors v
1
and v
2
with components (t
1
, r
1
) and
(t
2
, r
2
) is
v
1
, v
2
= c
2
t
1
t
2
r
1
.r
2
,
where c is a constant equal to the speed of light in vacuum and . represents the usual scalar product between
two spatial vectors. From the above denition of the inner product it is clear that v, v need not be positive
denite. The following terminology is used to describe the dierent 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 dened as, v =
|v, v|. Now, let us rst consider Minkowski space with only
one spatial dimension, i.e. with r = x x.
(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| uv if u and v are both time-like
v. |u, v| uv if u and v are both space-like
vi. |u, v| uv if even one is time-like
vii. |u, v| can be greater than, equal to or less than uv if one of out of u and v is time-like and the
other space-like.
(c) Now, consider Minkowski space with three spatial dimensions, i.e r = x x+y y +z z. Which of the relations
i-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. The
delta function can similarly also be thought of as a limiting case of sequence of Lorentzians. The sequence is
dened as
f
n
(x) =
1
n
n
2
x
2
+ 1
.
(a) Convince yourself that this notion makes sense by sketching the sequence of functions and observing
pictorially 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 function
in 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 dene a norm f and a metric d(f, g)
in the usual way.
(b) Calculate d(f
n
, f
m
).
(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 , you can nd an m such that for all n > m, d(f
n
, f
m
) < . Show that
this does not happen for the sequence of Lorentzians, i.e. d(f
n
, f
m
) cannot be made smaller than n > m
for any m.

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