Mathematical Methods in Nanophotonics: Problem Set 1, Exercises of Nanomaterial Engineering

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18.369 Mathematical Methods in Nanophotonics

Spring 2008

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18.325 Problem Set 1

Problem 1: Adjoints and operators

(a) We defined the adjoint t of states and opera-

tors by: (H11H2)= 1 ~ 1 ) ~IH2) and (Hll0lH2) =

( 0 t I HI))^ I Hz). Show that for a finite-

dimensional Hilbert space, where IH) is a col-

umn vector h, (n = I ,... ,d), 0 is a square

d x d matrix, and ( ~ ( ' 1I H ( ~ ) ) is the ordinary

conjugated dot product Enhi1)*hL2),the above

adjoint definition corresponds to the conjugate- transpose for both matrices and vectors.

(b) Show that if 0 is simply a number o, then

0+= o*. (This is not the same as the previ-

ous question, since 0 here can act on infinite-

dimensional (continuous) spaces.)

(c) If a linear operator 0 satisfies 0 t = 0-', then

the operator is called unitary. Show that a uni- tary operator preserves inner products (that is,

if we apply 0 to every element of a Hilbert space,

then their inner products with one another are unchanged). Show that the eigenvalues u of a unitary operator have unit magnitude (lul = 1) and that its eigenvectors can be chosen to be orthogonal to one another.

(d) For a non-singular operator 0 (i.e. 0-l exists),

show that (8-')t = (0t)-'. (Thus, if 0 is Her-

mitian then 0 - I is also Hermitian.)

Problem 2: Completeness

(a) Prove that the eigenvectors In) of a finite-

dimensional Hermitian operator 0 (a d x d

matrix) are complete: that is, that any d- dimensional vector can be expanded as a sum

En en In) in the eigenvectors with some coeffi-

cients en. It is sufficient to show that there are d linearly independent eigenvectors In):

(i) Show that every matrix

0 has at least one nonzero eigenvector 11)

(. .. use the fact that every polynomial with

nonzero degree has at least one (possibly complex) root).

(ii) Show that the space of Vl = {Iv) 1 (v 11) =

1. orthogonal to 11) is preserved (trans- formed into itself or a subset of itself) by

0. From this, show that we can form a

(d - 1) x (d - 1) Hermitian matrix whose

eigenvectors (if any) give (via a similar- ity transformation) the remaining (if any) eigenvectors of 0. (iii) By induction, form an orthonormal basis of d eigenvectors for the d-dimensional space.

(b) Suppose that we have an infinite-dimensional

Hermitian operator 0 that can be simulated on

a computer (with arbitrary, but finite, memory and time): its solutions can be approximated to arbitrary accuracy by a finite-dimensional Her-

mitian operator (e.g. 0 discretized on a finite

grid). Argue that the infinite-dimensional eigen- vectors form a complete basis for anything that we care about; can you give an example of a sense in which they do not form a complete basis?'

(c) Completeness is not automatic for eigenvectors in general. Give an example of a non-singular non-Hemitian operator whose eigenvectors are

not complete. (A 2 x 2 matrix is fine. This case

is also called "defective.")

Problem 3: Maxwell eigenproblems

(a) In class, we eliminated E from Maxwell's equa-

tions to get an eigenproblem in H alone, of the

form 8 1 H ) = $ I H ). Show that if you in-

stead eliminate H , you cannot get a Hermitian

eigenproblem in E except for the trivial case

E = constant. Instead, show that you get a generalized Hermitian eigenproblem of the form

A I E) = !$B I E), where both A and B are Her-

mitian operators.

l ~ o r a more precise discussion of the completeness of conti- nous Hermitian operators, see e.g. Courant & Hilbert, Methods of Mathematical Physics vol. 1.