Quantum Mechanics for Mathematicians

Problem 3:
By using the fact that any unitary matrix can be diagonalized by conjugation
by a unitary matrix, show that all unitary matrices can be written aseX, for
Xa skew-adjoint matrix inu(n).
By contrast, show that
A=

(

− 1 1

0 − 1

)

is in the groupSL(2,C), but is not of the formeXfor anyX∈sl(2,C) (this
Lie algebra is all 2 by 2 matrices with trace zero).
Hint: For 2 by 2 matricesX, one can show (this is the Cayley-Hamilton
theorem: matricesXsatisfy their own characteristic equation det(λ 1 −X) = 0,
and for 2 by 2 matrices, this equation isλ^2 −tr(X)λ+ det(X) = 0)

X^2 −tr(X)X+ det(X) 1 = 0

ForX∈sl(2,C),tr(X) = 0, so hereX^2 =−det(X) 1. Use this to show that

eX= cos(

√

det(X)) 1 +

sin(

√

det(X))

√

det(X)

X

Try to use this foreX=Aand derive a contradiction (taking the trace of the
equation, what is cos(

√

det(X))?)

Problem 4:

• Show thatMis an orthogonal matrix iff its rows are orthonormal vectors
  for the standard inner product (this is also true for the columns).


• Show thatMis a unitary matrix iff its columns are orthonormal vectors
  for the standard Hermitian inner product (this is also true for the rows).

B.3 Chapters 5 to

Problem 1:
On the Lie algebrasg=su(2) andg=so(3) one can define the Killing form
K(·,·) by
(X,Y)∈g×g→K(X,Y) =tr(XY)

1. For both Lie algebras, show that this gives a bilinear, symmetric form,
   negative definite, with the basis vectorsXjin one case andljin the other
   providing an orthogonal basis if one usesK(·,·) as an inner product.


2. Another possible way to define the Killing form is as

K′(X,Y) =tr(ad(X)◦ad(Y))

Here the Lie algebra adjoint representation (ad,g) gives for eachX∈ga

linear map

ad(X) :R^3 →R^3