Operators in Quantum Mechanics
Summer term 2017
A. Scrinzi
J. Bucher
Problem sheet 1
Thu, May 4
P1.1: Normed vector space
Let C[0,1] denote the continuous real-valued functions on [0, 1].
(a) Convince yourself that with pointwise addition this is a linear space.
(b) Show that the functionals
Z
||f ||∞ = max |f (x)| and ||f ||1 =
x∈[0,1]
1
dx|f (x)|
(1)
0
are norms.
(c) Generalize the norm and proof of the norm properties for the space continuous functions A(~r, p~) over a compact subset of phase-space Γ ⊂ R6 .
P1.2: Convergence, Cauchy sequence
Remember the definition of convergence: a sequence of elements {xn , n =
0, 1, . . . , ∞} from a normed (vector-) space V is said to converge to an element
x ∈ V if ||x −
xn || → 0 as n → ∞. Two other important concepts are open ball
B (x) = {y ||y − x|| < } and neighborhood of x, Nx : ∃ and B (x) ⊂ Nx ,
i.e. a set that contains x and also an open ball around x.
(a) Prove that any convergent sequence is a Cauchy sequence.
(b) xn → x if and only if for each neighborhood Nx of x there exists an M
such that m ≥ M implies xm ∈ Nx .
P1.3: Operator norm
Let V denote a vector space over the complex numbers C with a norm || · ||. Let
T denote a linear transformation V → V
T : T v = u,
(2)
(a) Show that linear transformations form themselves a linear space B, if one
defines addition of two maps by
T, S ∈ B : (T + S)v = T u + Su
and in similar way multiplication by a scalar.
1
(3)
(b) Assume now that V is finite-dimensional (i.e. there is a finite basis) Show
that
||T || := max ||T v||/||v||
(4)
v∈V
defines a norm on B (||T v|| is the norm on V).
(c) Consider the case where V is infinite-dimensional. Show that
||T || := sup ||T v||/||v||
(5)
v∈V
can diverge, i.e. not every linear operator is bounded. Give the simplest
possible examples for this with V = L2 (dx, R). On l2 (= “infintely long
vectors” (a0 , a1 , ...an , ...), can you give an infinite-dimensional matrix that
is unbounded?
Continuity is an essential property of maps, as it allows us to take limits, i.e.
to approximate, e.g. an idealized measurement by a sequence of increasingly
accurate measurements and assume that the perfect measurement would be
close to our “increasingly accurate” measurements. In a sense, it is the idea of
“continuity” makes the concept of “accuracy” meaningful.
For functions f between sets with a norm, continuity can be best understood
as “the limit of the function values is the function value of the limit”, i.e.
f is continuous ⇔ f (xn ) → f (x) if xn → x.
(6)
This is not the most general definition of continuity, but it serves all our purposes and matches intuition.
A bounded operator is an operator with finite operator norm (problem above).
In linear spaces, boundedness of a linear map and continuity are intimately related.
P1.4: Boundedness and continuity
b : E → F be a linear map.
Let E and F be two normed linear spaces and let B
Show that the following statemens are equivalent:
b is continuous at the point 0 = x ∈ E
1. B
b is bounded
2. B
b is continuous everywhere
3. B
w.r.t. to the supremum-norm.
Hint: This is a standard result for linear maps. For those who have not been
exposed to it, here a few hints: show 1 ⇒ 2 ⇒ 3 ⇒ 1. In the first step use
the fact that in a linear space, any vector ||x|| = 1 can be scaled to length
b
. Apply this to ||Bx||/||x||
that appear in the defintion of the norm. For the
2
step bounded ⇒ continuous, observe that ||B(xn − x)|| < ||B||||xn − n|| by
construction of the operator norm.
P1.5: Continuity of maps, completeness
Let E and F be two Banach spaces and let B(E, F) be the space of the continuous
linear maps from E to F.
(a) From the previous exercises we know that the linear maps themselves form
b B := supx∈E ||Bx||
b F /||x||E , B
b ∈ B is a norm
a linear space. Show that ||B||
for B. Here || · ||E , || · ||F denote the norms in the respective spaces.
(b) Show that B with || · ||B is a Banach space.
Hint: For completeness of B one needs to show that a Cauchy sequence
bn } defines a continuous linear map B
b := limn→∞ B
bn . Using complete{B
ness of F, one sees that it definitely defines a map into F. Boundedness
can be shown with a little effort, which then also implies continuity by the
previous problem.
P1.6: Resolvent
Let A ∈ A be an element of a C ∗ algebra. Then also A − z ∈ A, where we use
the notation A − z for A − z1, z ∈ C. We call (A − z)−1 : (A − z)−1 (A − z) =
(A−z)(A−z)−1 = 1 the resolvent RA (z) of A at z if it exists and (A−z)−1 ∈ A.
(a) Show that the formal series expansion
(z − A)−1 = z −1
∞
X
(A z −1 )n
(7)
n
converges for |z| > ||A||.
Hint: Use the properties of the norm on a C ∗ -algebra to show the the
series of finite sums is Cauchy.
(b) Conclude from the above that (A − z)−1 ∈ A for ||A|| < |z|.
3