Operators in Quantum Mechanics
Summer term 2017
A. Scrinzi
J. Bucher
Problem sheet 5
Thu, June 1
(The first example is re-posted from last time - study, it uses key techniques)
P5.1: Proof the B.L.T. theorem
for bounded operators in Hilbert space, which says, that any bounded operator A with a dense domain D(A) can be uniquely extended to an continuous
operator A on all of D(A) = H with the same norm.
(a) Construct such an extension explicitly by finding a “natural” map for
those elements, that are not in the dense domain, and make sure you got
a unique definition.
(b) Crosscheck linearity for the newly defined points of the operator.
(c) What is the reason the norms must agree?
(d) Explicitly convince yourself of the uniqueness.
P5.2: Bounded multiplication operator
Let fA (x) be a measurable function and A the multiplication operator on
L2 (dx, R)
(Af )(x) = fA (x)ψ(x),
D(A) = {ψ ∈ L2 (dx, R)|Aψ ∈ L2 (dx, R)}
(1)
Show that A is bounded iff fA is essentially bounded, i.e. there is a real number
a such that Ia = {x| |fA (x)| > a} has measure 0.
(a) Show that ||A|| ≤ ||fA ||∞ , remember
||f ||∞ = inf{a|µ(Ia ) = 0}
(2)
(which adjusts the supremum-norm for the case of only essentially bounded
functions).
(b) Show ||A|| <
6 ||fA ||∞ .
Hint: By contradiction: if we had ||A|| = c, but c < ||fA ||∞ we could
construct a test function φ ∈ H such that c < ||Aφ||that contradicts this.
P5.3: Hermitian operators and quadratic forms
Show that a densely defined operator A is hermitian if and only if its quadratic
form qA is real-valued.
Hint: Evaluate the quadratic form for a vector ψ + iφ
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P5.4: Closed operator - counter-example
Show that the operator (Aφ)(x) = φ(0) on D(A) = {φ ∈ L2 (dx, [−1, 1]), φ continuous}
is not closed.
Hint: Construct a series such that (φn , Aφn ) = (φn , 1), where φn → φ = 0 and
compare the limit of the pairs with the pair at φ, Aφ.
P5.5: Normal operators are closed
Show that.
Hint: Use the lemma that A∗ is closed (will be proven in the lecture). The
graph norms can be related using the definition of normality.
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