Mathematisches Institut der Universit¨at M¨unchen 3.5.2018
Ubungsblatt 4 zu Mathematische Quantenmechanik II ¨
Aufgabe 1:
Consider the N-fold Hilbert space tensor product H⊗Nb of a Hilbert space H with itself. Let σ∈ SN be a permutation of{1,2, . . . , N}, and pbσ :H⊗Nb → H⊗Nb defined by
pbσ[u1⊗. . .⊗uN] :=uσ−1(1)⊗. . .⊗uσ−1(N). a) Show thatpbσ defines a unitary operator.
b) Show thatpbσpbτ =pbτ σ for any two permutationsσ, τ ∈ SN. Define the two following operators,
SbN = (N!)−1 X
σ∈SN
pbσ, and AbN = (N!)−1 X
σ∈SN
sign(σ)pbσ.
c) Show thatSbN and AbN are orthogonal projections satisfying SbNAbN = 0 if N ≥2.
d) Show thatpbσ◦AbN = sign(σ)AbN for all permutation σ∈ SN. Aufgabe 2: ShowSbNh
H⊗Nb i
and AbNh H⊗Nb i
are Hilbert spaces.
Aufgabe 3:Let H1 and H2 be Hilbert spaces. Prove that for any sequence (An)n∈N inL(H1) and forA∈L(H1), such that,kAn[ψ]−A[ψ]kn→∞−→ 0 for allψ∈ H1 then
k(An⊗idb H2)[φ]−(A⊗idb H2)[φ]k →0
for allφ∈ H1⊗Hb 2, whereA⊗idb H2 is the closure ofA⊗idH2.
