Operator Algebra
4.6 Problems
with
ρ1(Ip)|e1 =V1 and ρ2(Ip)|e2 =V2 (4.21) as in Eq. (4.20). The composite mapS≡T2◦T−11mapsV1isomorphically ontoV2. We now show that
S◦ρ1(a)=ρ2(a)◦S for alla∈A,
and hence thatρ1∼ρ2. Applying the right-hand side of this equation on a
|v1 ∈V1, and noting that by (4.21)|v1 =ρ1(zp)|e1for somezp∈Ip, we get
ρ2(a)◦S|v1 =ρ2(a)◦
T2◦T−11
ρ1(zp)|e1 =ρ2(a)◦T2
T−11ρ1(zp)|e1
= ρ2(a)
◦T2(zp)=ρ2(a)ρ2(zp)|e2 =ρ2(azp)|e2, while the left-hand side gives
S◦ρ1(a)|v1 =
T2◦T−11
ρ1(a)ρ1(zp)|e1 =
T2◦T−11
ρ1(azp)|e1
=T2
T−11ρ1(azp)|e1
=T2(azp)=ρ2(azp)|e2.
We have shown that if two irreducible representations of a semi-simple al- gebra have the same kernel, then they are equivalent. The converse is much
easier to prove (see Problem4.36).
(a) Show that the operator L2≡L21+L22+L23 commutes with Lk, k= 1,2,3.
(b) Show that the set{L+,L−,L3}is closed under commutation, i.e., the commutator of any two of them can be written as a linear combination of the set. Determine these commutators.
(c) WriteL2in terms ofL+,L−, andL3. 4.5 Prove the rest of Proposition4.1.8.
4.6 Show that if[[A,B],A] =0, then for every positive integerk,
#Ak,B$
=kAk−1[A,B].
Hint: First prove the relation for low values of k; then use mathematical induction.
4.7 Show that forDandTdefined in Example2.3.5,
#Dk,T$
=kDk−1 and # Tk,D$
= −kTk−1.
4.8 Evaluate the derivative ofH−1(t )in terms of the derivative ofH(t )by differentiating their product.
4.9 Show that for anyα, β∈Rand anyH∈End(V), we have eαHeβH=e(α+β)H.
4.10 Show that(U+T)(U−T)=U2−T2if and only if[U,T] =0.
4.11 Prove that ifAandBare hermitian, theni[A,B]is also hermitian.
4.12 Find the solution to the operator differential equation dU
dt =tHU(t ).
Hint: Make the change of variable y =t2 and use the result of Exam- ple4.2.3.
4.13 Verify that d dtH3=
dH dt
H2+H
dH dt
H+H2
dH dt
.
4.14 Show that ifAandBcommute, andf andgare arbitrary functions, thenf (A)andg(B)also commute.
4.15 Assuming that[[S,T],T] =0= [[S,T],S], show that
#S,exp(tT)$
=t[S,T]exp(tT).
Hint: Expand the exponential and use Problem4.6.
4.16 Prove that
exp(H1+H2+H3)=exp(H1)exp(H2)exp(H3)
×exp −1 2
[H1,H2] + [H1,H3] + [H2,H3]
provided thatH1,H2, andH3commute with all the commutators. What is the generalization toH1+H2+ · · · +Hn?
4.17 Denoting the derivative ofA(t )byA, show that˙ d
dt[A,B] = [ ˙A,B] + [A,B˙].
4.18 Prove Theorem4.3.2. Hint: Use Eq. (4.11) and Theorem2.3.7.
4.19 LetA(t )≡exp(tH)A0exp(−tH), whereHandA0are constant opera- tors. Show thatdA/dt= [H,A(t )]. What happens whenHcommutes with A(t )?
4.20 Let|f,|g ∈C(a, b)with the additional property that f (a)=g(a)=f (b)=g(b)=0.
Show that for such functions, the derivative operatorD is anti-hermitian.
The inner product is defined as usual:
f|g ≡ b
a
f∗(t )g(t ) dt.
4.21 In this problem, you will go through the steps of proving the rigorous statement of theHeisenberg uncertainty principle. Denote the expectation (average) value of an operatorAin a state|ΨbyAavg. Thus,Aavg= A = Ψ|A|Ψ. Theuncertainty(deviation from the mean) in the normalized state
|Ψof the operatorAis given by
A=6
(A−Aavg)2
=
Ψ|(A−Aavg1)2|Ψ.
(a) Show that for any twohermitianoperatorsAandB, we have Ψ|AB|Ψ2≤ Ψ|A2|ΨΨ|B2|Ψ.
Hint: Apply the Schwarz inequality to an appropriate pair of vectors.
(b) Using the above and the triangle inequality for complex numbers, show that
Ψ|[A,B]|Ψ2≤4Ψ|A2|ΨΨ|B2|Ψ.
(c) Define the operatorsA′=A−α1,B′=B−β1, whereαandβ are realnumbers. Show thatA′andB′are hermitian and[A′,B′] = [A,B].
(d) Now use all the results above to show the celebrated uncertainty rela- tion
(A)(B)≥1 2
Ψ|[A,B]|Ψ.
What does this reduce to for position operatorxand momentum oper- Heisenberg uncertainty
principle
atorpif[x,p] =i?
4.22 Show thatU=expAis unitary ifAis anti-hermitian. Furthermore, if Acommutes withA†, then expAis unitary. Hint: Use Proposition4.2.4on UU†=1andU†U=1
4.23 FindT†for each of the following linear operators.
(a) T:R2→R2given by T
x y
= x+y
x−y
.
(b) T:R3→R3given by
T
⎛
⎝ x y z
⎞
⎠=
⎛
⎝
x+2y−z 3x−y+2z
−x+2y+3z
⎞
⎠.
(c) T:R2→R2given by T
x y
=
xcosθ−ysinθ xsinθ+ycosθ
,
whereθis a real number. What isT†T?
(d) T:C2→C2given by T
α1 α2
=
α1−iα2 iα1+α2
.
(e) T:C3→C3given by
T
⎛
⎝ α1 α2 α3
⎞
⎠=
⎛
⎝
α1+iα2−2iα3
−2iα1+α2+iα3 iα1−2iα2+α3
⎞
⎠.
4.24 Show that ifPis a (hermitian) projection operator, so are1−Pand U†PUfor any unitary operatorU.
4.25 For the vector
|a = 1
√2
⎛
⎜⎜
⎝ 0 1
−1 0
⎞
⎟⎟
⎠,
(a) find the associated projection matrix,Pa.
(b) Verify thatPadoes project an arbitrary vector inC4along|a. (c) Verify directly that the matrix1−Pais also a projection operator.
4.26 Prove Proposition4.4.6
4.27 Let|a1 ≡a1=(1,1,−1)and|a2 ≡a2=(−2,1,−1).
(a) Construct (in the form of a matrix) the projection operatorsP1andP2
that project onto the directions of|a1 and|a2, respectively. Verify that they are indeed projection operators.
(b) Construct (in the form of a matrix) the operatorP=P1+P2and verify directly that it is a projection operator.
(c) LetPact on an arbitrary vector(x, y, z). What is the dot product of the resulting vector with the vectora1×a2? What can you say about Pand your conclusion in (b)?
4.28 LetP(m)=m
i=1|eiei|be a projection operator constructed out of the firstmorthonormal vectors of the basisB= {|ei}Ni=1ofV. Show that P(m)projects into the subspace spanned by the firstmvectors inB. 4.29 What is the length of the projection of the vector(3,4,−4)onto a line whose parametric equation isx=2t+1,y= −t+3,z=t−1? Hint: Find a unit vector in the direction of the line and construct its projection operator.
4.30 The parametric equation of a lineLin a coordinate system with origin Ois
x=2t+1, y=t+1, z= −2t+2.
A pointP has coordinates(3,−2,1).
(a) Using the projection operators, find the length of the projection ofOP on the lineL.
(b) Find the vector whose beginning isP and ends perpendicularly onL.
(c) From this vector calculate the distance fromP toL.
4.31 Let the operatorU:C2→C2be given by U
α1
α2
=
⎛
⎝i√α1 2−i√α2
2 α1
√2+√α22
⎞
⎠.
FindU†and test ifUis unitary.
4.32 Show that the product of two unitary operators is always unitary, but the product of two hermitian operators is hermitian if and only if they com- mute.
4.33 LetSbe an operator that is both unitary and hermitian. Show that (a) Sis involutive (i.e.,S2=1), and
(b) S=P+−P−, whereP+andP−are hermitian projection operators.
4.34 Show that if a representation ρ:A→L(V)is surjective, then it is irreducible. Hint: The operator|aa|is inL(V)for any|a ∈V.
4.35 Show thatρ(eiej)=ρ(ei)ρ(ej)fori, j=0,1,2,3 in Example4.5.3.
4.36 Show thatanytwo equivalent representations ofanyalgebra have the same kernel.
4.37 To prove Proposition4.5.5, first show thatρ(A)|vis a subspace. Then prove thatρ(A)W⊂W. For the “only if” part of an irreducible representa- tion, take|vto be in any subspace ofV.