Qucstion 1 The set of functions Po = 1, P = I, 2 = * ; are orthogonal
3
with nit weight on the range -1 < < +I. Anothcr sct of functions that are also
ort hogonal with nit wcight and span tlhe same spare are Fo = c', F = , F =5r-3.
a. IVritc the normalizcd forms ofP and E for i == 1,2.3. 1|
b. Find the unitary matrix Uthat transforms from thc normalized P, basis to the nor
Imalized E basis. 3
C. Find the nitary matrix V that trans•orns fromthe normalizcd E, basis to the nor
malizcd P basis. [1])
d. Expand f(r) =5r? -3r + I in terns of the nomnalizcd versions of both bases, and
verify tlhat the transformnation matrix U coverts the P-basis cxpansion of f() into its
F-basis cxpansion. (2+2-+1]
Qucstion 2Thc ON set of functions (over a suitablc rangc of z, y, ) given by
B ={lo) = Crc-. Jo) = Cye-r, Jos) = Czc-r}}spans
spans aa threc
threc dimemsional voctor
space ol functions. V.
a. ind the matrix reprcscnt ation, M, of the operator Lr = -i(y - ) i n the B;
basis. (3)
b. IVhat are the cigenvalucs of M and the corresponding cigenvectors? (2
C. Writc the cxplicit form of M' which is thc matrix M writton in a dilcrent ON basis,
B = {lo)lo,)lo3)} using the unitary givcn below (The matrix elements of U are
Uij i= (0,, o;)) (2):
0
æ= 1//2 -i/V2 (1)
0 1/V2 i/2
d. Calculate the cxpcctation valucs of the matrix MM w.rt. lo), o,),lo). 3|
c. Calculatc the expectation valucs of the matrix Mw.r.t. lo),).l). (3
Qucstion 3 A, B.C arc finitc-dimensional Hormitian matrices such that C=A+B
and A, B) = 0. The ON cigonvectors of Aare u)u2).... u,).
a. Writc downthc mitary transformation to digonalize Cin terms of u;),i= 1,2,.... n.
b. Consider tlhe natrix D= A' - AB + A BA + B + B, cvaluatc the conmntator
|D,C1. (3|
c. Lct. A and B be positive-dcfnite matrices and the largcst cigenvalue of Abe Aa
and that for B be A 9osuch that A + Ap = 1with the same
corresponling cige
vcctor. Allother cigenval1es of A and B arc less thap 1/2. Find the cxpcct ation value
(uC)when ) is a unit-nornalizcd vector 3
3
with nit weight on the range -1 < < +I. Anothcr sct of functions that are also
ort hogonal with nit wcight and span tlhe same spare are Fo = c', F = , F =5r-3.
a. IVritc the normalizcd forms ofP and E for i == 1,2.3. 1|
b. Find the unitary matrix Uthat transforms from thc normalized P, basis to the nor
Imalized E basis. 3
C. Find the nitary matrix V that trans•orns fromthe normalizcd E, basis to the nor
malizcd P basis. [1])
d. Expand f(r) =5r? -3r + I in terns of the nomnalizcd versions of both bases, and
verify tlhat the transformnation matrix U coverts the P-basis cxpansion of f() into its
F-basis cxpansion. (2+2-+1]
Qucstion 2Thc ON set of functions (over a suitablc rangc of z, y, ) given by
B ={lo) = Crc-. Jo) = Cye-r, Jos) = Czc-r}}spans
spans aa threc
threc dimemsional voctor
space ol functions. V.
a. ind the matrix reprcscnt ation, M, of the operator Lr = -i(y - ) i n the B;
basis. (3)
b. IVhat are the cigenvalucs of M and the corresponding cigenvectors? (2
C. Writc the cxplicit form of M' which is thc matrix M writton in a dilcrent ON basis,
B = {lo)lo,)lo3)} using the unitary givcn below (The matrix elements of U are
Uij i= (0,, o;)) (2):
0
æ= 1//2 -i/V2 (1)
0 1/V2 i/2
d. Calculate the cxpcctation valucs of the matrix MM w.rt. lo), o,),lo). 3|
c. Calculatc the expectation valucs of the matrix Mw.r.t. lo),).l). (3
Qucstion 3 A, B.C arc finitc-dimensional Hormitian matrices such that C=A+B
and A, B) = 0. The ON cigonvectors of Aare u)u2).... u,).
a. Writc downthc mitary transformation to digonalize Cin terms of u;),i= 1,2,.... n.
b. Consider tlhe natrix D= A' - AB + A BA + B + B, cvaluatc the conmntator
|D,C1. (3|
c. Lct. A and B be positive-dcfnite matrices and the largcst cigenvalue of Abe Aa
and that for B be A 9osuch that A + Ap = 1with the same
corresponling cige
vcctor. Allother cigenval1es of A and B arc less thap 1/2. Find the cxpcct ation value
(uC)when ) is a unit-nornalizcd vector 3
