Question

DP, Wary i wię In ozeigenvector basis, we have oy = with eigenvalues = +1 and eigenvectors = 1 Fil G). Find the transformation matrix to transform from ozeigenvector basis to oyeigenvector basis. Then use it to find the representation of the Oy eigenvectors and oy in the new basis. Discuss is this expected?

Answer 1

When the basis is changed to that of sigma-y, the e-vectors of sigma-y become
to) and (o
. This is as expected because these e-vectors are a representation of the main basis vector and so the e-vectors of sigma-y must be  
to) and (o
.

In the basis of G2 eigenvectors, The e-vectors of sy in G2 basis too are, 1y+> = ( A lyos () The e-vector of ez a are, 1+> = () A 1->= .. The oferation of basis hansformation is done by a unitary mathix (0) called the hans formation mahix. It is given by, o g u= <y+l+> <4+1->) ky-lt> <y-1-3/ Now, <y+/+> = 20=1)(6.) = , 51+ (-i)eo] <y+1-) = *(1-19 0 ** (0-1) =

<y=1+) (i 1768]. - e co-ito] <y-t-> wa (1) *). - str. 40 +1) - įpi -171 همسرور و مة || ا - | J2 - matix. Now, to e-veitars of sy in the new basis are given as follows: ly to your = û got his A lyu hoa ôl you you of the new e-vitas of sy in og rasis be 17+> * IX-> 18+) = û ly+> />= Úly->

14+> = (1) 4 (7-5) * = <+k1 (9) = (8) (990:-) = <it 20%) = () : In the new besis, the e-vectors of sy one (O) (O) Now, the of hatar hansformation under change of basis is given by, - Â = û Â So, 'sy must transform into sy oor os, Sy oth (-1)(2) (:-) 26-27 ( ;) ** (-:-1); -; ) =( ) 560) ut

is the new form og og 5- 6:9) This is as expected because now ___& nor- nain. by is same - स्त्र