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Assume that: is identical to the identity operator called completeness rela- tion shown below, • Ja) is an arbitrary ket from the vector space V • operators A and B are linear operators acting on vectors N > T4:XA;l = î from V 1=1 • the set of all eigenvectors of  is given by |41), |A2), ... |AN), and form an orthonormal basis d; is eigenvalue of  that corresponds to the ket |A;) A¡; and Bij are matrix elements of the matrix represen- Using the completeness relation above and given any opera- tor B, show that N BijlA;)(A;| B = tations of the operators  and B a) Show that c) If Ê =  show that the equation above becomes N la) = > 14:XA¡|a) b) The result of above equation implies that the operator N

Assume that: is identical to the identity operator called completeness rela- tion shown below, • Ja) is an arbitrary ket from the vector space V • operators A and B are linear operators acting on vectors N > T4:XA;l = î from V 1=1 • the set of all eigenvectors of  is given by |41), |A2), ... |AN), and form an orthonormal basis d; is eigenvalue of  that corresponds to the ket |A;) A¡; and Bij are matrix elements of the matrix represen- Using the completeness relation above and given any opera- tor B, show that N BijlA;)(A;| B = tations of the operators  and B a) Show that c) If Ê =  show that the equation above becomes N la) = > 14:XA¡|a) b) The result of above equation implies that the operator N

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Assume that: is identical to the identity operator called completeness rela- tion shown below, |a) is an arbitrary ket from the vector space V • operators à and B are linear operators acting on vectors from V N > 14:XA¡| = Î • the set of all eigenvectors of  is given by |41), |A2), ... |AN), and form an orthonormal basis A; is eigenvalue of  that corresponds to the ket |A;) Aij and Bij are matrix elements of the matrix represen- Using the completeness relation above and given any opera- tor B, show that N N tations of the operators A and B B = >> Bijl4:{A;| a) Show that c) If B =  show that the equation above becomes N = <미 |A;X{A;|a) A = > 세4:XA;l b) The result of above equation implies that the operator N |A:) (A;| l=1
Transcribed Image Text:Assume that: is identical to the identity operator called completeness rela- tion shown below, |a) is an arbitrary ket from the vector space V • operators à and B are linear operators acting on vectors from V N > 14:XA¡| = Î • the set of all eigenvectors of  is given by |41), |A2), ... |AN), and form an orthonormal basis A; is eigenvalue of  that corresponds to the ket |A;) Aij and Bij are matrix elements of the matrix represen- Using the completeness relation above and given any opera- tor B, show that N N tations of the operators A and B B = >> Bijl4:{A;| a) Show that c) If B =  show that the equation above becomes N = <미 |A;X{A;|a) A = > 세4:XA;l b) The result of above equation implies that the operator N |A:) (A;| l=1
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