ive been revising all holidays, unfortunately ive just realised ive been finding eigenvalues using the ensemble when i may have to change basis for the exam. looks at homework questions, workshop questions... nothing!! anyway an example problem: rewrite |PSI> = a|0> + b|1> in the basis |phi1> = a|0> + b|1> |phi2> = b*|0> + a*|1> my attempt im assuming that to change basis you need to use the identity operator with the new basis, in this case I = |phi1><phi1| + |phi2><phi2| so I|PSI> = |PSI> which looks a bit strange to me. ANOTHER example write: |PSI> = 1/SQRT2 (|0> + |1>) in the basis: |3> = SQRT(1/3) |0> + SQRT(2/3) |1> |4> = SQRT(2/3) |0> + SQRT(1/3) |1> my attempt I = |3><3| + |4><4| I|PSI> = |PSI> Please could someone confirm my method is correct before i make an *** of it in the exam. any help once again appreciated!