projection operators (20 points) consider a hilbert space spanned by a basis or orthonormal state in) labeled by one discrete quantum number n. the states [n) are the eigenstates of some operator o, and n labels the various eigenvalues oln) λη1n). (for example, if 0 were the hamiltonian, the λn would be the energies. ) for each n we define the projection operator onto the state n) as thus, there is a different pn for each state n (a). demonstrate that p is hermitian, and that p2 -p, (b). what is the result of p,1 acting on an arbitrary state |切ーσrncrnlm)? explain why the name "projection operator" is justified. if there are n distinct values of n, all operators can be represented by n × n matrices. what does pa look like in this matrix representation? /1/㈣pale) is prop- (c). in general, pale) is not normalized. show that pn erly normalized (d). how are the number (ppalp) and the state palu)/v(蛚pal related to the result of making a measurement of o? relate them to the postulates of quantum mechanics now consider a system where the hilbert space is labeled by more than one quantum number: the hydrogen atom, with states [nlm) labeled by n, l, and m. the projection operator associated with just a given value of n now has a sum over all values of the other quantum numbers, in the following, consider the particular hydrogen atom wave function, 5) (1210) + v21200) | 100) = (e). consider a measurement of energy. which values of n might be observed? write down the projection operators associated with each possible result, and use them to calculate the probabilities of each outcome and the result of the collapse of the wave function. do these results agree with what you would have expected? (f). if you were instead to make a measurement of lz, how would you define the projection operator(s) you need? repeat the calculation of (e) for this case