Consider the second derivative operator on the space of functions which vanish at x = 0 and x = L:

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D2|f) = |d²f/dx²) (a) Find the matrix elements of the second derivative operator D2 in the x) basis. (b) Explicitly check that (r|D2|r') = ((r'|D2|x))*. This condition, together with our assumption about the boundary conditions, is sufficient to prove that D2 is a Hermitian 4 operator. (c) Compute the commutator of D2 with the X operator: [D2, X]. Recall that the action of the X operator in the r) basis is Xr) rr), i.e. r) is the eigenbasis of the operator X. Hint: act the commutator on a vector |f). = (d) Find the eigenvalues and eigenvectors of D2. Hint: you should use the assumed boundary conditions. D2|f) = |d²f/dx²) (a) Find the matrix elements of the second derivative operator D2 in the x) basis. (b) Explicitly check that (r|D2|r') = ((r'|D2|x))*. This condition, together with our assumption about the boundary conditions, is sufficient to prove that D2 is a Hermitian 4 operator. (c) Compute the commutator of D2 with the X operator: [D2, X]. Recall that the action of the X operator in the r) basis is Xr) rr), i.e. r) is the eigenbasis of the operator X. Hint: act the commutator on a vector |f). = (d) Find the eigenvalues and eigenvectors of D2. Hint: you should use the assumed boundary conditions.

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