Mathematical foundations

Problem:

Let Ω be the operator defined bψ Ω = |Φ><ψ| where |Φ> and |ψ> are two vectors in a vector space V.
(a)  Under what conditions is Ω Hermitian?
(b)  Calculate Ω2.  Under what conditions is Ω a projector?

Solution:

Problem:

Quantum mechanics is often conveniently formulated in terms of matrix operators.  Let {|i>} be an orthonormal basis for the state space.
(a)  Prove that for any operator A we have ∑ij|<i|A|j>|2 = Tr(AA), where Tr denotes the trace.
(b)  Derive the condition that must be satisfied for the product of two Hermitian operators to be itself a Hermitian operator.
(c)  Prove that the trace of a matrix operator is invariant under a change of representation, i.e. a change of basis.

Solution:

Problem:

Use the virial theorem for the eigenstates of atomic hydrogen, namely, <nlm|T|nlm> = -Enl, where Enl is the eigenenergy of the |nlm> bound state and T is the kinetic energy operator for an electron bound to a proton, to evaluate the sum ∑nlm|<nlm|p|n'l'm'>|2, where p is the momentum operator and the sum is over a complete set of bound and continuum states.  Take the proton mass to be infinite and give your answer in terms of En'l' and the mass me of the electron.

Solution:

Problem:

Consider the matrix representation of the operator

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(a)  Is T Hermitian?
(b)  Solve for the eigenvalues.  Are they real?
(c)  Determine the normalized eigenvectors.  Since eigenvectors are not unique to within a phase factor, arrange your eigenvectors so that the first component of each is positive and real.  Are they orthogonal?
(d)  Using the eigenvectors as columns, construct U, the inverse of the unitary matrix which diagonalizes T.  Use this to find this diagonalized version Td = UTU.
What is special about the diagonal elements?
(e)  Compare the determinant |T|, the trace Tr(T ), and eigenvalues of T to those of Td.

Solution:

Problem:

The Hamiltonian operator for a two state system is given by
H = a(|1><1|-|2><2|+|1><2|+|2><1|),
where a is a number with the dimensions of energy.  
(a)  Find the eigenvalues of H and the corresponding eigenkets |ψ1> and |ψ2> (as linear combinations of |1> and |2>).
(b)  A unitary transformation maps the {|1>, |2>} basis onto the {|ψ1>, |ψ2>} basis.  We have U|i> = |ψi>.  Write down the matrix of U and the matrix of U in the {|1>, |2>} basis.

Solution: