Eigenvalues and eigenfunctions

Find eigenvalues and eigenfunctions

Problem:

Consider a 3-dimensional state space with orthonormal eigengbasis {|a>, |b>, |c>}.  The action of the Hamiltonian on the basis vectors is
H|a> = -iE|b> + iE|c>,  H|b> = iE|a>,  H|c> = -iE|a>, where E is a real constant with units of energy.
(a)  Construct the matrix of the Hamiltonian in the {|a>, |b>, |c>} basis.
(b)  Find the energy eigenvalues and the corresponding normalized eigenvectors.

Solution:


Properties and consequences

Problem:

Suppose |i> and |j> are eigenkets of some Hermitian operator A.  Under what conditions can we conclude that |i> + |j> is also an eigenket?  Justify your answer.

Solution:

Problem:

Consider a quantum system for which the exact Hamiltonian is H.  Assume the quantum system is of bounded spatial extend, so that it is known rigorously that the eigenstates of H, {|Ψn>}, are complete. 
(a) Show that if |Ψn> and |Ψm> are two eigenstates of H with eigenvalues En and Em
with En ≠ Em, then <Ψnm> = 0.
(b) Suppose En = Em, with n ≠ m.  Can we still have <Ψnm> = 0 ?
(c) The problem H|Ψ> = E|Ψ> is very complicated but it is suggested that we use a trial function |Ψtrial> for |Ψ> and approximate E by E = <Ψtrial|H|Ψtrial>/<Ψtrialtrial>.
Show that E > E0, where E0 is the lowest eigenvalue of H.

Solution:

Problem:

Consider two solutions to the one-dimensional time-independent Schrödinger equation with same energy E: ψ1(x) and ψ2(x).
(a)  Prove that regardless of the potential energy function U(x),

ψ2(x) ∂ψ1(x)/∂x - ψ1(x) ∂ψ2(x)/∂x = C,

where C is a constant.
(b)  By considering the boundary conditions, show that if ψ1(x) and ψ2(x) are bound state solutions, then C = 0.  From this, show that ψ2(x)  = γ ψ1(x) for some constant γ, thus proving that there are no degenerate bound state solutions to the one-dimensional time-independent Schrödinger equation.
(c)  Why does this theorem fail for continuum eigenstates (i.e. unbound states)?  Give a specific example of degenerate continuum eigenstates in one dimension.

Solution:

Problem:

A particle of mass m moves in one dimension.  It is remarked that the exact eigenfunction for the ground state is Ψ(x) = A/cosh(λx), where λ is a constant, and A is the normalization constant.  Assuming that the potential U(x) vanishes at infinity, derive the ground state energy and also U(x). 

Solution: