- Consider the two-state system investigated in Section 7.2.
Show that the most general expressions for the perturbed energy eigenvalues and
eigenstates are

and

respectively. Here, . You may assume that , . - Consider the two-state system investigated in Section 7.2. Show that if
the unperturbed energy eigenstates are also eigenstates of the perturbing Hamiltonian then

and

to all orders in the perturbation expansion. - Consider the two-state system investigated in Section 7.2. Show that if
the unperturbed energy eigenstates are degenerate, so that
then
the most general expressions for the perturbed energy eigenvalues and
eigenstates are
- Calculate the lowest-order energy-shift in the ground state of the one-dimensional harmonic
oscillator when the perturbation
- Let
denote a properly normalized eigenstate of the hydrogen atom corresponding to the
conventional quantum numbers
,
, and
. Show that the only non-zero matrix elements of
the operator
between the various
states take the values

- Calculate the energy-shifts and perturbed eigenstates associated with the linear Stark effect in the
state of a hydrogen atom.
- Suppose that the Hamiltonian,
, for a particular quantum system, is a function of some parameter,
.
Let
and
be the eigenvalues and eigenkets of
. Prove
the
*Feynman-Hellmann theorem*[45] - According to Section 4.6, the Hamiltonian for the radial wavefunction of an energy eigenstate of the hydrogen atom corresponding to the conventional quantum numbers
,
, and
is written
- Demonstrate that
- The relativistic definition of the kinetic energy,
, of a particle of total energy
and rest mass
is
- The Hamiltonian of the valence electron in a hydrogen-like atom can be written
- According to Dirac's relativistic electron theory, there is an additional relativistic correction to the Hamiltonian
of a valence electron in a hydrogen-like atom that takes the form
*Darwin term*[24]. Treating as a small perturbation, deduce that, for the special case of a hydrogen atom, it causes an energy-shift in the energy eigenstate, characterized by the standard quantum numbers , , , of - Consider an energy eigenstate of the hydrogen atom characterized by the standard quantum numbers
,
, and
.
Show that the energy-shift due to spin-orbit coupling
takes the form
- Demonstrate that if the energy-shifts due to the electron's relativistic mass increase, the Darwin term, and
spin-orbit coupling, calculated in the previous three exercises for
an energy eigenstate of the hydrogen atom, characterized by the standard quantum numbers
,
, and
, are added together then the
net
*fine structure energy-shift*can be writtenShow that fine structure causes the energy of the states of a hydrogen atom to exceed those of the and states by .

- The linear Stark effect exhibited by the
states of a hydrogen atom depends crucially on the supposed
degeneracy of these states. However, this degeneracy is lifted by fine structure. Consequently, the expressions
for the Stark energy shifts derived in Section 7.6 are only valid when the energy splitting predicted by
the linear Stark effect greatly exceeds that caused by fine structure. Deduce that this is the case when
- Demonstrate that

- Taking electron spin into account, the
energy eigenstates of the hydrogen atom in the presence of an external
electric field are

(See Section 7.6.) Here, is an unperturbed energy eigenket corresponding to the standard quantum numbers , , and . Moreover, the energies of these states are

where is the external electric field-strength, and the unperturbed energy eigenvalue corresponding to . The fine structure Hamiltonian can be written

- Consider the
energy eigenstates of the hydrogen atom in the Paschen-Back limit. (See Section 7.8.) These
states are conveniently labeled using the standard quantum numbers
,
,
, and
.
Treating the fine structure Hamiltonian,
, defined in the previous exercise, as a small
perturbation, show that the perturbed energies of the various states are

Here, is the Bohr magnetron, and the external magnetic field-strength. - Justify Equation (7.143).