Question

Problem 16. The Hamiltonian of a particle can be expressed in the form $H = epsilon_1 a^+a + epsilon_2 (a + a^+)$, $(a, a^+) = 1$, where $epsilon_1$ and $epsilon_2$ are constants. (a) Find the energies of the stationary states. (You are not required to find the corresponding state functions.) (b) The same except that the commutator of $a$ and $a^+$ is $(a, a^+) = q^2$, where $q$ is a pure number. Hint: Keeping the harmonic oscillator in mind, introduce new annihilation and creation operators $b$ and $b^+$ by writing $b = alpha a + eta$, $b^+ = alpha a^+ + eta$ and choose the constants $alpha$ and $eta$ wisely.

          Problem 16. The Hamiltonian of a particle can be expressed in the form
$H = epsilon_1 a^+a + epsilon_2 (a + a^+)$, $(a, a^+) = 1$,
where $epsilon_1$ and $epsilon_2$ are constants.
(a) Find the energies of the stationary states. (You are not required
to find the corresponding state functions.)
(b) The same except that the commutator of $a$ and $a^+$ is
$(a, a^+) = q^2$,
where $q$ is a pure number.
Hint: Keeping the harmonic oscillator in mind, introduce new annihilation and creation operators $b$ and $b^+$ by writing
$b = alpha a + eta$,
$b^+ = alpha a^+ + eta$
and choose the constants $alpha$ and $eta$ wisely.

Problem 16. The Hamiltonian of a particle can be expressed in the form
H = epsilon1 a^+a + epsilon2 (a + a^+), (a, a^+) = 1,
where epsilon1 and epsilon2 are constants.
(a) Find the energies of the stationary states. (You are not required
to find the corresponding state functions.)
(b) The same except that the commutator of a and a^+ is
(a, a^+) = q^2,
where q is a pure number.
Hint: Keeping the harmonic oscillator in mind, introduce new annihilation and creation operators b and b^+ by writing
b = alpha a + eta,
b^+ = alpha a^+ + eta
and choose the constants alpha and eta wisely.

Added by Nicholas J.

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