q-Oscillators and a q-deformed Debye Model for Lattice Heat Capacity
81
also holds. The eigenstates |n q are orthonormal.
|n q =
(a
†
q )
n
[n] q !
|0 q
(59)
Also
a q |n q =
[n] q |n − 1 q
(60)
a
†
q |n q =
[n + 1] q |n + 1 q
(61)
q n + 1|a
†
q |n q =
[n + 1] q
(62)
q n − 1|a q |n q =
[n] q
(63)
The Hilbert space spanned by {|n q } is positive definite only if |q| ≤ 1. For larger values of |q|, states with negative squared norm arise and the probability interpretation
of quantum mechanics is lost.
1.4.2 Energy Spectrum of q-deformed Harmonic Oscillator
The Hamiltonian of the q-deformed harmonic oscillator is
H q =
p
2
q
2m
+
1
2
mω
2 x
2
q
(64)
where the q-position x q and the q-momentum p q of the oscillator are related to the
q-creation and q-annihilation operators a
†
q and a q as
x q =
2mω
(a
†
q + a q )
(65)
p q = i
mω
2
(a
†
q − a q )
(66)
where a q and a
†
q satisfy the q-oscillator algebra (38) and (39). Then the Hamiltonian
reads
H q =
1
2
ω(a
†
q a q + a q a
†
q ) =
1
2
ω([N ] q + [N + 1] q )
(67)
Here we are using the boson realisation of the q-oscillator algebra in which N q = N .
The number and energy eigenstates of the q-oscillator are then the same as those of the