Nonlinear Coherent States for Anisotropic 2D Dirac Materials
321
which are also a linear combination of Landau states (4),
Ψ
f
α (x, y) =
∞
n=0
a n Ψ n (x, y),
(15)
with the coefficients of expansion a n verifying the relations
a 1 f (1) =
√
2 ˜
αa 0 , a n+1 f (n + 1)
√
n + 1 = ˜
αa n , ˜
α = α exp(−iδ).
(16)
Hence, Θ
−
f introduces a phase δ in the eigenvalue α and a n is a parameter depending
on whether f (n + 1) = 0 or f (n + 1) = 0.
3.2 Some Examples
We now consider some examples for f (N) with the aim of obtaining coherent states
as similar to the standard ones, as well as to discuss the effects of strain on the
NLCSs [14, 20]. It is worth to mention that the functions f (N) are chosen according
to the relations in Eq. (16).
Case for f (1) = 0 Choosing f (N + 1) = 1, that satisfies f (1) = 0, the
corresponding NLCSs, their probability density ρ α (x) and the mean values of the
operators S q and S 2
q in such states (see Fig. 2) are
Ψ
f
α (x, y) =
1
2 exp
| ˜
α| 2
− 1
Ψ 0 (x, y) +
∞
n=1
√
2 ˜
α n
√
n!
Ψ n (x, y)
,
(17a)
ρ α (x) =
1
2 exp
| ˜
α| 2
− 1
⎡
⎣ ψ
2
0 (x) +
∞
n=1
˜
α n
√
n!
ψ n (x)
2
+
∞
n=1
˜
α n
√
n!
ψ n−1 (x)
2
+ 2 Re
∞
n=1
˜
α n
√
n!
ψ n (x)ψ 0 (x)
⎤
⎦ ,
(17b)
q α =
˜
α + (−1) q ˜
α ∗
√
2i q (2 exp
| ˜
α| 2
− 1)
exp
| ˜
α|
2
+
∞
n=1
| ˜
α| 2n
√
(n − 1)!(n + 1)!
,
(17c)
2
q α =
1
2(2 exp
| ˜
α| 2
− 1)
1 + 4| ˜
α|
2 exp
| ˜
α|
2
+ (−1)
q ( ˜
α
2
+ ˜
α
∗2 )