Download Bohr-Sommerfeld quantization rule in noncommutative space T. V.

Bohr-Sommerfeld quantization rule in
noncommutative space
T. V. Fityo, I. O. Vakarchuk and V. M. Tkachuk
Chair of Theoretical Physics,
Ivan Franko National University of Lviv,
12 Drahomanov St., Lviv, Ukraine
Several independent lines of theoretical physics investigations (string theory#1, black holes#2) imply
h̄
1
∆X ≥
+ β∆P .
(1)
2 ∆P
√
It means that ∆X ≥ ∆Xmin = h̄ β. Kempf#3 proposed to modify
canonical commutation relation [x, p] = ih̄
[X, P ] = ih̄(1 + βP 2).
(2)
Heisenberg inequality for this commutation relation is equivalent to (1).
#1 Gross
D. J., Mende P. F. String theory beyond the Planck scale // Nucl. Phys. B.—
1988.— V. 303, P. 407–454
#2 Maggiore
M. A generalized uncertainty principle in quantum gravity // Phys. Lett.
B.— 1993.— V. 304, P. 65–69
#3 Kempf
A. et al. Hilbert space representation of the minimal length uncertainty relation
// Phys. Rev. D.— 1995.— V. 52, P. 1108–1118
Known approximate methods
1. Perturbation theory#4.
2. ?
Bohr-Sommerfeld quantization rule
To find spectrum of the problem
"
P2
#
+ U (X) ψ = Eψ,
2m
Quasi-coordinate representation
[X, P ] = ih̄f (P ).
d
X = x, P = P (p), p = −ih̄ ;
dx
(3)
dP (p)
= f (P ).
dp
Eigenfunction and action of P 2 on it
i
ψ(x) = exp S(x) ,
h̄
#4 Brau
ih̄
P 2ψ = P 2(S 0) − [P 2(S 0)]00S 00 + . . . ψ.
2
F. Minimal length uncertainty relation and the hydrogen atom // J. Phys. A.—
1999.— V. 32, P. 7691–7696
In linear approximation over h̄
1
i
q
ψ(x) =
C1 exp
h̄
|P f (P )|
P (p) =
q
Z x
i
pdx + C2 exp −
h̄
Z x
pdx
,
2m(E − U (x)). Matching
Zx2
pdx = πh̄(n + δ),
n = 0, 1, 2, . . .
x1
or in initial variables
−
I
XdP
= 2πh̄(n + δ).
f (P )
WKB approximation is valid if
d
P 2 h̄ dx P f (P ) .
For f (P ) = 1 + βP 2
2
∆Xmin
aλ
.
a
a — characteristic size of the system, λ = 2πh̄/P .
(4)
1D Examples
Hamiltonian
P 2 + X2
P 2, −a < X < a
α
P2 − X
P 2 − Xγ2
[X, P ] = ih̄(1 + βP 2).
Eigenvalues En
1
2
(2n + 1) + β n + n + 2 + O(β 2) #5
2
4
2
πn
πn
+
β
+ O(β 2) #6
2a
3
2a
r
2
√
1 1 − 1 + 2α
#7
− 4β
β
n+δ
—
En − EnWKB
1 β + O(β 2 )
4
O(β 2)
0
√
4
−π(n+δ)/
γ
−β e
#5 Kempf
A. et al. Hilbert space representation of the minimal length uncertainty relation
// Phys. Rev. D.— 1995.— V. 52, P. 1108–1118
#6 Detournay
S. et al. About maximally localized states in quantum mechanics // Phys.
Rev. D.— 2002.— V. 66.— 125004
#7 Fityo
T. V. et al. One dimensional Coulomb-like problem in deformed space with
minimal length // J. Phys. A.— 2006.— V. 39.— P. 2143–2149.
3D Examples
[Xi, Pj ] = ih̄((1 + βP 2)δij + β 0PiPj ).
L2 → (l + 1/2)2.
γ
.
X
H = P2 −
Its spectrum in linear approximation over β, β 0 is #8
En,l ≈ −
|
γ2
γ4
"
#
"
#
β − β 0/2
!
1
1
2
1
0
−
−
+
β
+
β
+
.
2
3
4n
8n
l + 1/2 n
l + 1/2 n
l(l + 1)(l + 1/2)
{z
WKB approximation
}
H = P 2 + X2
β 0 #9
0
2
0
2
En,l ≈ 2n + 3 + (β + β )(n + 3/2) + (β − β )(l + 1/2) +2β − .
{z
}
|
2
WKB approximation
#8 Benczik
S. et al. The hydrogen atom with minimal length // Phys. Rev. A.— 2005.—
V. 72.— 012104
#9 Chang
L. N. et al. Exact solution of the harmonic oscillator in arbitrary dimensions with
minimal length uncertainty relations // Phys. Rev. D.— 2002.— V. 65.— 125027
Let us consider more general case of deformed space
[X, P ] = ih̄f (X, P ).
It is unknown if we can express initial operators
X = X(x̂, p̂),
P = P (x̂, p̂),
where [x̂, p̂] = ih̄. Classical variables x and p always exist that
{X, P }x,p =
∂X ∂P
∂P ∂X
−
= f (X, P ).
∂x ∂p
∂x ∂p
I
pdx =
Z
dpdx =
H≤En
=
Z
H(P,X)≤En
dXdP
= 2πh̄(n + δ).
f (X, P )
(5)
Example
Let us consider harmonic oscillator eigenvalue problem
(P 2 + X 2)ψ = Eψ
in deformed space
[X, P ] = i(1 + αX 2 + βP 2).
WKB approximation (5) gives
√ 2
√ 2
√
√
√
( α + β) 2(n+δ) αβ
( α − β) −2(n+δ)√αβ α + β
En =
e
+
e
−
.
4αβ
4αβ
2αβ
Leading term is underlined. It coincides with leading term of exact solution #10.
#10 Quesne
C., Tkachuk V. M. Harmonic oscillator with nonzero minimal uncertainties in
both position and momentum in a SUSYQM framework // J. Phys. A.— 2003.—
V. 36.— 10373–10389.