Zero energy correction method for non-Hermitian Harmonic oscillator with simultaneous transformation of co-ordinate and momentum:Wave function analysis under Iso-spectral condition

0.08 MB·English

by  Biswanath Rath

Zero energy correction method for non-Hermitian Harmonic oscillator
with simultaneous transformation of co-ordinate and momentum:Wave
6 function analysis under Iso-spectral condition.
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Biswanath Rath and P. Mallick

r
p
Department of Physics, North Orissa University, Takatpur, Baripada -757003,
A
Odisha, INDIA
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]
h
p
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t
n We present acomplete analysis onenergy andwavefunction of Harmonic oscillator
a
u with simultaneous non-hermitian transformation of co-ordinate (x (x+iλp) ) and
q → √(1+βλ)
[ momentum (p (p+iβx) ) using perturbation theory under iso-spectral condition.

→ √(1+βλ)
2
v Further we notice that two different frequency of oscillation (w1,w2)correspond to
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6 same energy eigenvalue ,which can also be verified using Lie algebraic approach [
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6 Zhang et.al J.Math.Phys 56 ,072103 (2015) ]. Interestingly wave function analysis
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1. usingsimilaritytransformation[F.M.Fernandez, Int. J.Theo. Phys. (2015)(inPress)]
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5 refers to a very special case.
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: PACS(2008) : 03.65.Db
v
i
X
r Key words- Non-Hermitian Harmonic oscillator, Perturbation theory, Wavefunc-
a
tion
I.INTRODUCTION
In quantum mechanics, Harmonic oscillator plays a major role in understanding
limitations of various approximation methods such as variational method, W.K.B.
method, perturbation method etc. The basic advantage of this oscillator is that
its energy levels and eigenfunctions are exactly known [1]. However, when the co-
ordinate and momentum simultaneously under goes a non-Hermitian transformation
[2-4], its energy eigenvalue can be iso-spectral to original Harmonic oscillator. Un-
der iso-spectral condition, the wavefunctions of the transformed Hamiltonian differ
drastically from that of the original Hamiltonian. However, a complete picture on
wavefunction is still a missing subject. Further, no such explicit calculations on wave-
functionsareavailableatpresent. Hence theaimofthispaperistopresent acomplete
picture on wavefunction and energy under iso-spectral condition using perturbation
theory.
II.Energylevelsand Wavefunction ofSimple Harmonic Oscillator (SHO)
Let’s consider the Hamiltonian of SHO [1]
p2 x2
H = + (1)
HO
2 2
whose exact energy eigenvalues is
1
E = (n+ ) (2)
n
2
and corresponding wavefunction is
ψn = s√π12nn!Hn(x)e−x22 (3)
where H (x) is the Hermite polynomeal.
n
III. SHO under non-Hermitian transfromation of co-ordinate (x) and
momentum (p)
Let us consider the non-Hermitian transformation of x and p as [2-4]
x+iλp
x (4)
→ (1+βλ)
q
and
p+iβx
p (5)
→ (1+βλ)
In this transformation we noticed thqat the transformed co-ordinate and momen-
tum preserve the commutation relation i.e.
[x,p] = i (6)
1
Now the new Hamiltonian with transformed x and p becomes non-Hermitian in
nature and is
(p+iβx)2 (x+iλp)2
H = + (7)
2(1+λβ) 2(1+λβ)
IV. Second Quantization and Hamiltonian
InordertosolvetheaboveHamiltonian(Eqn. (7)), weusethesecondquantization
formalism as
(a+a+)
x = (8)
√2ω
and
ω
p = i (a+ a) (9)
2 −
r
where the creation operator, a+ and annhilation operator a satisfy the commuta-
tion relation
[a,a+] = 1 (10)
and ω is an unknown parameter. The Hamiltonian in Eqn. (7) can be written as
H = H +H (11)
D N
where
(1 β2) (2a+a+1)
H = [(1 λ2)ω + − ] (12)
D
− ω 4(1+λβ)
and
a2 (a+)2
H = U +V (13)
N
4(1+λβ) 4(1+λβ)
(1 β2)
V = [ ω(1 λ2)+ − 2(λ+β)] (14)
− − ω −
(1 β2)
U = [ ω(1 λ2)+ − +2(λ+β)] (15)
− − ω
V.(a). Zero Energy Correction Method (Case Study for U=0)
Now we solve the the eigenvalue relation:
2
HΨ (x) = ǫ Ψ (x) (16)
n n n
using perturbation theory as follows. Here we express
k
ǫ = ǫ(0) + ǫ(m) (17)
n n n
m=1
X
The zeroth order energy ǫ(0) satisfies the the following eigenvalue relation
n
H ψ >= H n >= ǫ(0) n > (18)
D| n D| n |
where ψ(0) isthezeroth order wave function andǫ(m) is themth order perturbation
n n
correction.
(2n+1) (1 β2)
ǫ(0) = [(1 λ2)ω + − ] (19)
n 4(1+λβ) − ω
and
k
ǫ(m) = ǫ(1) +ǫ(2) +ǫ(3) +....... (20)
n n n n
m=1
X
The energy correction terms will give zero contribution if the parameter is deter-
mined from non-diagonal terms of H [2]
N
Let the coefficient of a2 is zero [5] i.e.
(1 β2)
U = [ ω(1 λ2)+ − +2(λ+β)] = 0 (21)
− − ω
which leads to( considering positive sign)
(1+β)
ω = ω = (22)
1 (1 λ)
−
In this case,
1
ǫ(0) = (n+ ) (23)
n 2
Now the perturbation correction term is
(a+)2 (λ+β)
H = V = (a+)2 (24)
N
4(1+λβ) −1+λβ
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In this case one will notice that
n(n 1)
< n H n 2 >= V − (25)
| N| − q4(1+λβ)
< n 2 H n >= 0 (26)
N
− | |
Hence it is easy to note that all orders of energy corrections will be zero. Let us
consider explicitly corrections up to third order using standard perturbation series
given in literature[1,2,6-8], which can be written as
ǫ(1) =< ψ H ψ >= 0 (27)
n n N n
| |
< ψ H ψ >< ψ H ψ > < ψ H ψ >< ψ H ψ >
ǫ(2) = n| N| k k| N| n = n| N| n+2 n+2| N| n = 0
n (ǫ(0) ǫ(0)) (ǫ(0) ǫ(0) )
kX6=n n − k n − n+2
(28)
< ψ H ψ >< ψ H ψ >< ψ H ψ >
ǫ(3) = n| N| p p| N| q q| N| n = 0 (29)
n p,q (ǫ(n0) ǫ(p0))(ǫ(n0) ǫ(q0))
X − −
or
< ψ H ψ >< ψ H ψ >< ψ H ψ >
ǫ(3) = n| N| n+2 n+2| N| n+4 n+4| N| n = 0 (30)
n (ǫ(0) ǫ(0) )(ǫ(0) ǫ(0) )
n − n+2 n − n+4
Here second order correction is zero due to < ψ H ψ >= δ and third
n N n+2 n,n+4
| |
order correction is zero due to < ψ H ψ >= δ . Similarly one can notice
n+4 N n n+4,n+2
| |
all correction terms ǫ(m) will be zero. Hence
n
1
ǫ = ǫ(0) = E(0) = (n+ ) (31)
n n n 2
|ψn >= (√√π2ωn1n!)21Hn(√ω1x)e−ω1x22 (32)
with
< ψ ψ >= 1 (33)
n n
|
4
Form Eq.(2) and Eq.(23), one can see that the total energy of the Harmonic
oscillator and the Harmonic oscillator with non-hermitian transformation remains
the same.
V.(b). Coresponding Wavefunction using Perturbation Theory
Here we find the wavefunction as
(n+2)! (n+4)! (n+6)!
Ψ(k) = ψ > +f ψ > +(f )2 ψ > +(f )3 ψ > +......
n | n λ,βq2√n! | n+2 λ,β q8√n! | n+4 λ,β q48√n! | n+6
(34)
where f = (λ+β) . In its compact form one can write,
λ,β (1+λβ)
(λ+β) (n+2k)!
Ψ(k) = [ ]k ψ > (35)
n (1+λβ) s n! | n+2k ω1
k=0
X
The normalization condition here can be written as[6]
< ψ Ψ(k) >= 1 (36)
n| n
So also the eigenvalue relation
1
< ψ H Ψ(k) >= E = (n+ ) (37)
n| | n n 2
VI.(a). Zero Energy Correction Method (Case Study for V=0)
Let the coefficient of (a+)2 is zero [2] i.e.
(1 β2)
V = [ ω(1 λ2)+ − 2(λ+β)] = 0 (38)
− − ω −
which leads to
(1 β)
ω = ω = − (39)
2 (1+λ)
In this case, we would like to state that ω calculated using similarity transformation
[3] remains the same as ω . Now the perturbation term becomes
2
a2 (λ+β)
H = U = a2 (40)
N
4(1+λβ) 1+λβ
In this case one will notice that
[(n+1)(n+2)]
< φ H φ >= U (41)
n| N| n+2 q 4(1+λβ)
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< φ H φ >= 0 (42)
n+2 N n
| |
Hence we have
ǫ(1) =< φ H φ >= 0 (43)
n n N n
| |
< φ H φ >< φ H φ > < φ H φ >< φ H φ >
ǫ(2) = n| N| k k| N| n = n| N| n−2 n−2| N| n = 0
n (ǫ(0) ǫ(0)) (ǫ(0) ǫ(0) )
kX6=n n − k n − n−2
(44)
< φ H φ >< φ H φ >< φ H φ >
ǫ(3) = n| N| p p| N| q q| N| n = 0 (45)
n (ǫ(0) ǫ(0))(ǫ(0) ǫ(0))
p,q n p n q
X − −
or
< φ H φ >< φ H φ >< φ H φ >
ǫ(3) = n| N| n−2 n−2| N| n−4 n−4| N| n = 0 (46)
n (ǫ(0) ǫ(0) )(ǫ(0) ǫ(0) )
n − n−2 n − n−4
Here second order correction is zero due to < φ H φ >= δ and third
n N n−2 n,n−2
| |
order correction is zero due to < φ H φ >= δ . Similarly one can notice
n−4 N n n−2,n−4
| |
all correction terms ǫ(m) will be zero. Hence
n
1
ǫ = ǫ(0) = E(0) = (n+ ) (47)
n n n 2
which is the same as the energy level of harmonic oscillator as given in Eq(2) and
|φn >= (√√π2ωn2n!)21Hn(√ω2x)e−ω2x22 (48)
VI.(b). Coresponding Wavefunction using Perturbation Theory
Here we consider the wavefunction as
√n! √n! √n!
Φ(k) = φ > +f φ > +(f )2 φ > +(f )3 φ > +.....
n | n λ,β2 (n 2)!| n−2 λ,β 8 (n 4)!| n−4 λ,β 48 (n 6)!| n−6
− − −
q q q (49)
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In its compact form, one can write
(λ+β) √n!
Φ(k) = ( )k φ > (50)
n (1+λβ) (n 2k)!2kk!| n−2k ω2
k=0
X −
q
Here we notice that for x i.e.
→ ∞
φ (x ) 0 (51)
n
→ ∞ →
and
Φ(k)(x ) 0 (52)
n
→ ∞ →
In this case, the normalization condition can be written as[6]
< φ Φ(k) >= 1 (53)
n n
|
So also eigenvalue relation
1
< φ H Φ(k) >= E = (n+ ) (54)
n| | n n 2
VII. Comparision with Similarity Transformation using Lie-algebra [4].
In the above we notice that two different frequency( w ,w )corresponds to same
1 2
energy eigenvalue .Now we compare our results with that of Zhang et.al [4] using
Lie-algebra as follows.Previous authors consider the Hamiltonian
1
H = s (a+a+ )+s (a+)2 +s a2 +s a+ +s a (55)
0 1 2 3 4
2
having energy eigenvalue
1 s s2 +s s2 s s s
ǫ = s2 4s s (n+ )+ 2 3 1 4 − 0 3 4 (56)
n 0 − 1 2 2 s2 4s s s
q 0 − 0 1 2
Case-I w = w = 1+β.
1 1−λ
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In this case we have the Hamiltonian
1 (λ+β)
H = (a+a+ ) (a+)2 (57)
2 − 2(1+λβ)
Now comparing we get
s = 1 (58)
0
(λ+β)
s = (59)
1 −2(1+λβ)
s = 0 (60)
2
s = 0 (61)
3
s = 0 (62)
4
Hence the
1
ǫ = (n+ ) (63)
n
2
which is same as given in Eq(31).
Case-II w = w = 1−β.
2 1+λ
In this case we have the Hamiltonian
1 (λ+β)
H = (a+a+ )+ a2 (64)
2 2(1+λβ)
Now comparing we get
s = 1 (65)
0
(λ+β)
s = (66)
2 2(1+λβ)
s = 0 (67)
1
s = 0 (68)
3
s = 0 (69)
4
Hence the
1
ǫ = (n+ ) (70)
n
2
8
which is same as given in Eq(46). Now we see results of Lie-algebra matches with
that of perturbation theory on energy level calculation.
VIII. Comparision with Similarity Transformation [3]
It is worth to mention that Fernandez[3] has calculated groundstate wave func-
tion of this oscillator using similarity transformation.The explict expression for wave
function [3] is
|φ0 >= (ωπ2)41e−ω2x22 (71)
It is easy check that our result in Eq. (45) with n = 0 remains the same as that of
Fermamdez [3]. However, we do not have literature for further comparison.
IX. Conclusion In this paper, we suggest a simpler procedure for calculating
energy levels and wave function of the non-Hermitian harmonic oscillator under si-
multaneous transfromation of co-ordinate and momentum using perturbation theory
. Further energy eigenvalue calculated using perturbation theory matches nicely with
that of Lie algebric method [4.At this point it is necessary to mention that the ground
state wave function calculated by Fernandez [3] refers to zeroth order wave function
as reflected in VI(b).However we present a complete picture on wave function. In all
the cases, we show that the energy levels remain the same as that of simple Harmonic
oscillator. Here we would like to state that if the parameter ω is determined using
variational principle [9] i.e. dǫ(n0) = 0, then one has to calculate all orders of perturba-
dω
tioncorrectionsi.e. k ǫ(m) because < n H n+2 >= 0so also < n+2 H n >= 0
m=2 n | N| 6 | N| 6
and it will be a cubPersome process. However, if the parameter is determined either
using < n H n+2 > or < n+2 H n > as done in above procedure then the energy
N N
| | | |
levels and wave ftnction are obtained easily. Further we notice that
1
< φ H φ >=< ψ H ψ >=< φ H Φ(k) >=< ψ H Ψ(k) >= (n+ ) (72)
n| D| n n| D| n n| | n n| | n 2
This directly reflects that non-commuting operators i.e. [H,H ] = 0 corresponds
D
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