bohmian trajectories from coherent states, istanbul 3 july 2013
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Bohmian quantum trajectories from coherent states
Sanjib Dey
Pseudo-Hermitian Hamiltonians in Quantum Physics XIIKoc University, Istanbul 2-6 July 2013
Based on arXiv:1305.4619, with Prof. Andreas Fring (City University)
Sanjib Dey (City University London) Bohmian trajectories from coherent states 1 / 22
Classical mechanics in complex plane
Two possibilities to obtain complex Hamiltonian :1 p2 + x2 + ix3
2 p2 + x2 ⇐ p = pr + ipi, x = xr + ixi
Direct connection : complex Hamiltonians⇐⇒ P T symmetry.
Solve canonical equations of motion :
xr =12
(∂Hr
∂pr+
∂Hi
∂pi
), xi =
12
(∂Hi
∂pr− ∂Hr
∂pi
),
pr = −12
(∂Hr
∂xr+
∂Hi
∂xi
), pi =
12
(∂Hr
∂xi− ∂Hi
∂xr
)
Sanjib Dey (City University London) Bohmian trajectories from coherent states 2 / 22
Example : Poschl-Teller potential
H =p2
2m+
V0
2
[λ(λ−1)
cos2(x/2a)+
κ(κ−1)sin2(x/2a)
]− V0
2(λ+κ)2 for 0≤ x≤ aπ
Complexify : x⇒ xr + ixi, p⇒ pr + ipi
Real and imaginary part
Hr =p2
r −p2i
2m− V0
2(λ+κ)2
+V0
[(λ2−λ)
[cosh
( xia
)cos( xr
a
)+1][
cosh( xi
a
)+ cos
( xra
)]2
−(κ2−κ)
[cosh
( xia
)cos( xr
a
)−1][
cos( xr
a
)− cosh
( xia
)]2
]
Hi =pipr
m+V0
[(λ2−λ)sinh
( xia
)sin( xr
a
)[cosh
( xia
)+ cos
( xra
)]2 − (κ2−κ)sinh( xi
a
)sin( xr
a
)[cos( xr
a
)− cosh
( xia
)]2]
P T : xr→−xr, xi→ xi, pr→ pr, pi→−pi, i→−iSanjib Dey (City University London) Bohmian trajectories from coherent states 3 / 22
Classical trajectory : Poschl-Teller potential
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0
00
500
500500
(a)xi
xr
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- 2
- 1
0
1
2
3
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00
0
0 0
0
0
500
500
500
500
(b)xi
xr
- 6 - 4 - 2 0 2 4 6
- 3
- 2
- 1
0
1
2
3
Blue : x0 = 4.5, p0 = 41.8376i, E =−31.7564Black : x0 = 3+1.5i, p0 =−30.1922+0.385121i, E =−6.55991−13.5182i
Sanjib Dey (City University London) Bohmian trajectories from coherent states 4 / 22
Can we explain the motion of the quantum particle in thesame way??
Sanjib Dey (City University London) Bohmian trajectories from coherent states 5 / 22
Bohmian mechanics
Quantum theory⇒ Solution of Schrodinger equation : ψ⇒ Probabilitiesof actual result.
Is it possible to find some other interpretation?
David Bohm(1952)⇒ Alternative trajectory based interpretation.
Undoubtedly successful : photodissociation problems, tunnellingprocess, atom diffraction by surfaces, high harmonic generation etc.
Bohmian mechanics =⇒ Still ongoing and controversial.Keeping interpretational issues aside =⇒ Apply it.
Sanjib Dey (City University London) Bohmian trajectories from coherent states 6 / 22
Bohmian mechanics (real case)
Time dependent Schrodinger equation :
ih∂ψ(x, t)
∂t=− h2
2m∂2ψ(x, t)
∂x2 +V(x)ψ(x, t)
WKB polar decomposition :
ψ(x, t) = R(x, t)eih S(x,t), R(x, t),S(x, t) ∈ R
Substitute ψ(x, t) into Schrodinger equation and separate real and imaginarypart :
St +(Sx)
2
2m+V(x)− h2
2mRxx
R= 0 ⇐ Quantum Hamilton-Jacobi equation
mRt +RxSx +12
RSxx = 0 ⇐ Continuity equation
Sanjib Dey (City University London) Bohmian trajectories from coherent states 7 / 22
Real Bohmian
∗ Velocity :
mv(x, t) = Sx =h2i
[ψ∗ψx−ψψ∗x
ψ∗ψ
]∗ Quantum potential :
Q(x, t) =− h2
2mRxx
R=
h2
4m
[(ψ∗ψ)2
x
2(ψ∗ψ)2 −(ψ∗ψ)xx
ψ∗ψ
]
∗ Effective potential Veff(x, t) = V(x)+Q(x, t).∗ Two options to compute quantum trajectories :
1 Solve⇒ v(x, t)2 Solve⇒ mx =−∂Veff/∂x
Sanjib Dey (City University London) Bohmian trajectories from coherent states 8 / 22
Bohmian mechanics (complex case)
∗ Decompose :ψ(x, t) = e
ih S(x,t), S(x, t) ∈ C
∗ Substitute ψ(x, t)⇒ time dependent Schrodinger equation :
St +(Sx)
2
2m+V(x)− ih
2mSxx = 0
∗ Velocity :
mv(x, t) = Sx =hi
ψx
ψ
∗ Quantum potential :
Q(x, t) =− ih2m
Sxx =−h2
2m
[ψxx
ψ− ψ2
x
ψ2
]∗ Less explored in the literature.
Sanjib Dey (City University London) Bohmian trajectories from coherent states 9 / 22
What do we learn from Bohmian mechanics??
Unlike the usual interpretation, it gives us a system in a precisely definablestate, whose dynamics are determined by definite laws, analogous to classical
equations of motion.
Sanjib Dey (City University London) Bohmian trajectories from coherent states 10 / 22
Generalised Klauder coherent state
Klauder coherent state
ψJ(x, t) :=1
N (J)
∞
∑n=0
Jn/2 exp(−iωten)√ρn
φn(x), J ∈ R+0
ρn := ∏nk=1 ek, N 2(J) := ∑
∞
k=0 Jk/ρk, ρ0 = 1
SummaryCoherent states ψJ(x, t)⇒ Bohmian scheme⇒ Bohmian trajectories.
Draw classical trajectories.
How close !!! coherent states⇐⇒ classical case.
Sanjib Dey (City University London) Bohmian trajectories from coherent states 11 / 22
Application : Poschl-Teller model (real case)
φn(x) =1√Nn
cosλ
( x2a
)sinκ
( x2a
)2F1
[−n,n+κ+λ;k+
12
;sin2( x
2a
)]Stationary state Bohmian :
v(t) = 0 ⇐ Not the behaviour of a classical particle.
Klauder coherent state :
ψJ(x, t) :=1
N (J)
∞
∑n=0
Jn/2 exp(−iωten)√ρn
φn(x)
ρn = n!(n+κ+λ)n, N 2(J) = 0F1 (1+κ+λ;J)
Classical solution :
x(t) = a arccos
[α−β
2+√
γcos
(√2Em
ta
)], α, β, γ constant
Sanjib Dey (City University London) Bohmian trajectories from coherent states 12 / 22
0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 02 . 0 0
2 . 0 1
2 . 0 2
2 . 0 3
2 . 0 4
2 . 0 5
x ( t )
t
( a )
0 5 10 15 20 252
3
4
5
6
(c)
J = 20 J = 10 J = 2 J = 20.2846
x(t)
t
Qualitatively not identical with classical trajectories !!
Let us look at the uncertainty !!
Let us look at the behaviour of|ψ(x, t)|2 with time too.
Sanjib Dey (City University London) Bohmian trajectories from coherent states 13 / 22
0 5 1 0 1 5 2 0 2 50
1
2
3
4
5
6
7
Q = - 0 . 3 0 7 5 9 3 Q = - 0 . 1 4 9 5 2 3 Q = - 0 . 0 4 2 5 5 5
∆x ∆p
t
( a )
0 1 2 3 4 5 60.0
0.2
0.4
0.6
0.8
t = 0t = 1t = 10t = 20t = 30
|(x
,t)|2
x
(b)
Not a squeezed coherent state, ∆x∆p ≫ }/2 !!Shape of the wave packet changes with time, i.e. not a classical particle!!
Need to localise the wavepacket.
How can we do that??
Sanjib Dey (City University London) Bohmian trajectories from coherent states 14 / 22
Mandel parameter
ψJ(x, t) := 1N (J)
∞
∑n=0
Jn/2 exp(−iωten)√ρn
φn(x)
ψJ(x, t) :=∞
∑n=0
cn(J)e−iωten |φn〉, cn =Jn/2
N (J)√
ρn⇐ weighting function
We need, ψJ(x, t)⇒ to be well localised.
To examine : check weighting probability, |cn|2⇒ Poissonian.
Deviation of |cn|2 from Poissonian is captured by Mandel parameter, Q .
If ψJ is strongly weighted around 〈n〉, Q = ∆n2
〈n〉 −1 = J ddJ ln d
dJ lnN 2
Q = 0 ⇒ Pure Poissonian, Q > 0 ⇒ Super-Poissonian.Q < 0 ⇒ Sub-Poissonian, |Q | � 1 ⇒ Quasi-Poissonian.
Sanjib Dey (City University London) Bohmian trajectories from coherent states 15 / 22
Sub-Poissonian regime
0 5 1 0 1 5 2 0 2 50
1
2
3
4
5
6
7
Q = - 0 . 3 0 7 5 9 3 Q = - 0 . 1 4 9 5 2 3 Q = - 0 . 0 4 2 5 5 5
∆x ∆p
t
( a )
0 1 2 3 4 5 60.0
0.2
0.4
0.6
0.8
t = 0t = 1t = 10t = 20t = 30
|(x
,t)|2
x
(b)
Q =−0.307593,−0.149523,−0.042555
We are in sub-Poissonian regime !!!What happens in the quasi-Poissonian, Q→ 0 regime??
Sanjib Dey (City University London) Bohmian trajectories from coherent states 16 / 22
Q(J,κ+λ) = J2+κ+λ
0F1(3+κ+λ;J)0F1(2+κ+λ;J) −
J1+κ+λ
0F1(2+κ+λ;J)0F1(1+κ+λ;J)
Let us control κ, λ and J, so that Q→ 0
0 5 10 15 20 25
0.5100
0.5103
0.5106
0.5109
0.5112
x p
t
Q= -0.000054529 Q= -0.000013634 Q= -0.000002726
(a)
0.0 0.4 0.8 1.2
0.5000055
0.5000070
0 1 2 3 4 5 60
1
2
3
t = 0 , J = 0 . 0 0 2 2 9 0 6 t = 0 . 6 5 , J = 0 . 0 0 2 2 9 0 6 t = 0 , J = 2 t = 4 , J = 2|Ψ
(x,t)|2
x
( b )
Two sets : κ = 90, λ = 100, J = 2,0.5,0.1 andκ = 2, λ = 3, J = 2,0.5,0.1
Sanjib Dey (City University London) Bohmian trajectories from coherent states 17 / 22
Quasi-Poissonian regime
0.0 0.2 0.4 0.6 0.8 1.02.00
2.01
2.02
2.03
2.04
2.05 (a)
J = 2.0 J = 0.5 J = 0.1
x(t)
t 0 5 10 15 20 25 302.00
2.01
2.02
2.03
2.04
2.05
2.06 (b)
J = 0.0022906 J = 0.00057265 J = 0.000114531
x(t)
t
Sanjib Dey (City University London) Bohmian trajectories from coherent states 18 / 22
Stationary state : complex case
ψn(x) =1√Nn
cosλ
( x2a
)sinκ
( x2a
)2F1
[−n,n+κ+λ;k+
12
;sin2( x
2a
)]
-6 -4 -2 0 2 4 6
-2
-1
0
1
2x
0= ±0.1
x0= ±1.5
x0= ±2.0
x0= ±2.45
x0= 5.0
x0= 5.5
x0(t)
t
(a)
-6 -4 -2 0 2 4 6-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
x0= ±0.1x0= ±0.3x0= ±0.9x0= ±1.5x0= ±2.7x0= ±3.6x0= ±4.5x0= ±5.0x0= ±5.5
x5(t)
t
(b)
Sanjib Dey (City University London) Bohmian trajectories from coherent states 19 / 22
Classical and Klauder state
- 0.9
- 0.9 - 0.9
- 0.9 - 0.9
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- 0.8 - 0.8
- 0.8 - 0.8- 0.8
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- 0.3- 0.3- 0.3 - 0.3 - 0.3
- 0.3- 0.2- 0.2
- 0.2- 0.2
- 0.2 - 0.1- 0.1- 0.1- 0.1 - 0.1
xi
xr
(a)
- 20 - 15 - 10 - 5 0 5
- 3
- 2
- 1
0
1
2
3
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- 0.8- 0.8
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- 0.4 - 0.4
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- 0.2 - 0.2
- 0.2- 0.2
- 0.2 - 0.2
- 0.2
00
00
0
0
00 0
0
0
0
0
0
0.2
0.20.2
0.2
0.20.2
0.2 0.2
0.2
0.4
0.4
0.4
0.4
0.4 0.4
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0.60.6
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0.80.80.8
0.8
0.8
0.8
0.8
0.8
0.8
xr
xi (b)
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- 3
- 2
- 1
0
1
2
3
Sub-Poissonian regime, Q < 0
Sanjib Dey (City University London) Bohmian trajectories from coherent states 20 / 22
Classical and Klauder state
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0
00
500
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(a)xi
xr
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- 2
- 1
0
1
2
3
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00
0
0 0
0
0
500
500
500
500
(b)xi
xr
- 6 - 4 - 2 0 2 4 6
- 3
- 2
- 1
0
1
2
3
Quasi-Poissonian regime, Q → 0Perfect matching : Classical⇐⇒ Klauder coherent state
Sanjib Dey (City University London) Bohmian trajectories from coherent states 21 / 22
ConclusionDirect connection : quantum⇐⇒classical, through Bohmiantrajectories.
Q→ 0, Klauder state is a perfect coherent state for both real andcomplex cases.
Klauder state is canonical and squeezed state for Harmonic Oscillator.
Must take Klauder state for generalised models, instead of Glauber state.
OutlookOne can study different potentials especially the complex case.
Also noncommutative models where one has direct connection with P T ,dealing with pseudo-Hermitian Hamiltonians.
Interesting to explore how the conventional quantum mechanicaldescription can be reproduced from Bohmian scheme.
Thank you for your attentionSanjib Dey (City University London) Bohmian trajectories from coherent states 22 / 22
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