Bohmian trajectories from coherent states, Istanbul 3 July 2013

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

)


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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00

500

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(a)xi

xr

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0

1

2

3

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0

0 0

0

0

500

500

500

500

(b)xi

xr

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- 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


Can we explain the motion of the quantum particle in thesame way??


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.


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


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


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.


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.


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.


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


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


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.


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??


Mandel parameter

ψJ(x, t) := 1N (J)

∞

∑n=0


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.


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??


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


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


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)

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-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)


Classical and Klauder state

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xi

xr

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xi (b)

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Sub-Poissonian regime, Q < 0


Classical and Klauder state

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Quasi-Poissonian regime, Q → 0Perfect matching : Classical⇐⇒ Klauder coherent state


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

Bohmian trajectories from coherent states, Istanbul 3 July 2013

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