Qualitative measurement of Klauder coherent states using Bohmian Mechanics Sanjib Dey City University London December 03, 2013 Based on Phys. Rev. A 88, 022116 (2013), with Prof. Andreas Fring Sanjib Dey (City University London) Bohmian trajectories from coherent states 1 / 23
23
Embed
Qualitative measurement of Klauder coherent states using Bohmian machanics, City December 3
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Qualitative measurement of Klauder coherent states usingBohmian Mechanics
Sanjib Dey
City University London
December 03, 2013
Based on Phys. Rev. A 88, 022116 (2013), with Prof. Andreas Fring
Sanjib Dey (City University London) Bohmian trajectories from coherent states 1 / 23
What is a coherent state?
Superposition of large no of quantum states⇒ Classical particle.For example, Glauber coherent state :
|α〉= N (α)∞
∑n=0
αn√
n!|n〉, N (α)⇒ e−
|α|22
Sometimes called the minimum uncertainty wavepacket ∆x∆p≈ }/2
Sanjib Dey (City University London) Bohmian trajectories from coherent states 3 / 23
Procedure
One can analyse all the properties mathematically. Which is notsufficient to realise the quality precisely.
How would you measure the precise quality?
Draw the classical trajectories by solving :
x =∂H∂p
, p =−∂H∂x
(1)
Draw the dynamics of the coherent states of the particle and compare.
How would you draw the trajectories of the coherent states?
Sanjib Dey (City University London) Bohmian trajectories from coherent states 4 / 23
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 5 / 23
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 6 / 23
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 7 / 23
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
]
Sanjib Dey (City University London) Bohmian trajectories from coherent states 8 / 23
Summarize
Solve canonical equations =⇒ Classical trajectoryCoherent state =⇒ Bohmian scheme =⇒ Trajectoriesof coherent stateCompare these two =⇒ Quality of coherent states
Sanjib Dey (City University London) Bohmian trajectories from coherent states 9 / 23
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,γ) :=1
N (J)
∞
∑n=0
Jn/2e−iγen
√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 10 / 23
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 !!
Look at the uncertainty of X & P
Look at the behaviour of |ψ(x, t)|2 with time too.
Sanjib Dey (City University London) Bohmian trajectories from coherent states 11 / 23