THE CLASSICAL LIMIT OF BOHMIAN MECHANICS
THROUGH COHERENT STATESBY CHRISTOPHER RACKAUCKAS
Coherent states have been studied in Quantum Mechanics as acandidate for states that exhibit classical-like behavior because such states
minimize the uncertainty relation making ∆x∆p = ~2 . The purpose of thispaper is to examine the Bohmian trajectories of particles with wavepackets incoherent states. Through this analysis we see that the trajectories particles
with wavefunctions in the coherent state give the classical result for the freeparticle, a particle under constant force, and the harmonic oscillator.
Abstract.

1.

Introduction

Bohmian Mechanics is a mechanics for point particles with mass. It uses the basic
formulas of Quantum Mechanics along with an extra equation known as the Guidance Equation. The Guidance Equation requires the use of of particle positions
and gives a complete description of each particle's trajectory. Thus, in Bohmian
Mechanics, if given the particle position and its corresponding wavefunctionψ,then using the equations of Bohmian Mechanics one can compute the trajectoryof the particle for all time. Furthermore, the motions of an ensemble of particles
in Bohmian Mechanics lead to the statistical results one would expect when using
Quantum Mechanics. For these reasons, Bohmian Mechanics has become known as
the causal interpretation of Quantum Mechanics.
Due to the prominence of Quantum Mechanics and the diculty of Bohmian
Mechanics, most publications examine Bohmian Mechanics through a high-levelview: looking in detail at proofs of the compatibility of Bohmian Mechanics withthe experimental results that have been used to conrm Quantum Mechanics orproving the viability of Bohmian Mechanics through the derivation of Bohmian
Mechanics from accepted rst-order principles. Another topic of interest has been
the computation of trajectories for ensembles of particles in systems of interest fromQuantum Mechanics, for example the innite square well. However, this paper is
not concerned with these topics. Instead it will examine the closed form solutions for
Bohmian particles with Gaussian wavepackets in systems of interest from Quantum
Mechanics.
2.

Foundations of Bohmian Mechanics

Bohmian Mechanics is a hidden-variable theory with well-dened particle trajectories that is compatible with non-relativistic Quantum Mechanics. A Bohmiansystem is dened by the positions of its N particles Q1 ,...QN , Qt ∈ R

3togetherwith the mass of the k th particle mk , and the system's corresponding wavefunction

ψ , which is dened as a function
ψ : R3N × R → C
1

THE CLASSICAL LIMIT OF BOHMIAN MECHANICS THROUGH COHERENT STATES
which governs the motion of the particles on conguration space R

3N

2

through theformuladQk~∇ψk= vk =={(Q, t)}dtmkψk∇ψk∇ψkwhere ={ψk (Q, t)} denotes the imaginary part of ψk (Q, t) [3]. The last require-ment is that the wavefunction obeys the Schrodinger equation, that is∂ψi= − Ĥψ∂t~where Ĥ is the non-relativistic Hamiltonian.
The last important aspect of Bohmian Mechanics is known as the Quantum
Equilibrium Hypothesis. The Quantum Equilibrium Hypothesis states that for an
ensemble of identical systems each having a wavefunction ψ , the typical empirical distribution of the congurations of the particles is given approximately by2

the probabilities p = |ψ|(this can also be taken as a theorem, derivations fromBohmian Mechanics [4, 3] ). Thus the statistical outcomes for large ensembles ofparticles in identical systems for Bohmian Mechanics is in accord with Born's Law
from Quantum Mechanics and is thus experimentally equivalent.
It is important to notice that Bohmian Mechanics is basically an extension ofQuantum Mechanics where one takes the claim that particle positions exist (thoughare unknown) and have well-dened trajectories.

The Foundational Equations in One Dimension

3.

For the purposes of our investigation, we will look at a single particle in onedimension. For such systems,∂ψ(x,t)v(x, t) =~={ ∂x }m ψ(x, t)where ψ is subject to the Schrodinger Equation∂ψi= − Ĥψ.
∂t~With just these two equations we can solve the trajectories of single particles in
one dimension with their associated wavefunction.
4.

Coherent States and The Gaussian Wavepacket

Glauber Coherent States, or Coherent States, have been a subject of study since
Schrodinger rst identied them as being the states which minimize the uncertainty~
2 [1, 2]. This leads to the speculation that coherent states in
Bohmian Mechanics may have classical trajectories.
relation ∆x∆p =≥Our analysis will utilize what is known as the Gaussian wavepacket. The Gaussian wavepacket is dened as a wavepacket with an initial distribution as a Gaussian,that isA −x2
ψ(x, 0) = √ e 2σ2σ1

where A = 4 √π

due to the normalization of the wavefunction. It is well-known thatGaussian wavepackets the coherent states of systems such as the free particle, theparticle under a constant force, and the harmonic oscillator. Thus since we expect

THE CLASSICAL LIMIT OF BOHMIAN MECHANICS THROUGH COHERENT STATES

3particles in coherent states to exhibit classical-like behavior, we expect particles
with a Gaussian Wavepacket to have classical trajectories in Bohmian Mechanics.
5.

A Free Particle With A Gaussian Wavepacket

Take an initial Gaussian Wavefunction

A −x2 ip0
ψ(x, 0) = √ e 2σ2 + ~ xσwith a HamiltonianĤ = −~2 ∂ 22m ∂x2

In order to solve for the time-dependent wavefunction, we will rst transform ψ
into the momentum basis by the relationψ̃(p, 0) = √

12π~

ˆ ∞

pe−i( ~ )x ψ(x, 0) dx.
−∞

Solving this integral givesrψ̃(p, 0) = A

σ −(p−p0 )2 σ2 /2~2e~

By solving the Schrodinger equation we get that
ip2

ψ̃(p, t) = ψ̃(p, 0)e−( ~ ) 2musing the relation

ψ(x, t) = √

12π~

ˆ ∞
e−i(p/~)x ψ̃(p, t) dp−∞

and dening

β =1+i~tmσ 2

we get thatA
ψ(x, t) = √ ef (x,t)σβ
wheref (x, t) =

ip0 x
ip2 t
x2
xp0 t
p20 t2− 0 − 2 ++~2m~ 2σ β2mσ 2 β
4m2 σ 4 β 2Knowing the time-dependent wavefunction, we can now solve for the Bohmianvelocity eld. We know thatA∂f (x, t)∂ψ(x, t)= √ ef (x,t)
∂x
∂xσβ
Thus

∂ψ(x,t)∂xψ(x, t)=∂f (x, t)
ip0x
p0 t=− 2 +∂x~σ β
2mσ 2 βTherefore={p0∂ψ(x, t)}=∂x~

Thus we get from the guidance equation thatvx (t) =p0m

THE CLASSICAL LIMIT OF BOHMIAN MECHANICS THROUGH COHERENT STATES
and thusx(t) =4p0 t.
mTherefore the motions of a free particle with a Gaussian wavepacket is equivalent
to the motion of a classical particle.
6.

A Particle With A Constant Force And A Gaussian Wavepacket

Take an initial Gaussian wavefunction

A −x2 ip0
ψ(x, 0) = √ e 2σ2 + ~ xσ
under a constant force F . The Hamiltonian for such a system will beH̄ = −1 ∂2− Fx2m ∂x2

and thus by the Avron-Herbst formula [6] we know thatψ(x, t) = ψ(x −i F2 3ii
F 2t , 0)e− ~ 6m t + ~ tF x+ 2m~ t2m

Therefore by substitution of our wavepacket into this equation we receive thefollowing wavefunction
F2 2i
A −x2 ip0 −( 2m t ) i~ F 2 i F 2 3 i
ψ(x, t) = √ e 2σ2 + ~ x+ 2σ2 + p0 2m t − ~ 6m t + ~ tF x+ 2m~ tσThus∂ψ(x, t)xip0i
= (− 2 ++ tF )ψ(x, t)∂xσ~~and∂ψ(x,t)

vx (t) =~p0tF={ ∂x } =+m ψ(x, t)mm

Taking F = ma0 we can write this equation asv(t) =p0+ a0 tmwhich is the velocity of the classical particle.
7.

The Shifted Ground State Gaussian Wavefunction in the HarmonicOscillator
Take an initial Gaussian Wavefunction(x−a)2A
ψ(x, 0) = √ e− 2σ2σwith a Hamiltonian ofĤ = −Take the case where~2 ∂ 21+ mω 2 x22m ∂x22r~σ=mωTHE CLASSICAL LIMIT OF BOHMIAN MECHANICS THROUGH COHERENT STATES

5This wavepacket is equal to the shifted ground state of the harmonic oscillator.
Thus the time-dependent wavefunction is found to be [5]A
ψ(x, t) = √ ef (x,t)σ
wherea2a1xa
1 x
f (x, t) = − ( − cos(ωt))2 − i( ωt + 2 sin ωt − 2 sin 2ωt)
2 σ σ
2σ4σ
Notice that

∂ψ∂xψ

=aia
∂f (x, t)x= − 2 − 2 cos(ωt) − 2 sin(ωt)∂xσσσand thus∂ψvx =~~asin(ωt)={ ∂x } = −mψmσ 2

By plugging in σ we get thatv = aω sin(ωt).
which matches the classical result.
8.

Conclusion

Bohmian Mechanics is a causal interpretation of Quantum Mechanics that wasdeveloped to answer philosophical issues dealing with the formulation of Quantum Mechanics.

Coherent states of quantum systems have been analyzed before

in Quantum Mechanics as being the states of least uncertainty and as state which
exhibit classical-like behavior. By looking at these cases we notice that the coherent states of the free particle, the particle under a constant force, and harmonic
oscillator all give trajectories that match the classical trajectory.

This leads us

to speculate that the trajectories of particles in coherent states may have classicaltrajectories.

References[1] Bernstein, Jeremy. "More About Bohm's Quantum." American Journal of Physics 79.6 (2011):601-06.
[2] Bowman, Gary E. "Bohmian Mechanics as a Heuristic Device: Wave Packets in the Harmonic
Oscillator." American Journal of Physics 70.3 (2002): 313-18.
[3] Dürr, Detlef Christopher, and Stefan Christopher Teufel. Bohmian Mechanics: The Physics
and Mathematics of Quantum Theory. Berlin: Springer, 2009.
[4] Holland, Peter R. The Quantum Theory of Motion: An Account of the De Broglie-BohmCausal Interpretation of Quantum Mechanics. Cambridge [England: Cambridge UP, 1993.
[5] Schi, Leonard I. Quantum Mechanics. New York: McGraw-Hill, 1968.
[6] Thaller, Bernd Christopher. Visual Quantum Mechanics: Selected Topics with ComputerGenerated Animations of Quantum-Mechanical Phenomena. Page 193. New York:Springer/TELOS, 2000.