Bohm Trajectories for a Coupled Two-Dimensional Harmonic Oscillator
Bohm Trajectories for a Coupled Two-Dimensional Harmonic Oscillator
Exact analytic solutions of the stationary Schrödinger equation enable the exploration of detailed properties of a quantum system. We note also that the integrability of a classical dynamical system need not necessarily imply its quantum integrability[1].
This Demonstration considers two-dimensional Bohm trajectories in a harmonic potential + perturbed by a coupling term . Analytic expressions of the Schrödinger equation for this potential are known for a limited number of eigenstates. There exist solutions for the ground state and the first and second excited states[1–3]. Obviously, as , the coupled harmonic oscillator behaves asymptotically as two independent one-dimensional oscillators. Here the one-dimensional harmonic oscillator is referred to as a single quantum particle. The motion of two particles is represented in a two-dimensional configuration space by the coordinates along a trajectory. A superposition state leads to an entangled or nonlocal behavior in a two-dimensional configuration space. In general, an entangled wavefunction cannot be factorized into a product of two single-particle wavefunctions.
1
2
2
x
2
y
γ(xy)
ψ
n,m
γ->0
(x,y)
ϕ
The motion ranges from motionless, periodic to quasi-periodic to fully chaotic, depending on the parameter and the constant phase shift . In the de Broglie–Bohm (or causal) interpretation of quantum mechanics[4–6], the particle position and momentum are well defined, and the motion can be described by continuous evolution according to the time-dependent Schrödinger equation.
γ
α
i
The velocity vector of a superposition state , as expressed in the guiding equation, for one particle will depend upon the positions of the other, whenever the total wavefunction is not a product of single-particle wavefunctions (nonfactorizability)[4]. The projection of the trajectory onto two-dimensional configuration space leads to a decomposition of two spatially divided motions in one-dimensional real space. This is quantum entanglement in the de Broglie–Bohm interpretation. In our case, the one-dimensional quantum particles in a superposition state behave manifestly nonlocally, because of the coupling factor . The coupling factor determines the shapes of the orbits.
ϕ
ϕ
γ
γ
The graphic shows the trajectories (white/blue), the velocity vector field (red), the absolute value of the wavefunction and the initial and final points of the trajectories. The initial point is shown as a white square, which you can drag. The final point is shown as a small white or blue dot. The coupled harmonic potential (if enabled) is shown with blue/black contour lines.
Details
Details
The two-dimensional stationary Schrödinger equation with potential , a harmonic potential with a coupling term, can be written:
V(x,y)
-+V(x,y)(y,y,t)=(x,y,t)=iℏ(x,y,t)
2
ℏ
2μ
2
∇
(i)
ψ
n,m
E
n
(i)
ψ
n,m
∂
t
(i)
ψ
n,m
with the potential
V(x,y)=k++γ(xy)
1
2
2
x
2
y
with reduced mass , Planck's constant , the constants and the Laplacian operator in Cartesian coordinates. For simplicity, set , and equal to 1.
μ
ℏ
γ,k∈
2
∇
μ
k
ℏ
With the variable transformation , , we get two different quantum harmonic oscillators with frequencies = and =.
X=(x+y)
1
2
Y=(x-y)
1
2
ω
X
1+γ
ω
Y
1-γ
Each single part of the superposition state can be expressed by an unnormalized product state
ϕ
ψ
n,m
ϕ
n
ϕ
m
4
ω
X
1
2
ω
X
2
X
H
n
ω
X
4
ω
Y
1
2
ω
Y
2
Y
H
m
ω
Y
where
H
n
The solution of the Schrödinger equation will return a set of eigenvalues and coupled eigenfunctions in the original coordinates .
(i)
ψ
n,m
(x,y)
With the energy eigenvalue =m++n+, the unnormalized, time-dependent wavefunction gives[2]:
E
n,m
1
2
ω
X
1
2
ω
Y
(i)
ψ
n,m
a
i
4
ω
X
ω
Y
1
4
2
x
2
y
ω
X
ω
Y
1
2
ω
X
ω
Y
H
m
(x+y)
ω
Y
2
H
n
(x+y)
ω
X
2
-it(+)
E
n,m
α
i
e
with and ,∈. the energy becomes complex valued.
0≤γ<1
a
i
α
i
For
γ>1
This special equation , obeys the Schrödinger equation only for or 1.
(i)
ψ
n,m
n,m=0
An unnormalized wavefunction for two one-dimensional particles, from which the trajectories are calculated, can be defined by a superposition state :
ψ
n,m
ϕ
ϕ==(x,y,t)+(y,x,t)+(x,y,t)
3
∑
i=1
(i)
ψ
n,m
(1)
ψ
1,0
(2)
ψ
0,1
(3)
ψ
1,1
with ==0; ; ==1 and
α
1
α
3
0≤≤1
α
2
a
1
a
2
0≤≤1
a
3
or in detail:
ϕ=x+y+x-y+2-2
1/4
()
ω
X
ω
Y
2
it+it
A
2
A
3
e
ω
X
2
it+it
A
2
A
3
e
ω
X
2
it+it+i
A
1
A
3
α
2
e
ω
Y
2
it+it+i
A
1
A
3
α
2
e
ω
Y
it+it
A
1
A
2
e
2
x
a
3
ω
X
ω
Y
it+it
A
1
A
2
e
2
y
a
3
ω
X
ω
Y
-it-it-it-xy(-)-+(+)
A
1
A
2
A
3
1
2
ω
X
ω
Y
1
4
2
x
2
y
ω
X
ω
Y
e
with
A
1
3
ω
X
2
ω
Y
2
A
2
ω
X
2
3
ω
Y
2
and
A
3
3
ω
X
2
3
ω
Y
2
For the frequencies , are equal, with ==1.
γ=0
ω
X
ω
Y
ω
X
ω
Y
Some examples:
For =γ==0 the quantum particles are at rest.
a
3
α
2
For =γ=0 and ≠0 and ≠z with , the velocity field becomes autonomous and obeys the time-independent part of the continuity equation
a
3
α
2
α
2
z∈
∇
v
with ; the trajectories reduce to circles with the velocities and :
ρ=ψ
*
ψ
v
x
v
y
v
x
ysinα
cosα-++
2
x
2
y
2
x
2
y
v
y
v
x
For =0, and ≠0 and ≠zπ with , the velocity field becomes time dependent, but the orbits reduce to fractions or multiples of a circle:
a
3
0<γ<1
α
2
α
2
z∈
v
x
ω
X
ω
Y
ω
X
ω
Y
ω
X
ω
Y
2
x
2
y
ω
X
ω
Y
2
x
2
y
v
y
v
x
When PlotPoints, AccuracyGoal, PrecisionGoal and MaxSteps are increased (if enabled), the results will be more accurate. The initial distance between the two starting trajectories is determined by the factor . You can change the coupling constant potential , the constant phase shift and the superposition factor parameters in the program.
δx
γ
α
2
a
3
Special thanks to Vikram Athalye from Cummins College of Engineering for Women, Pune, India for his support.
References
References
[1] R. S. Kaushal, "Quantum Mechanics of Noncentral Harmonic and Anharmonic Potentials in Two-Dimensions," Annals of Physics, 206(1), 1991 pp. 90–105. doi:10.1016/0003-4916(91)90222-T.
[2] I. H. Naeim, J. Batle and S. Abdalla, "Solving the Two-Dimensional Schrödinger Equation Using Basis Truncation: A Hands-on Review and a Controversial Case," Pramana, 89(5), 2017 70. doi:10.1007/s12043-017-1467-z.
[3] R. M. Singh, F. Chand and S. C. Mishra, "The Solution of the Schrödinger Equation
for Coupled Quadratic and Quartic Potentials in Two Dimensions," Pramana, 72(4), 2009 pp. 647–654. doi:10.1007/s12043-009-0058-z.
for Coupled Quadratic and Quartic Potentials in Two Dimensions," Pramana, 72(4), 2009 pp. 647–654. doi:10.1007/s12043-009-0058-z.
[4] P. R. Holland, The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics, New York: Cambridge University Press, 1993.
[6] S. Goldstein, "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy (Summer 2017 Edition). (Jan 31, 2023) plato.stanford.edu/entries/qm-bohm.
External Links
External Links
Permanent Citation
Permanent Citation