Download Bohmian Mechanics

Bohmian Mechanics
For the seminar: ‘Ausgewählte Probleme der Quantenmechanik’
Faculty of Physics, University of Vienna, WS 2011/2012
Christian Knobloch a0846069
1 Introduction
In the following lines the concept of Bohmian mechanics should be explained. Although it is
nowadays well established as Bohmian mechanics the better name would be De-Brougli-Bohm-Theory, because long before Bohm, De Broglie had nearly the same ideas about this topic. The
reader, who is used to the usual interpretation of the quantum theory, which is called the
Copenhagen interpretation, will maybe be a bit confused, because this view goes in a totally
new direction, more backwards to classical physics than completely new points of view.
2 Background
As said in the lines above the first, who thought about new interpretations of Schrödinger’s
equation was de Broglie in 1926. He gave up his studies about it because of criticisms made
by Pauli and additional objections made by himself. Although he published this point of view
first in 1930 in a book called “An Introduction to the Study of Wave Mechanics”, but it stayed
largely unknown. The following years were full of interesting discussions, because in 1935
Albert Einstein et al. published their famous paper about the, in their opinion, incomplete
quantum mechanics. Niels Bohr was the main opponent in this discussion, which should last
until 1964 when John S. Bell published his paper, introducing his inequality, which should
be able to solve the discussion, whether quantum mechanics is incomplete or not. In the
time between this paper by Einstein et al. and the solution by Bell an other paper can
be found, that tries to solve the discussion by introducing a new interpretation of quantum
theory by David Bohm. It is completely different from the nowadays accepted Copenhagen
interpretation. As Papers about the completeness of quantum mechanics were published,
adherents of Bohmian mechanics and also himself tried to find counterarguments, as we will
see. Today just a small minority of physicists maintain this theory.
3 ‘New physical interpretation of Schrödinger’s equation’[1]
That is, what Bohm called the fourth chapter of his paper about his view of quantum theory.
The thing, that both theories, the copenhagen interpretation and the Bohmian one, share
is the well known Schrödinger equation. The ‘usual’ interpretation says, that in the wave
function, which is treated and described in this equation, the whole state of a particle or
better the whole information about it is contained. In contrast to that Bohm says, that it can
not be the whole information since we are not even able to describe the particles position in
a doubleslitexperiment. The starting point of Bohm’s view is, that for a theory explaining the
quantum phenomena, this should be possible, at least in principle. The new theory should,
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however, be possible to reproduce all results of the ‘usual’ theory verified by a physical
experiment.
4 Theory
As mentioned, the starting point is Schrödinger’s equation as well:
i~
∂ψ
~2 2
=−
∇ ψ + V (x )ψ
∂t
2m
As representation for the wave function we use the polar form written as
i
ψ (x , t ) = R (x , t ) exp
S (x , t ) ,
~
(1)
(2)
where R and S are real functions and R = |ψ |. At this point we note, that the probability
distribution of a certain event is given by P = ψ ∗ ψ = R 2 . To get an evolution in time for the
two, yet unknown functions S and R , we insert the polar form into Schrödinger’s equation:
i~
∂ ~2 2 i
i
i
R e~S = −
∇ R e ~ S + V (x ) R e ~ S
∂t
2m
(3)
To get the time derivatives for R and S it is necessary to apply the gradient and to split
into a real and an imaginary part, which is calculated in detail in Appendix A and gives us:
∂R
1
=−
(R ∇2 S + 2∇R ∇S )
∂t
2m
(4)
∂S
(∇S )2
~2 ∇2 R
=−
+ V (x ) −
∂t
2m
2mR
(5)
Here we get an image of why this theory is called Bohmian-mechanics. Equation (5), namely
is the so called Hamilton-Jacobi equation from classical mechanics, up to a term arising in
our equation calculated for quantum theory. This term got the name Quantum potential by
Bohm and can be denoted as:
Q =−
~2 ∇2 R
2mR
(6)
This certain potential just arises for quantum phenomena. Now we want to have a look at
the imaginary part of the splited Schrödinger equation. We start from (4) and re express it in
terms of P = ρ and j , where j is the probability current density, derived from Schrödinger’s
equation. For a detailed calculation see Appendix B. The result of this calculation can be
written as:
∂
ρ + div~j = 0
∂t
(7)
This is the well known equation of continuity for the probability current density for a certain
quantum system, often called quantum equilibrium hypothesis.
So the theory takes the Schrödinger equation using the polar representation of quantum
states and splits it into an imaginary and a real part. So the real part of a quantum
system that evolves in time obeying Schrödinger’s equation is represented by the classical
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Hamilton-Jacoby-equation supplemented by the quantum potential, which arises for quantum
phenomena. The imaginary part of such a system can be described by the continuity of
probability density and its current.
So it seems to be the case, that Bohm solved the question, what the imaginary part in the
Schrödinger equation stands for, nevertheless he provides no explanation what an imaginary
probability density should be.
Particle Trajectories: Reading papers about Bohmian mechanics ([2],[3]) one will soon
arrive at ideas about trajectories for particles. The physical reason for it is, in Bohms view,
that it should be possible to determine the location of a certain particle in space for each
time t , at least in principle. To get the position of such a particle it is necessary to know
which trajectory it follows. At this point we note, that it is indeed possible to determine the
position at each time, but for this just holds for exactly known initial conditions. This can
never be guaranteed for practical reasons and so we again have an uncertainty in our results,
but Bohm stresses that this is actually not inherent in the formalism and just arises in the
experiment. To get such trajectories now we have to know the particles velocity v , which can
be denoted as:
v =
∇S
= ẋ (t )
m
(8)
This is already the so called Leitgleichung, which we can get via inserting the polar form of
ψ into the following relation:
i~ ∂x∂ ψ ψ ∗ − ψ ∂x∂ ψ ∗
ẋ (t ) = −
(9)
2m
ψ ∗ψ
This equation can also be derived from Schrödinger’s equation (e.g.[2]). The interpretation of
such trajectories is much more complicated than equation (8) suggests. One famous example
is the doubleslitexperiment, which includes both, the trajectory hypothesis as well as the
quantum potential.
Doubleslit:Taking this topic under investigation, we have to distinguish between the ‘usual’
quantum view of reality, and the one Bohm, Einstein and de Broglie suggest us. They say
that there has to be a uniquely defined reality, without which physics would be reduced
to a set of formulas. For that we have to assume, that there has to be a physically real
wave satisfying Schrödinger’s equation together with a particle that follows the trajectories
defined in (8). The wave is lead by the quantum potential, which does not fall to zero at
long distances, even when the wave intensity becomes negligible. This is a result of the fact
that Q is not altered, when ψ gets multiplied by a constant. It follows, that it is possible to
declare the so called “spooky action at a distance” (R. A. Bertlmann), which we have for an
EPR-pair of particles, in terms of this quantum potential. Especially in this case the quantum
potential includes strong interactions between the two particles, which are not seperabel
any more.
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Figure 1 Left: Calculated quantum potential after a doubleslit. Right: Trajectories for a particle passing this doubleslit.
The bunching of the trajectories can be seen in the middle. | D. J. Bohm and B. J. Hiley, “The de Broglie Pilot Wave Theory
and the Further Development of New Insights Arising Out of It”, Foundation of Physics, Vol. 12, No. 10, 1982
The quantum potential also has the whole information about the space, in which it exists,
so that it makes a different if one or both slits are open for a doubleslitexperiment. B.J.
Hiley [5] explains this case with three qualities of information, active, inactive and passive
information. These informations are distributed to each path depending on whether both slits
are open or just one.
5 Objections against Bohmian mechanics
Thinking about such theorems, that would knock out a Bhmian-like-theory one can bring the
theorem by John S. Bell (1964), who gave us a inequality, which can not be fulfilled by local
realistic theories, or such that want to introduce ‘hidden-variables’ to make quantum mechanics
local or realistic. Contrary to that, nevertheless Bohm named his theory “in terms of hidden
variables”, an argument for Bohmian-mechanics is, that the quantum potential acts highly
nonlocal and immediately on a particle. However, a big fault is that in Bohmian-mechanics
this is not possible without a preferred inertial system. Nevertheless in the theory they argue,
that the violation of Bell’s inequality is a proof, that it would not make sense to search for a
Einstein-realistic Bohmian-theory.
6 Conclusion
Bohmian-mechanics often led to debates and confrontations on many conferences and is
nowadays just represented by a small minority of physicists. Although the majority uses the
formulas mathematical apparatus the Copenhagen interpretation induced, it is interesting to
see, what other interpretations of the quantum formalism and its roots can and what they
can not achieve. Generally reading papers about Bohmian-theory remains often a confused
impression, because many authors express things in their own picture, they have about it.
All in all there are at least three assumptions that can be make for a Bohmian-theory:
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•
The ψ -field satisfies Schrödinger’s equation.
•
If we write ψ = R exp(iS /~), then the particle momentum is restricted to p = ∇S (x )
•
We have a statistical ensemble of particle positions, with a probability density P =
|ψ (x )|2 .
7 Appendix A:
Derivation of time evolution of S and R :
Our starting point is (3). The functions R and S , however, depend just on x and t , which
gives us:
∂R i S
i
∂S
~2
i
i
i
i
i
S
S
S
i~
e ~ + i~ R e ~
=−
∇ ∇R e ~ + R e ~ ∇S + V (x ) R e ~ S
(10)
∂t
~
∂t
2m
~
Further dissolving of the gradient gives us:
i~
~2
−
2m
i
∇2 R e ~ S
∂R i S
i
∂S
i
e ~ + i~ R e ~ S
=
∂t
~
∂t
(11)
i iS
i iS
i2
i
i
i
i
S
S
2
+ e ~ ∇R ∇S + e ~ ∇S ∇R + 2 R e ~ ∇S ∇S + R e ~ ∇ S +V (x ) R e ~ S
~
~
~
~
This equation we can now split in a real and an imaginary part, which gives us:
∂R
1
=−
(R ∇2 S + 2∇R ∇S )
∂t
2m
(12)
∂S
(∇S )2
~2 ∇2 R
=−
+ V (x ) −
∂t
2m
2mR
(13)
8 Appendix B:
Derivation of the continuity equation:
For this derivation we start at equation (4). From dissolving and multiplying R on the
right side we get:
∂R
1 R 2 ∇2 S
2R ∇R ∇S
=−
−
∂t
2m
R
2mR
(14)
Now we want a relation for the probability density R 2 = |ψ ∗ ψ | = ρ and form the following
derivative:
∂ 2
∂R
R = 2R
∂t
∂t
(15)
Transformation gives us:
2R
∂R
1
=−
R 2 ∇2 S + 2R ∇R ∇S
∂t
m
5
(16)
To this we apply the product rule for the gradient as well as equation (15):
∂ 2
1
R = − ∇(R 2 ∇S )
∂t
m
(17)
∂
(R 2 ∇S )
ρ = −∇
.
∂t
m
(18)
And with R 2 = ρ we get:
2
For a further expression of the term (R m∇S ) we need the Schrödinger equation again, from
which a formula for the probability current density follows:
~j = i~ (ψ ∇ψ ∗ − ψ ∗ ∇ψ )
2m
Again we insert the polar form
~j = i~ R e ~i S ∇R e− ~i S −
2m
of our wavefunction and get:
i
i
i
i
i
i
−
S
−
S
S
S
R e ~ ∇S − R e ~
∇R e ~ + R e ~ ∇S
~
~
(19)
(20)
The only remaining term is:
~j = 1 R 2 ∇S ,
m
(21)
which is the one we were searching for. Putting this into (18), we get
∂
ρ + div~j = 0
∂t
6
(22)
9 Think abouts
•
What is particle-wave duality in Bohmian-mechanics?
•
How does the quantum potential interact with the particle itself?
•
Why does the electron move in orbits, when there is no uncertainty relation?
10 References
[1] David Bohm, “A Suggested Interpretation of the Quantum Theory in Terms of
"Hidden" Variables. I & II”, Physical Review, Jan. 15, 1952, Vol. 85 Nr. 2
[2] Gerd Ch. Krizek, “Einführung in die Bohmsche Interpretation der Quantenmechanik”, Jan. 31, 2006
[3] D. J. Bohm and B. J. Hiley, “The de Broglie Pilot Wave Theory and the Further
Development of New Insights Arising Out of It”, Foundation of Physics, Vol. 12,
No. 10, 1982
[4] D. Dürr, S. Goldstein, R. Tumulka, N. Zanghì, “Bohmian Mechanics”, Dec. 31,
2004
[5] B.J. Hiley, “Active Information and Teleportation”, Epistemological and Experimental Perspectives on Quantum Physics, eds. D. Greenberger et al. 113-126,
Kluwer, Netherlands, 1999
[6] Ilja Schmelzer, “Bohmsche Mechanik”, Uni München, April 10, 2001
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