An Alternative Formulation of Quantum Mechanics: Implications of Bohmian Mechanics in Chemistry

Of the several perplexing features of quantum mechanics, perhaps the most challenging to our notions of common sense is that particles do not have locations until they are observed. This is exactly what the standard view of quantum mechanics(SQM), popularly known as the Copenhagen interpretation tells us. Instead of a definite position and momentum assigned to a particle in Newtonian physics, we now have a cloud of probabilities described by a mathematical structure known as a wave function. This wave function evolves over time, which is governed by precise rules codified in something called the Schrödinger equation. The mathematics are clear enough but the actual whereabouts of particles aren’t so. Until a particle is observed, an act that causes the wave function to “collapse,” we have no definitive say about its location. Albert Einstein, among others, objected to this idea. As his biographer, Abraham Pais wrote: “We often discussed his notions of objective reality. I recall that during one walk Einstein suddenly stopped, turned to me and asked whether I really believed that the moon exists only when I look at it.”

“Since we cannot change reality, let us change the eyes which see reality.” ~ Nikos Kazantzakis.

But there’s another view — one that’s been around for almost a century — in which particles really do have precise positions at all times. This alternative formalism, known as the Bohmian mechanics(BM) never became as popular as the Copenhagen Interpretation. Surprisingly, this alternative formalism moderates the long-standing incompatibility between the chemical description of atoms and molecules and the description provided by the standard view of quantum mechanics. I will try to illustrate this parity problem between the descriptions of some very fundamental concepts in modern chemistry like orbitals, chemical bond in chemistry etc. and the descriptions supplied by SQM, and how that gets moderated from the BM perspective.

Bohmian Formalism of Quantum Mechanics:

A simplified formalism following the original paper (Bohm 1952)[1] is explained. The canonical form of the Schrödinger equation is

Eq. 1

where V is the potential and ψ the wavefunction. Since ψ is a complex function, we can write it as

Eq. 2

where P is the modulus and ϕ is the phase of ψ.

If we introduce eq. (2) into (1), we obtain two equations for P and ϕ:

Eq.3
Eq. 4

Eq. (3) ensures that probability is conserved and eq. (4) governs the dynamics of the quantum system. By comparing it with the Hamilton-Jacobi equation of classical mechanics, it is easy to see that eq. (4) has an extra term added to the potential V. Then, it is possible to imagine that this term plays the role of a new potential: the so-called quantum potential Uq can be defined as:

Eq. 5

Therefore if the total potential Vtotal = V + Uq, is introduced in eq. (5), an equation completely analogous to the classical Hamilton-Jacobi eq. (i) is obtained:

Eq. 6

The idea of David Bohm was to use Hamilton-Jacobi formalism in order to compute trajectories for quantum systems; From this perspective, the only difference between quantum mechanics and classical mechanics remains to be the quantum potential instead of a classical one.

By means of eq. (6), the trajectory of a quantum particle provided we know the initial conditions. The problem is that we cannot know the exact initial conditions of a quantum particle. For this reason, it is necessary to apply a statistical treatment to the possible initial conditions and, as a consequence, to the possible trajectories. The Bohmian Measurement Theory (Bohm 1953)[2] explains why the probability of finding the particle in a region of space is |ψ|2. Thus, according to BM, quantum phenomena are the result of statistical analysis over the possible trajectories of a particle. In this way, the new theory can reproduce all the results of SQM, but with the addition to that now both precise position and precise momentum can be simultaneously assigned to quantum systems. This theory supplies correct predictions for the results of measurements but offers a novel “classical” view of quantum phenomena. The price to be paid for accepting this theory is the introduction of a new strange force to account for the quantum potential.

The Notion of Orbital

The Oxford Dictionary of Chemistry says, both the wave function of the electron, as well as its corresponding spatial region of high electron density(computed as the square of the electron’s wavefunction), is called an ‘orbital’. The discontinuity between the two meanings of ‘orbital’ is not mathematical(for squaring a wave function isn’t ambiguous anyway) but belongs to a much deeper conceptual level.

The electronic density can be roughly interpreted as a mean of the definite positions occupied by an electron in its motion around the nucleus. This picture is based on conceiving electrons as traditional individual objects, with definite positions and velocities, whose only difference with the classical particles is that their behaviour is not governed by classical equations of motion but by an equation that determines their position only in a statistical way. But this view is incompatible with the Heisenberg principle of SQM, according to which quantum “particles” can have no trajectories. This is because the uncertainty principle forbids us to assign a position and momentum simultaneously to a quantum particle. It can be shown that: any assignment of a definite momentum and a definite position is logically forbidden by the very structure of the theory.

BM, by contrast, offers a completely different and less conflictive view where: the electron always has definite properties of position and velocity, so it can be conceived as a localized particle. In the ground state, the electron is at rest at a distance equal to the Bohr radius. This fact justifies the size of the atom that chemists write in the periodic table. Additionally, this picture offers an alternative explanation for the stability of matter: to say in Holland words: “if the particle is at rest relative to the nucleus, it is evidently not accelerating, and hence does not radiate. Therefore, it does not lose energy and it will not spiral into the nucleus, the outcome predicted by classical electrodynamics When the atom is in an excited state the electron is in an orbit around the nucleus.”

Structure of an Atom

In the framework of SQM, Quantum systems do not behave like “traditional” individual objects when the subsystems of a composite system are tried to be identified. To develop this argument, we do not necessarily need to consider many-electron systems, since the fact is manifested even in the simplest system, whose equation has a completely analytical solution: the hydrogen atom. Consider a free electron and a free proton are two quantum systems where each one can be represented by its own Hamiltonian. But in the hydrogen atom, they interact through a Coulombic potential, so the Hamiltonian of the Hydrogen atom becomes,

Eq. 7

where e is the charge of the electron, m denotes the mass of the respective particle, Q its position, P its momentum, and the subscripts e and p denotes electron and proton respectively. Alternatively, the Hamiltonian of the atom can also be expressed as

Eq.8

where K is the kinetic energy and U is the internal energy of the system. By means of a change of variables in terms of the centre of mass coordinates QCOM and PCOM, and the relative coordinates QREL and PREL :

Eq. 9
Eq. 10

where M is the total mass, and m is the reduced mass.

In this new coordinate system, the Hamiltonian can be written as,

Eq 11

where, K and W are:

Eq.12a and 12b

Till this point, this seems a mere change of variables, as that commonly used in classical mechanics. But now the quantum feature appears: since the kinetic energy K only depends on the total momentum, and the internal energy U only depends on differences of positions and on their derivatives, [K, U] = 0 and, allowing us to write the total Hamiltonian H of the atom as:

Eq. 13

where H is the Hamiltonian(w: Internal Energy and k: Kinetic Energy) and I the identity operators of the Hilbert spaces. Moreover, in a reference frame at rest with respect to the centre of mass, the Hamiltonian of the atom becomes just W . Therefore, it can be supposed that there are two equivalent ways of conceiving the hydrogen atom: one, as an electron and a proton in interaction, and the other as a single system characterized by the internal energy and represented by the Hilbert space HW. However, the two pictures are not equivalent, after the interaction between electron and proton, the hydrogen atom becomes a single system, represented by the Hilbert space HW in which the electrons and protons are no longer recognizable, but are inextricably “mixed” to constitute a new entity. Quantum “particles”, by contrast, do not preserve their identity after an interaction, and this fact does not depend on the complexity of the system, but on the very nature of SQM.

Bohmian quantum particles are individual objects in the “traditional” sense. They preserve their identity through time to the extent that they follow definite trajectories. They also preserve their identity as components of a composite system: for instance, the hydrogen atom is always composed of a proton in a fixed position and an electron at a definite distance from it. In this sense, this picture is in agreement with the view of atoms implicit in chemistry: the components of an atom are considered as “traditional” objects, whose only difference with respect to classical particles is that their behaviour is not governed by the classical equations of motion.

The Concept of Molecular Structure

The molecular structure is a 3D arrangement of atoms which constitutes the molecules. In classical terms, assigning definite spatial position for the atomic nuclei corresponding to the minimum energy which can be easily found from potential energy curves. This kind of figures, which represent the potential as a function of the distances between the nuclei of a molecule, is obtained on the basis of the assumption of the clamped nuclei strategy underlying the B-O approximation where: electrons are considered as moving under the influence of Coulomb potential produced by nuclei at rest, “clamped” at definite positions. This move simplifies calculations to a great deal, but at the cost of contradicting the Heisenberg principle, which forbids a quantum particle from simultaneously having definite values of position and of momentum. “In this clamping-down approximation, the atomic nuclei are treated essentially as classical particles having fixed positions and fixed momenta (to be precise, zero) assigned to them which violates the uncertainty principle.” (Chang 2015).

In BM, the difficulties related to the clamped nuclei assumption simply disappear. Since quantum particles are individual objects, they always have a definite position and velocity; therefore, the picture of nuclei at rest in precise locations is natural in the Bohmian framework. Therefore, the total potential can be computed without conceptual difficulties and the molecular structure which corresponds to the minima of the curve can be deduced.

The Chemical Bond

“The Chemical Bond” is rightly said to be the language of logic for chemists. But there is a great disparity between the simplistic picture familiar to chemists(atoms linked by their bonds) and the picture obtained from SQM. Let’s consider the case of the simplest molecule, the Hydrogen molecule. According to the traditional way of speaking in chemistry, two hydrogen atoms can be linked by a covalent bond to form a hydrogen molecule. But the problem is to make sense of the bond in the context of SQM.

In order to study the formation of a hydrogen molecule by means of BM, it is necessary to begin by considering two hydrogen atoms initially far apart and electrons in 1s state. When the atoms are brought closer, the wavefunction of the whole system can be computed by the following time-independent Schrödinger equation:

Eq. 14

Eq. (14) can be solved with the usual approximations: as well-known, it has a bonding and an antibonding solution. If we consider the σ bonding orbital of this molecule in the light of BM, the result is not less surprising than that of the hydrogen atom. For the case of nuclei, there are two kinds of interactions: repulsive electromagnetic interaction and attractive quantum interaction. There is an equilibrium point where electromagnetic and quantum forces cancel with each other So, the quantum force is responsible for the attraction between atoms and the formation of a stable bond. For the electrons, calculations show that they are at rest in the region close to the middle point of the line joining the two nuclei. Surprisingly, BM offers a picture of a chemical bond similar to the Lewis bond, but in this case on the basis of a fully articulated theory and of detailed computations.

Despite the conceptual advantages that BM may offer in the context of the foundations of chemistry, certain shortcomings cannot be ignored. Perhaps the main difficulty is to account for the quantum potential, which is responsible for the non-classical effects on the motion of the particles but, by contrast with any other physical potential, has no clear physical source. Of course, this fact does not affect the empirical equivalence of BM and SQM; however, it is a feature that makes Bohmian formulation unpalatable for many physicists, who are used to link any potential with a field, and the field with its source.

[Classical Mechanics :
Of the several equivalent formulations of classical mechanics, one such (which will be required later) is the Hamilton-Jacobi formalism. In this formulation, the motion is governed by the Hamilton-Jacobi equation,

where S is the action and V the potential. With this equation, the function S can be computed, and with it, the Hamilton-Jacobi formalism can be applied to compute the position and momentum of a particle for all times.]

– by Aniruddha Seal

References:

  1. Bohm, D. (1952). “A suggested interpretation of the quantum theory in terms of hidden variables.” Physical Review, 85 : 166-179.
  2. Bohm, D. (1953). “Proof that probability density approaches |ψ|2 in causal interpretation of the quantum theory.” Physical Review, 89 : 458-466.
  3. Woolley, R. G. (1978). “Must a molecule have a shape?” Journal of the American Chemical Society, 100 : 1073-1078.
  4. Landau, L. D. and Lifshitz, E. M. (1975). Mechanics. Amsterdam: Elsevier
  5. Goldstein, S. (2016). “Bohmian mechanics.” In E. Zalta (ed.), The Stanford Encyclopedia of Philosophy.
  6. Holland, P. (1993). Quantum Theory of Motion. Cambridge: Cambridge University Press.
  7. Chang, H. (2015). “Reductionism and the relation between chemistry and physics.” Relocating the History of Science: Essays in Honor of Kostas Gavroglu. New York: Springer.
  8. Fortin, Sebastian, Olimpia Lombardi, and Juan Camilo Martínez González. (2017). “The relationship between chemistry and physics from the perspective of Bohmian mechanics.” Foundations of Chemistry 19.1 : 43-59.

Image Sources:

Cover: https://www.forbes.com/sites/briankoberlein/2016/11/28/nasas-emdrive-and-the-quantum-theory-of-pilot-waves/#ed4ca7c4d2b1

About the Author

Aniruddha is an undergrad majoring in Chemistry with a Physics minor at NISER Bhubaneswar. He was among the finalists selected to represent India for the Chemistry Olympiad 2018. Scholarly pursuits aside, he spends a lot of time reading about history when he is not trying to do science. The Byzantine Empire is his long-standing interest.

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