The Quantum Mechanics of a Rolling Molecular “Nanocar”

We formulate a mathematical model of a rolling “molecular wheelbarrow”—a two-wheeled nanoscale molecular machine—informed by experiments on molecular machines recently synthesized in labs. The model is a nonholonomic system (briefly, a system with non-integrable velocity constraints), for which no general quantization procedure exists. Nonetheless, we successfully embed the system in a Hamiltonian one and then quantize the result using geometric quantization and other tools; we extract from the result the quantum mechanics of the molecular wheelbarrow, and derive explicit formulae for the quantized energy spectrum. We also study a few variants of our model, some of which ignore the model’s nonholonomic constraints. We show that these variants have different quantum energy spectra, indicating that in such systems one should not ignore the nonholonomic constraints, since they alter in a non-trivial way the energy spectrum of the molecule.


A Nonholonomic Chaplygin Systems
In brief, nonholonomic systems are mechanical systems with non-integrable velocity constraints (a precise definition will be presented shortly), and nonholonomic Chaplygin systems are nonholonomic systems with special (translation) symmetries. These concepts are defined in terms of a "mechanical system" on a smooth manifold. Definition 1. Let Q be a smooth n-dimensional Riemannian manifold with (Riemannian) metric g, and suppose that it is also connected and orientable. By a mechanical system on Q we will mean a pair (Q, L), where L : T Q → R is a regular Lagrangian of mechanical type: L = T − V , where T : T Q → R is the kinetic energy given by T (q,q) = 1 2 g ij (q)q iqj -where i, j = 1, . . . , n and g ij are the components of g-and V : Q → R is a smooth function-the potential energy (we identify V with its lift to T Q).
We note that we will adhere to the Einstein summation convention for repeated indices throughout.
Let us now add constraints to our mechanical system. Suppose that we now define a constraint distribution D ⊂ T Q by the one-forms {ω a } k a=1 , k < n, as We will assume that the constraints are linear and homogeneous, so that locally ω a (v) = c a j (q)q j , that the constraints are non-integrable, and that D has constant rank. Then the triple (Q, L, D) is known as a nonholonomic mechanical system [1], or simply a nonholonomic system for short. Now, suppose that a k-dimensional Lie group G acts on Q such that Q := Q/G is a manifold; this happens, for example, if G acts freely and properly on Q. Let g be the Lie algebra of G, and ξ Q the infinitesimal generator on Q corresponding to ξ ∈ g. We assume that its lifted action leaves L and D invariant, and that at each q ∈ Q, the tangent space T q Q can be decomposed as is the tangent to the orbit through q ∈ Q [1, Section 2.8]. Then we will call (Q, L, D, G) a Chaplygin nonholonomic mechanical system [1,2]. Chaplygin systems give rise to a principal bundle π : Q → Q, with principal connection A : T Q → g such that ker A = D. This connection can then be used to decompose any tangent vector v q ∈ T q Q into horizontal and vertical parts: We can now form the reduced velocity phase space T Q/G, and the Lagrangian L induces the reduced Lagrangian l : T Q/G → R satisfying L = l • π T Q , where π T Q : T Q → T Q/G is the standard projection. Furthermore, the decomposition (A.3) gives rise to the reduced constrained Lagrangian l c : T Q → R given by l c (r,ṙ) := L(q, hor(q)), where r = π(q) andṙ = T q π(q). Locally, we will write the reduced constrained Lagrangian as where henceforth Greek indices will range from 1 to m := dim Q = n−k, the indices a, b, c will range from 1 to k = dim G, and where V : Q → R is defined by V = V •π. Since we will be dealing exclusively with the reduced constrained Lagrangian, we will drop the overbar on V henceforth. The G αβ are the components of the metric on the reduced space Q induced by g according to G r (v r , w r ) := g q (hor(v q ), hor(w q )), where r = π(q).
In our paper we deal exclusively with the well-studied subclass where G = R l ×S k−l , where 0 ≤ l ≤ k, and such that L is G-invariant. These are called abelian Chaplygin nonholonomic mechanical systems [2]. (These nonholonomic systems have translational symmetry in some of the configuration variables.) We will henceforth refer to these systems simply as "Chaplygin systems." Since L is assumed to be Ginvariant, we have that l = L. We will therefore denote the corresponding reduced constrained Lagrangian l c by L c .
To arrive at the local equations of motion of a Chaplygin system we pick a local trivialization Q = Q × G, coordinatized by q = (r, s). The action of G is given by left translation on the second factor; the equations of motion then consist of a system of second-order ordinary differential equations on Q (the reduced system), together with a system of first-order constraint equations [1]: Here the star indicates that we have substituted the constraints (A.5b) into (A.5a) after differentiation, and are the components of the curvature of A. Since we have assumed that the constraints (A.5b) are nonintegrable, it follows that at least one of the components B a αβ is nonzero [1].
Now, from (B.1) it follows that the only possibly nonzero contribution from these inner products are from the terms where cos ϕ results from differentiation in µ x and sin ϕ from differentiation in µ y . But these inner products contain the integral which is zero since k ∈ Z. We conclude that the full energy of the molecular wheelbarrow is, to first-order in µ x and µ y , E (k,n) .

D Quantization of the Reduced System
Recall that Q 1 = S 1 × S 1 . The reduced Lagrangian L c : T Q 1 → R obtained by substituting the nonholonomic constraints into the Lagrangian in equation (5) of the paper is: The kinetic energy metric here is g c = diag{J, β}, and so all three pre-quantization requirements outlined in our article are trivially satisfied. The Hamiltonian operator defined by equation (15) in the article becomes:Ĥ We note that R = 0 (since the components of g c are constant). The corresponding time-independent Schrödinger equationĤ c (ψ r ) = Eψ r is: This rearranges to equation (19) in our article without the ma 2 2 /(24βJ) term (this is the R-correction, which is zero this time around). As such, all subsequent results are the same, except for the omission of ma 2 2 /(24βJ) in all formulas.

E Quantization of the Holonomic System
Whenφ(0) = 0, ϕ(t) = ϕ 0 , as explained below equation ( Because the constraints are integrable, x = a(cos ϕ 0 )θ + C 1 , y = a(sin ϕ 0 )θ + C 2 , (E.2) the system (E.3) is no longer nonholonomic, it is holonomic. In such systems the relations between the generalized coordinates (e.g., (E.2)) are used to reduce the system to a lower-dimensional one free of constraints. Indeed, without loss of generality we can set C 1 = C 2 = 0 (this effectively defines the origin of the coordinate system used to locate the center of mass of the wheelbarrow) and substitute the resulting relations from (E.2) into the Lagrangian in (E.3). Simplifying the result yields The quantum mechanics of this system is identical to that of a quantum particle on a one-dimensional ring. The associated reduced energy spectrum is equation (33) in our article.
The kinetic energy metric of L nr is g nr = diag(I 1 , I, I/4, m, m) (relative to the coordinate ordering (ϕ, θ, θ d , x, y)) and is positive definite. A straightforward check of the other requirements for geometric quantization shows that they are satisfied. The Hamiltonian operator is once again given by (15), but in this case the R-correction is zero. A straightforward calculation yields an energy spectrum similar to (26) for the same µ x = µ y = 0 case: c n (γ nr ), γ nr = Iα 2 , k, l ∈ Z, n = 0, 1, . . . , E nr (0,0,0) ≈ where for the last approximation we assumed large γ and again used the approximations (27). In the zero-potential case (i.e., α = 0), we obtain the energies E nr,0 (k,l,j) = 2 2 k 2 I 1 + l 2 I + 4j 2 I , k, l, j ∈ Z, E nr,0 (0,0,0) = 0. (F.4) (We have appended the superscript "nr,0" to distinguish this unconstrained, zero potential energy spectrum from the others.)