Kinematic geometry of mass-triangles and reduction of Schrödinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters

Abstract

Schrödinger’s equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, ℝ⁹, naturally equipped with Jacobi’s kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger’s equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger’s equation to corresponding PDEs solely defined on triangular parameters—i.e., at the level of ℝ⁶/SO(3)—has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),ℝ⁶) which enable us to obtain such a reduction of Schrödinger’s equation of three-body systems to PDEs solely defined on triangular parameters.

In atomic and molecular physics, one naturally encounters the quantum mechanics of three-body systems such as helium atom He, the hydrogen molecular ion H₂⁺, the negative ion of hydrogen atom H⁻, etc. The fundamental equation for studying the physics of such systems is Schrödinger’s equation, namely

1

where U is the potential function, E is the energy level of a stationary state, and μ_j and Δ_j are, respectively, the individual mass and the Laplace-operator with respect to the coordinates of the individual particle, j = 1, 2, 3.

The study of the above Schrödinger’s equation of three-body systems has a long history, dating back to Hylleraas’ work on this subject in the 1930s, and there are an extensive literature and various approaches in the study of its solutions. The book of Bethe and Salpeter (1) and the Festschrift volume of Review of Modern Physics in honor of Hylleraas (2) contain many basic references on this topic.

The problem of maximizing the use of angular momentum conservation has, however, never been adequately treated. The usual procedure is to introduce approximations such as approximate reduced masses, etc. In this short article, I shall describe a geometric approach to the three-body problem which does not use any approximation or perturbation and hence applies just as well to general mass distributions as to lopsided ones. Following Jacobi, it is natural to introduce a Riemannian metric on the configuration space of a given mechanical system such that the kinetic energy of a motion is given by half of the square of speed. Such a kinematic Riemannian manifold is actually the geometric encoding of the kinetic energy of the given system and it always inherits an SO(3)-isometry from the rotational symmetry of the physical space. Moreover, the space of SO(3)-orbits also has an induced metric which measures the distances among neighboring orbits. We thus call it the orbital distance metric. In the special case of three-body systems, the orbit space, M̄, is exactly the “space of triangles” (i.e., each point of M̄ represents a class of congruent triangles) and the above orbital distance metric, ds̄², provides a natural geometric measurement of the difference in size and shape among triangles. Let I be the moment of inertia. Then ρ = is equal to the distance between the given triangle and the point-triangle, thus ρ provides a natural measurement of the size, while the 2-dimensional subspace M* ⊂ M̄ defined by setting ρ = 1 is the natural geometric representation of all possible shapes of triangles. It is a remarkable intrinsic beauty of the kinematic geometry of triangles that M* is always isometric to the euclidean hemisphere of radius ½, S₊²(½), (i.e., independent of the mass distribution).

The Schrödinger equation is naturally SO(3)-invariant, and this is the origin of angular momentum conservation in quantum mechanics. The above setting of SO(3)-orbital geometry of the configuration space enables us to maximize the use of angular momentum conservation [or rather, the SO(3)-invariance of three-body system] to reduce the Schrödinger equation to a system of l + 1 (resp l) PDEs at the level of (M̄, ds̄²), namely, solely in terms of triangular parameters, cf. Eqs. 20 and 20*.

A combination of the sphericality of M* and the above reduced equations will enable us to further reduce the solution of Eq. 1 to that of an infinite system of algebraic linear equations (3, 4).

Kinematic Geometry of Triangles and Specific Choice of Coordinates of the Configuration Space

Let μ = ∑ μ_j be the total mass, m_j = μ_j/μ be the percentages of mass, j = 1, 2, 3, and a_j be the position vectors. Following Jacobi, the kinematic metric on ℝ³ ⊕ ℝ³ ⊕ ℝ³ is defined by setting

2

Moreover, fixing the origin at the center of gravity enables us to reduce the configuration space to the 6-dimensional linear subspace ℝ⁶ given by ∑ m_ja_j = 0. The Laplace-operators on both ℝ⁹ and the above ℝ⁶ are given by

3

Thus, Schrödinger’s equation (1) can be rewritten as

1`

An important starting point for achieving the “SO(3)-reduction” of Eq. 1′ is the following specific, intrinsic choice of coordinate system in ℝ⁶, namely, setting

4

Then

5

Thus ℝ⁶ ≅ {(x, y) = (ξ₁, ξ₂, ξ₃, η₁, η₂, η₃), ξ_j, η_j ∈ ℝ} provides a nice cartesian coordinate system of (SO(3), ℝ⁶) which is advantageous for the SO(3)-harmonic analysis on its function space. Set

6

It is well known that every SO(3)-invariant polynomial of {ξ_j, η_j} can be uniquely expressed as a polynomial of {f₁, f₂, f₃}. Thus (f₁, f₂, f₃) also constitutes a natural coordinate system of the orbit space M̄. Set

7

and let (g_ij) be the inverse matrix of (g^ij). Then the orbital distance metric, ds̄², on M̄ is given by

8

Lemma 1. Let Ψ = Ψ(f₁, f₂, f₃) be an SO(3)-invariant function and Δ, Δ̄ be the Laplace operator on (ℝ⁶, ds²), (M̄, ds̄²) respectively. Then

9

where v = [det(g^ij)]^1/2 is a constant multiplier of the volume function of SO(3)-orbits.

Theorem (ref. 5). Set

10

Then

11

Moreover, the subset of θ = π/2 represents degenerate m-triangles and it contains three specific points with φ = φ_i, i = 1, 2, 3, representing the three shapes of binary collisions, where

12

Furthermore, the distance, r_ij, between the ith and jth particles is given by

13

Remarks. There are three sets of basic coordinates (i.e., triangular parameters) on M̄, namely, {I_j} is the natural one for kinematics, {f_j} is the natural one for algebra [such as SO(3)-representation] while (ρ, θ, φ) is geometrically the optimal one which reveals the sphericality of (M̄, ds̄²). Therefore, the coordinates (ξ_j, η_j) on ℝ⁶ and (f_j) on M̄ that are used in the next section are the advantageous ones for obtaining the reduction of Schrödinger’s equation from the level of ℝ⁶ to the level of M̄ = ℝ⁶/SO(3). However, after such a reduction has already been achieved, it is then natural to change to the spherical coordinates (ρ, θ, φ). For example, Lemma 1 can be rewritten as follows.

Lemma 1′. Let f = F(ρ, θ, φ) be an SO(3)-invariant function of ℝ⁶. Then

9`

The Reduction of Schrödinger’s Equation (1′) to PDE Solely in Terms of Triangular Parameters

The reduction of Schrödinger’s equation (1′) from the level of ℝ⁶ to the level of M̄ can be regarded as an improvement and a refinement of the usual version of “angular momentum conservation” in quantum mechanics. In terms of the above coordinates, the angular momentum operator is the following vector-value operator, namely

14

Moreover,

15

are some of its associated operators often useful in the study of angular momentum in quantum mechanics. Set

16

Then, it is easy to check that

17

and both h_α,β and h*_α,β are common zeros of Δ_ξ, Δ_η, and Δ_ξη. Hence, it follows from Eqs. 16 and 17 that all h_α,β^(k) and h_α,β^∗(k) are common zeros of Δ_ξ, Δ_η, and Δ_ξη.

Theorem 1 (ref. 6). Let ℂ[ξ, η] be the ring of complex coefficient polynomials in {ξ_j} and {η_j} and ℛ be the sub-ring of SO(3)-invariant ones (i.e., polynomials in {f_j}). Then ℂ[ξ, η] is a “vector space” over ℛ with {h_α,β^(k); h_α,β^∗(k)} as a specific basis, meaning that every element of ℂ[ξ, η] can be uniquely expressed as a linear combination of {h_α,β^(k); h_α,β^∗(k)} with coefficients in ℛ.

Remarks. The above specific family of basis-functions is characterized by the following properties up to constant factors:

(i) biharmonicity—i.e., common zeros of Δ_ξ, Δ_η, and Δ_ξη,

(ii) bihomogeneity in {ξ_j} and {η_j},

(iii) common eigenfunctions of J² and J₃, namely

18

Theorem 2 (ref. 6). For each given pair of azimuthal and magnetic quantum numbers (ℓ, m), |m| ≤ ℓ > 0, there exist two separate quantum types whose wave functions are respectively given by

19

where k = ℓ − m and {ψ_j} (resp {ψ_j^∗}) are (ℓ + 1)-tuple (resp ℓ-tuple) of SO(3)-invariant functions satisfying the following systems of PDE, respectively,

20

20*

where Δψ_j and Δψ_j^∗ are given by formula 9).

Remarks. C. N. Yang pointed out the following correlation between the above separation into two quantum types and the parity separation. Since the invariant functions are always even functions while h_α,β^(k) and h_α,β^∗(k) are even or odd according to (α + β) being even or odd, the two types of wave functions Ψ and Ψ* of Eq. 19 are always of different parity.

The proofs of the above results as well as the next stage of reduction to a system of algebraic linear equations will occur in succeeding papers, namely refs. 3–6.

ABBREVIATION

PDE

partial differential equation