Transition to N particles as final step to non-relativistic quantum theory

In this section the present derivation of non-relativistic QT is completed by deriving Schrödinger’s equation for an arbitrary number $N$ of particles or, more precisely, for statistical ensembles of identically prepared experimental arrangements involving $N$ particles.

In order to generalize the results of sections 2 and 5, a convenient set of $n = 3N$ coordinates $q1, ...qn$ and masses $m1, ...mn$ is defined by

(q1, q2, q3, ...qn - 2, qn - 1, qn ) = (x1, y1, z1, ..., xN , yN , zN ) , (m1, m2, m3, ...mn - 2, mn - 1, mn ) = (m1, m1, m1, ..., mN , mN , mN ) .

(105)

The index $I = 1, ...N$ is used to distinguish particles, while indices $i, k, ..$ are used here to distinguish the $3N$ coordinates $q1, ...qn$ . No new symbol has been introduced in (105) to distinguish the masses $mI$ and $mi$ since there is no danger of confusion in anyone of the formulas below. However, the indices of masses will be frequently written in the form $m (i )$ in order to avoid ambiguities with regard to the summation convention. The symbol $Q$ in arguments denotes dependence on all $q1, ...qn$ . In order to generalize the results of section 8 a notation $xI ,k$ , $⃗xI$ , and $mI$ (with $I = 1, ..., N$ and $k = 1, 2, 3$ ) for coordinates, positions, and masses will be more convenient.

The basic relations of section 2, generalized in an obvious way to $N$ particles, take the form

∂ ρ (Q, t ) ∂ ρ (Q, t) ∂ S (Q, t) -------------- + ----------------- -------------- = 0 (106) ∂ t ∂ q m ∂ q k (k ) k -d------ ---1-------- qk = pk (107) dt m (k ) d -------------- ------- pk = Fk (Q, t ) (108) dt ∫ ---- q = dQ ρ (Q, t)q (109) k k

The following calculations may be performed in complete analogy to the corresponding steps of section 2. For the present $N -$ dimensional problem, the vanishing of the surface integrals, ocurring in the course of various partial integrations, requires that $ρ$ vanishes exponentially in arbitrary directions of the configuration space. The final conclusion to be drawn from Eqs. (106)- (109) takes the form

n ( )2 ∫ ∑ 1 ∂ S ∂ S ∂ ρ ---------- ------ + -----+V = L , dQ ------L = 0, 0 0 2m (j) ∂ qj ∂ t ∂ qk j=1

(110)

The remaining problem is the determination of the unknown function $L0$ , whose form is constrained by the condition defined in Eq. (110).

$L0$ can be determined using again the principle of minimal Fisher information, see I for details. Its implementation in the present framework takes the form

∫ ∫ δ dt dQ ρ (L - L0 ) = 0 (111) Ea = 0, a = S, ρ, (112)

The following calculation can be performed in complete analogy to the case $N = 1$ reported in section 5. All relations remain valid if the upper summation limit $3$ is replaced by $3N$ . This is also true for the final result, which takes the form

[ ] 2 2 ℏ--- -1-- --1-----∂-ρ---∂-ρ-- ---1-------∂--ρ----- L0 = - + . 4 ρ 2 ρ m (j )∂ qj ∂ qj m (j ) ∂ qj ∂ qj

(113)

If a complex-valued variable $ψ$ , defined as in (52), is introduced, the two basic relations $Ea = 0$ may be condensed into the single differential equation,

( ) ( ) N ℏ ∂ ∑ 1 ℏ ∂ ℏ ∂ [------ + ---------- ------------ ------------ + V ]ψ = 0, i ∂ t 2m i ∂ x i ∂ x I =1 (I ) I ,k I ,k

(114)

which is referred to as $N -$ particle Schrödinger equation, rewritten here in the more familiar form using particle indices. As is well-known, only approximate solutions of this partial differential equation of order $3N + 1$ exist for realistic systems. The inaccessible complexity of quantum-mechanical solutions for large $N$ is not reflected in the abstract Hilbert space structure (which is sometimes believed to characterize the whole of QT) but plays probably a decisive role for a proper description of the mysterious relation between QT and the macroscopic world.

Let us now generalize the Arunsalam-Gould method, discussed in section 8, to an arbitrary number of particles. We assume, that the considered $N -$ particle statististical ensemble responds in $N 2$ ways to the external electromagnetic field. This means we restrict ourselves again, like in section 6, 7 to spin one-half. Then, the state function may be written as $ψ (x , s ; ....x , s ; ...x , s ) 1 1 I I N N$ where $sI = 1, 2$ . In the first of the steps listed at the beginning of section 8, a differential equation, which is equivalent to Eq. (114) for single-valued $ψ$ but may give non-vanishing contributions for multi-valued $ψ$ , has to be constructed. The proper generalization of Eq. (103) to arbitrary $N$ takes the form

( ) ( ) N ℏ ∂ ∑ 1 ℏ (I ) ∂ ℏ (I ) ∂ [------+ ---------- ---σ --------- --σ --------- +V ]ψ = 0, k l i ∂ t 2m (I ) i ∂ xI ,k i ∂ xI ,l I =1

(115)

where the Pauli matrices $(I ) σ k$ operate by definition only on the two-dimensional subspace spanned by the variable $s I$ . In the second step we perform the replacement

N 3 ∫ i ∑ e ∑ ⃗xI ,t [ ] ψ ⇒ exp { - --- -I-- dx ′ A (x ′ , t′) - cdt ′ϕ (x ′, t ′) } ψ, I,k k I I ℏ c I =1 k=1

(116)

using a multi-valued phase factor, which is an obvious generalization of Eq. (104). The remaining steps, in the listing of section 8, lead in a straightforward way to the final result

( ) ∑N ∑N ∑ 3 2 ℏ ∂ 1 ℏ ∂ eI [-------+ eI ϕ (xI , t ) + ---------- ------------ - ----Ak (xI , t) i ∂ t 2m i ∂ x c I =1 I =1 k=1 (I ) I ,k , ∑N (I ) (I ) + μ σ B (x , t ) + V (x1, ..., x , t)]ψ = 0 B k k I N I =1

(117)

where $(I ) μ = - ℏe ∕2m c B I I$ and $B⃗ = rot A⃗$ . The mechanical potential $V (x , ..., x , t ) 1 N$ describes a general many-body force but contains, of course, the usual sum of two-body potentials as a special case. Eq. (117) is the $N -$ body version of Pauli’s equation and completes - in the sense discussed at the very beginning of this paper - the present derivation of non-relativistic QT.

9 Transition to N particles as final step to non-relativistic quantum theory