The ’deterministic limit’ of quantum theory

5 The ’deterministic limit’ of quantum theory

In QT, the coupling term in QHJ [see (9)] prevents a deterministic limit of the kind found for the PHJ. To see this, we start from the assumption that a quantum mechanical system exists which admits a solution of the form (11) for arbitrary $t$ . Inserting (11) into (8),(9) leads to two equations. The first is the continuity equation which takes the same form (13) as before. The second is the QHJ, which takes the form
$( ) ( )2 ( )2 { 3 } ∂ S 1 ∑ ∂ S 1 ℏ ∑ -----+ ------ ------- +V + ------ --- 3 ϵ - [x - r (t )]2 = 0. ∂ t 2m ∂ x 2m ϵ ( k k ) k k k=1$ (22)

Eq. (22) shows that the coupling term diverges (for finite $ℏ$ ) in the limit $ϵ → 0$ . Consequently, there is no reason to expect, that the second derivative of $S$ with respect to $x k$ [see Eq. (13)] is regular at $ϵ → 0$ and that a delta-function-like $ρ (x, t)$ , as given by Eq. (11), can be a solution of (8), (9). Thus, the deterministic limit of QT (if it exists) cannot be obtained in the same way as in the PHJ.

We next consider several concrete solutions of QT which lead to probability densities ’similar’ to (11). As a first example we consider an ensemble of free particles which are distributed at $t = 0$ according to a probability density (11) centered at $r (0 ) = 0 k$ [set $r (t ) = 0 k$ in (11)]. The initial value for $S (x, t)$ is given by $S (x, 0 ) = p x 0,k k$ , i.e. $S (x, t )$ fulfills at $t = 0$ the ’deterministic’ relation (20). These initial values describe for small $ϵ$ a localized, classical particle in the sense that there is no uncertainty with respect to position or momentum. A calculation found in many textbooks leads to the following solution of Schrödinger’s equation for $ρ$ :
$( ) ( ) 3- { 3 } 1 2 1 ∑ 2 ρ (x, t) = ---------- exp - -------- [x - r (t) ] , πA (t ) ( A (t) k k ) k=1$ (23)

where $mr (t ) = p t k 0,k$ and $A (t ) = A (t ) = ϵ [1 + ( ℏ ∕ ϵ)2 (t ∕m )2 ] f$ . We see from Eq. (23) that the peak of $ρ$ moves in agreement with NM, but the width of the wave packet increases with increasing time as well as with decreasing $ϵ$ . A complete localization can only be achieved at $t = 0$ . At later times the quantum uncertainty, due to the finite $ℏ$ , dominates the behavior of the ensemble completely, despite our choice of ’deterministic’ initial conditions.

As a second example, we consider an ensemble of particles moving in a harmonic oscillator potential $V (x ) = (m ω2 ∕2 )x x k k$ using exactly the same initial conditions as in the above example of force-free motion. The result for $ρ$ takes the same form (23) as for the force free ensemble, but with $mr (t ) = (p ∕ ω ) sin ωt k 0,k$ and
$[ ] ( )2 2 1 ℏ 2 A (t ) = A (t ) = ϵ cos ωt + ---------- --- sin ωt . h m2 ω2 ϵ$ (24)

The width $A (t ) h$ increases again with decreasing $ϵ$ and prevents again a deterministic limit. We mention, without going into details [23], that a third example showing the same behavior may be found, namely an ensemble of particles moving under the influence of a constant force.

The three examples considered in this section correspond to three potentials proportional to $n x k$ , where $n = 0, 1, 2$ . For these potentials the expectation values of the corresponding forces fulfill the relation $---------- --- Fk (x ) = Fk (x )$ . Therefore, equations of motions for $----- xk$ and $---- pk$ exist as a consequence of Ehrenfest’s theorem. Despite of these classical features, even these ’optimal’ states do not permit a deterministic limit of QT. We conclude, in agreement with common wisdom, that this limit does not exist.

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