6 The ’combined limit’ of quantum theory

Let us summarize what has been achieved so far. In section 3 the limit ℏ  →    0 has been performed for arbitrary states (including wave packets with fixed width ϵ ). The result of this first ’standard limiting process’ was a classical statistical theory referred to as PHJ. In section 4 the limit ϵ →    0 of PHJ has been performed. The result of this second ’deterministic limit’ was NM. Therefore NM is a subset of the classical limit PHJ of QT but NM is not the classical limit of QT, since we cannot neglect almost all of the (statistical) states of PHJ. Thus, the two limiting processes performed in this order have not led us from QT to NM in the sense that NM can be said to be the classical limit ℏ  →    0 of QT. In section 5 it has been shown that inverting the order of the two limiting processes (first ϵ  →    0 then ℏ  →     0 ) does not solve the problem either since the limit ϵ  →    0 (with ℏ fixed) does not exist. The two limiting processes clearly do not commute. Thus, it is impossible to obtain NM as the classical limit of QT, no matter which order of the two (separate) limiting processes is chosen.

Fortunately, we have still the option to combine both limits; i.e. we could assume that the width of the wave packets is a monotone function of ℏ . This means that the localization of wave packets (the ’deterministic limit’) and the change of the basic equations of QT (the ’standard limit’) takes place simultaneously in the limit ℏ  →     0 . Such states seem artificial from the point of view of experimental verification since the numerical value of ℏ is not under our control. Nevertheless, a construction of NM from QT along these lines would certainly provide a kind of justification for Dirac’s claim that QT reduces to NM in the limit ℏ  →     0 . Note also, that the subject of our study is essentially of a formal nature. We are asking whether or not all predictions of NM can be obtained by means of some limiting process ℏ  →     0 , from the basic equations of QT. There are no in-principle constraints how to perform his limit.

A brief look at the above examples for A  (t) shows that a linear relation between ℏ and ϵ seems most promising. Thus, we set ,
ϵ  =   k ℏ,

were k is an arbitrary constant. In order to use a notation similar to section 4 [see (11)], ϵ will be used instead of ℏ as small parameter; it may be identified with ℏ in most of the following relations. Let us perform the identification (25) for the two examples considered in section 5, with potentials V  (x  ) =   Vf   (x )  =   0 and                                    2
V  (x  ) =   Vh   (x )  =   (m  ω   ∕2  )xk   xk , respectively. Using the same initial conditions as in section 5, the solutions for ρ and S of (8), (9) take essentially the same form in both cases, namely

                                        (                                       )
                 (           )  3-      {               3                       }
                       1        2                1    ∑                        2
ρ (x,   t)  =       ---------     exp       -  ------       [x    -   r   (t) ]      ,                                        (26)
                    π  ϵ(t )            (      ϵ (t)           k        k       )

                              3                                                                     3
                 m   ϵ˙(t ) ∑                             1                                       ∑      ϵ               r (t )
S (x,   t)  =    ----------       [x    -  r  (t ) ]2 -   --p   (t )r   (t ) +   p   (t)x     -         ---- tan  -  1 ---i---(27.)
                                     i       i                k       k            k       k
                  4  ϵ (t )                               2                                             2k             kpi   (t)
                            i=1                                                                   i=1
The solutions ρf  , Sf and ρh,   Sh for particle ensembles in force-free regions and linear-force fields may be obtained from Eqs. (26), (27) by using different widths ϵ (t ) =   ϵf  (t) and ϵ (t ) =   ϵ   (t)
             h , as given by
               (                  )
                             2                             [                                          ]
                           t                                      2                1            2
ϵf (t )  =   ϵ    1  +   ---------   ,       ϵh (t ) =   ϵ    cos    ωt  +   ------------- sin    ωt     ,
                         k2m2                                                k2m2     ω2

and different trajectory components                (f )
r  (t )  =   r     (t )
  k           k and               (h )
r  (t )  =   r
 k            k (as well as momentum components p   (t)  =   m  r˙  (t )
  k               k ), as given by
  (f )          p0,k             (h )      p0,k
r     (t )  =   ------t,       r      =   -------- sin  ωt.
  k                              k
                 m                        m   ωt

As Eq. (28) shows, both widths are time-dependent; for the free-particle ensemble the width increases quadratically, for the bounded motion of the harmonic oscillator it varies periodically. However, both widths vanish in the limit ϵ  →    0 for arbitrary (finite) times t . This means that the deterministic probability density we were looking for is, in fact, created in this limit. The solutions for S are well-behaved at ϵ  =   0 . The limiting process in the continuity equation (8) can be performed in a similar way as in section 4 (an additional term due to the time-dependence of ϵ (t ) is regular at ϵ  =   0 ). The remaining steps, the derivation of Newton’s equations and their field-theoretic derivation from the QHJ can be performed in the same way as in section 4. Note also that the QHJ is regular at ϵ =   0 and differs in this limit from the HJ equation [cf. the discussion following Eq. (7)]. In view of a recent discussion [16][10] it should be noted that this field-theoretic limit is not equivalent to its projection on the trajectory.

The above solutions, with ϵ and ℏ considered as independent parameters (as in section 5), have been reported many times in the literature. It has also been pointed out that the special value ϵ =   ℏ ∕m    ω , in the harmonic oscillator example, produces the coherent states found by Schrödinger [22]. On the other hand, the relevance of the weaker statement ϵ =   k  ℏ for the classical limit problem has apparently not been recognized. It is not necessary to restrict oneself to the coherent states of the harmonic oscillator (the special case k  =   1 ∕m   ω ) in order to obtain deterministic motion; the latter may be obtained for a much larger class of force-free states, harmonic oscillator states, and constant-force states (this last example has not been discussed explicitly) as shown above. Summarizing this section, we found three potentials                 n
V  (x  ) ~    x   , n  =   0, 1,  2 which allow for a derivation of NM from QT in the limit ℏ   →    0 . For these potentials equations of motion for x¯  , p¯
  k     k exist, as mentioned already. Home and Sengupta [9] have shown that for these potentials the form of the quantum-mechanical solution may be obtained with the help of the classical Liouville theorem.