4. Classical interpretation of Schroedinger equation
4.1. Milne's nonlinear superposition formula
Eq. (21) is a nonlinear Ermakov-Milne-Pinney (EMP) ODE (Ermakov, 1880; Milne, 1930; Pinney, 1950) depending on the parameter which, for a stationary system, is the dimensionless constant of integration related to (cf. Eqs. (8), (15) and (22)). It can be recast into the well-known real-valued Schroedinger radial eigenproblem (Landau, 1966):
through the `nonlinear phase'
that depends on the arbitrary initial phase related to the parameter B. The dependence of the solution to the nonlinear ODE (21) on the independent parameters E and A has been highlighted by ad-hoc labels.
By defining the irregular (i.e. exploding) solution:
to the Schroedinger equation (23), Eqs. (24) and (26) yield the following nonlinear superposition formula:
Therefore the solution to the nonlinear ODE (21) is defined by the superposition of the regular Schroedinger solution (which is exponentially convergent because of the choice of E as an eigenvalue) and the irregular Schroedinger solution (which is exponentially divergent at the bounday of the system). These two functions and form a fundamental set of solutions to the Schroedinger equation (23) (Leonhardt & Raymer, 1996; Richter & Wunsche, 1996). Therefore their Wronskian remains constant with respect to the spatial variable . Moreover, this Wronskian invariant is equal to the constant of integration (22) (Milne, 1930):
This yields the alternative definition of the fundamental free parameter A of our theory as the invariant Wronskian of the stationary Schroedinger equation (23).
Eqs. (25-28) show that the phase effects in a stationary quantum system are unambiguously related to the existence of the irregular solution (26) of the Schroedinger equation (23). We recover a general property of 1-dimensional stationary QM (Reinisch, 1994, 1997): a complete physical (i.e. dynamical) description of the eigenstate (3) demands the account of quantum phase effects; and these phase effects demand to take into account not only the regular (normalized) eigenfunction that defines the particular discrete eigenvalue E, but also the irregular (divergent) solution related to by the Wronskian invariant (28). Then the `nonlinear phase' defined by Eq. (25), which is simply the polar angle of the `trajectory' of the system in the complex plane , allows us to build the momentum field of the classical Hamilton-Jacobi type as shown below.
4.2. Hamilton-Jacobi definition of the radial matter flow
The real-valued regular radial Schroedinger eigenfunction appears through Eq. (24) to be the steady-state superposition of two unbound problems that consist of incoming and outcoming partial-scattering radial waves along the radial degree of freedom . Hence the divergence of . Moreover the corresponding two-branch DeBroglie momentum field , which describes the radial velocity field in the two equally probable outcoming and incoming radial directions, is simply the gradient of the action of each of these waves, namely
(cf. Eqs. (20), (22) and (25)).
This is the fundamental result of our theory and it defines the nonlinear radial-velocity soliton field:
in terms of the (square of the) nonlinear eigenstate that is the solution to the EMP ODE (21), i.e. in terms of the (squares of the) regular eigenfunction and the irregular solution to the Schroedinger eigenproblem (23), as shown by Eq. (27).
As a matter of fact, never vanishes because of Eqs. (27-28) and it always diverges at the boundary of the system, due to the presence of the irregular mode in Eq. (27). As a consequence, the momentum field (29) has indeed a soliton profile whose `wings' do actually describe the tunnel effect. Observe finally that Eq. (29) is but the well-known classical Hamilton-Jacobi definition:
of the radial momentum field in terms of the action S defined by Eqs. (3) and (6). Indeed, by use of the spherical polar coordinates, we have:
(cf. Eqs. (6), (8), (13), (15) which provides the undeterminacy, and Eq. (17)). Note that the two last expressions are not to be confused with and which are the canonical momenta respectively conjugate to the spherical polar coordinates and . Only equals , which will be of great importance for the quantization of the system along the radial degree of freedom (cf. Sect. 5 below).
Eq. (30) yields:
Therefore measures the actual time interval that is spent by the particle between the spheres of radii r and during either its incoming or its outcoming radial motion described by the momentum field (29-32). In terms of (cf. Eq. (3)), we have (since ):
(cf. Eqs. (8), (15) and (32)). Integrating Eq. (34) over the spherical polar angles , and taking into account the normalization condition (12), together with Eqs. (3) and (32), yields:
Hence, by use of Eq. (33), we finally obtain:
This demonstrates the (local) classical statistical information born by the wavefunction square modulus in terms of the actual time interval dt that is being spent by the particle between the spheres of radii r and . Moreover, Eq. (34) displays the expected angular dependence of the statistical time-of-stay dt if the averaging over the polar angles is not performed.
4.3. Schroedinger equation and classical mechanics
Let us first briefly recall the present state of the art concerning the link between QM and CM. This is basically the so-called semiclassical limit of QM, and it is described by the WKB (Wentzel, Kramers, Brillouin) approximation that was elaborated the same year as the discovery of the Schroedinger equation itself (Messiah, 1962; Landau & Lifchitz 1966; Rae, 1992).
In the WKB description of QM, one assumes that is a function of and that its phase may be asymptotically expanded as a polynomial in . Inserting this polynomial into the Schroedinger equation and equating powers of yields a `classical wavefunction' that is compatible with the classical equation of motion obtained from Eqs. (4) and (31) by neglecting the second-derivative term in Eq. (4) (). The WKB approximation works all the better as the action S of the system is large compared with the quantum of action h, or equivalently as the energy eigenvalue E is large compared with the average energy gap between the energy levels.
It is, however, well-known that the conditions of validity of the WKB approximation are neither necessary nor sufficient to obtain classical motion: the basic drawback with the WKB method is that it attempts to formulate the classical limit in terms of properties of the external potential and the DeBroglie wavelength which do not make reference to the quantum state of the system (Holland, 1993). It is indeed fairly obvious that not all physically relevant Schroedinger wavefunctions do vary slowly within the space of a DeBroglie wavelength, and hence that their corresponding second-derivative terms in Eq. (4) cannot always be regarded as small compared with the rest of this equation. There is formally in QM a lack of such a parameter in the WKB approximation that would allow one to adjust the wavefunction amplitude to the WKB conditions of a slowly varying profile within a DeBroglie wavelength. Said otherwise: in standard QM, there is a formal missing link with CM.
We believe that this missing link can be provided by the irregular Schroedinger mode that yields the additional parameter A in accordance with Eq. (28). Indeed, in the classically allowed region, the art of using the Milne transformation (23-28) is to choose both the free Wronskian parameter A and the initial conditions of the EMP equation (21) for in such a way that:
On the other hand, in the classically forbidden region, is obviously dominated by the exploding irregular solution (cf. Eq. (27)). This yields the tunnel effect by use of Eqs. (29-32): there is indeed a non-zero, although vanishingly small, momentum field in the classically forbidden region where .
4.4. Choice of initial conditions
The technical procedure that performs the Milne transformation is the following. Assume that, for a particular value of the parameter , a specific choice of the initial conditions and at a given is performed that keeps the nonlinear mode smooth and slowly varying in the classically allowed region. Therefore: there and Eq. (21) yields:
We shall call the right-hand-side of Eq. (37) `the classical wavefunction' for reasons that are explained below.
The problem is now to determine , , and .
Eqs. (22) and (29-32) define the reduced radial momentum field throughout the classically allowed region:
Obviously, this yields the classical equation of motion (hence the label `cl' for the function defined by Eq. (37)):
Now recall that the normalized Schroedinger eigenfunction , associated with the corresponding discrete eigenvalue E, satisfies the Schroedinger equation (23) with prescribed (regular) boundary values. Therefore this mode is completely defined and, in particular, we have and where and are two finite real values. We shall regard the solution of the ODE (21) as an initial-value problem, according to the following choice of parameters:
(thus we demand a local extremum of the nonlinear amplitude at the abscissa ). Eqs. (27-28) and (40) yield the following system of three equations for the definitions of the two unknown and at :
There is no solution, except for the following particular choice of the parameters:
Therefore the initial conditions for the definition of the nonlinear mode are taken at a local extremum of both the normalized Schroedinger eigenfunction and this nonlinear mode, with a common amplitude given by Eqs. (37) and (40) (Reinisch, 1994; 1997).
4.5. Classical quantization
The angular momentum of the system, defined in accordance with Eq. (31), yields:
by use of Eqs. (6) and of the spherical polar coordinates. By considering for the solar system the largest possible z-component of the angular momentum , namely where N is the main quantum number of the Kepler system, and by changing into (cf. Eq. (62)), Eq. (44) allows us to recover Nottale's remarkable conclusion that it is the quantization of the ratio , rather than that of , which yields the distribution of angular momentum in the solar system (Nottale, 1993, 1996a,b, 1997).
On the other hand, the quantization along the radial degree of freedom yields, by use of Eq. (29):
where the radial quantum number is equal to the number of nodes of the Schroedinger eigenfunction defined by Eq. (23) (Reinisch, 1994; 1997).
This radial quantization is of course reminiscent of the `good old' Bohr-Sommerfeld quantization that reads (White, 1931, 1934; Messiah, 1962; Landau & Lifchitz 1966). But recall that this latter is obtained as the result of the semiclassical WKB approximation of QM (cf. Sect. 4.3) that assumes the action S of the system to be large compared with the quantum of action h (or the system energy E to be large compared with the energy gap between two adjacent levels), while such approximations are not being done here. Therefore the radial quantization Eq. (45) is exact , regardless of the values of S or E. Hence the remarkable conclusion that the `vacuum state' corresponding to bears one single quantum of action h instead of the well-known semiclassical value . This is due to the proper account of the tunnel-effect divergence of the radial nonlinear mode at the boundary of the system (Reinisch, 1994; 1997).
© European Southern Observatory (ESO) 1998
Online publication: August 6, 1998