## 4. Classical interpretation of Schroedinger equation## 4.1. Milne's nonlinear superposition formulaEq. (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): by use of the following transformation (Milne, 1930; Alijah et al., 1986; Reinisch, 1994): through the `nonlinear phase' that depends on the arbitrary initial phase related to the
parameter 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 This yields the alternative definition of the fundamental free
parameter 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 ## 4.2. Hamilton-Jacobi definition of the radial matter flowThe 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
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 (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 (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 ## 4.3. Schroedinger equation and classical mechanicsLet 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 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 - i) the normalized Schroedinger eigenfunction satisfies regular physical boundary conditions;
- ii) the mode that is irregular at the boundary of the system oscillates just out of phase with so as to keep the nonlinear eigenstate smooth and slowly varying (Milne, 1930; Alijah et al., 1986; Reinisch, 1994).
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 conditionsThe 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 (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: which yields: 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 quantizationThe 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 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
© European Southern Observatory (ESO) 1998 Online publication: August 6, 1998 |