Does Schrödinger’s Wave Mechanics Completely Determine a System's Motion, or Only Statistically?

Berlin, 5 May 1927

The opinion prevails that a complete temporal-spatial description of the motion of a mechanical system according to quantum mechanics does not exist.

It does not make sense to specify the instantaneous configuration and the instantaneous velocities of an atom’s electrons.

In contrast, Schrödinger’s wave mechanics suggests unambiguous assignment of the system’s motions to any solution to the wave equation.

Do these assignments do justice?

Let ψ be a solution to a Schrödinger equation belonging to a given potential energy function Φ … (1)

If a system with just one degree of freedom is involved, then ψ defines at every point of the trajectory the kinetic.

If ψ is given, then by (1), is defined at each configuration point, i.e., the kinetic energy L.

If a system of just one degree of freedom is involved, then L defines the velocity only ambiguously. The motion is fully defined if the condition is added that the velocity should change only continuously.

For systems with several degrees of freedom, this method fails, because the direction of the motion is not known. However, the following consideration leads to the goal.

We assume that it is possible to assign unambiguously different directions to function ψ at each point of the n-dimensional configuration space n, and to decompose the kinetic energy into n summands, each of which is unambiguously assigned to one of those directions.

One could then also assign to each of these directions a velocity in that direction corresponding to that summand. The resultant of all these velocities would then be the system’s velocity vector in configuration space.

The symbol in (1), according to Schrödinger, refers to a metric of the configuration space, which is characterized by

. … (2)

Then

Δψ 8π2- h2 --------( E Φ)ψ – + 0. = E Φ – Δψ 2L gμνqμqν · · ds2 dt2 ------- - =  (3)

holds, where signifies the second covariant spatial differential quotient of ψ in configuration space,

. … (3a) 

We want to denote as the “tensor of the ψ-curvature,” as the scalar of this tensor, as the “scalar of the ψ-curvature.”

Our first task is then to assign unambiguously n directions to the tensor of the ψ-curvature. If is a unit vector, which accordingly satisfies the equation

… (4)

then this direction vector together with defines the scalar.

We ask about the directions for which becomes an extremum. Lagrange’s method produces for this the conditions[6] . … (5)

They involve the determinant equation … (5a) which has n ¢real positive² roots .

If all of these are real and differ from one another, then the equations (5) define n directions, i.e., n unit vectors except for the sign (principal directions).

These directions are perpendicular to each other, as one recognizes if for the equations one multiplies the first by , the second by , and then substracts them both from each other.

These n directions determine an orthogonal local system of coordinates of the ’s, in which the metric (at the origin) is Euclidean ( ).

In general, we shall denote with an overline quantities that refer to the local coordinate system.

In this local coordinate system, the tensor of the ψ-curvature can be decomposed into summands, each of which is allocated to one of the principal directions, as, according to (3), . … (3b) Δψ gαβψαβ = ψαβ ψαβ ∂2ψ ∂qα∂qβ —————– αβ σ ¯ ¿∂qσ ® ¾ ­ ½∂ψ –= ψαβ Δψ Aα gμνAμAν 1= ψαβ ψμνAμAν ψA = A(μα) ψA ψμν λgμν)Aν – ( 0= ψμν λgμν – 0, = λα) ( A(μα) ψμν λ(α)gμν)A(να) – ( 0= ψμν λ(β)gμν)A(β) – ( ν 0= A(β) μ A(μα) ξα gμν δμν = ψαβ Δψ α ¦ψαα

Each corresponds, according to (1), to a summand of the system’s kinetic energy, which we assign to the pertinent principal direction.

In the local system, ac- cording to (2), . … (2a)

According to (1) and (3b), furthermore, . … (6)

I introduce the hypothesis that the velocity components in the principal directions correspond to their due proportions of the kinetic energy. Therefore, we set accordingly . … (7)

Thus the velocity components of the system (except for the sign) are determined by the wave function ψ, and there only remains the task of transferring this result to the original coordinate system.

Through application of (5) to the local system and to the fundamental direction, assigned to index α, one obtains .

But since is equal to 1 for , but otherwise vanishes, it follows from this or . … (8)

On the other hand, the ’s are the components of the unit vector in the principal direction α. Hence, is that portion of the μ-component of the velocity that emerges from the velocity of the system in the principal direction α.

Through summation over all the principal directions, we obtain … (9) From (9), taking (7) and (8) into account, one gets ψαα 2L 2· α ¦qα = 2L h2-ψαα 4π2 ψ - –—————–· ¹ § α ¦© = qα 2· h2-ψαα 4π2 —————– ψ - –= ψμν λ(α)δμν)A(να) – ( 0= A(να) ν α = ψαβ λ( α) δαβ = ψαα λ( α) = ψαβ 0 = (if α β) ≠ ( A(μα) qαA(μα) · qμ · · α ¦qα A(μα)

… (10) This equation, in connection with (5a), (5), and (4), solves the given task— provided the ’s are ¢everywhere² negative.

There are 2n possible velocities at- tached to each spot in the configuration space. This ambiguity is to be expected a priori, in view of the quasi-periodic motions.

These show that the assignment of fully defined motions to solutions of the Schrödinger differential equation is, at least from the formal stand-point, just as possible as the assignment of definite motions to solutions of the Hamilton-Jacobi differential equation in classical mechanics.

Postscript to the correction proofs.

Mr. Bothe[7] has meanwhile calculated the example of an anisotropic two-dimensional resonator according to the reasoning provided here and thereby found results that, from the physical point of view, must certainly be rejected.

Stimulated by this, I found out that this schema does not prop- erly take into account one general condition that must be set for a general law of motion of the systems.

Specifically, a system Σ is considered, which consists of two energetically mutually independent partial systems and this means that the potential energy as well as the kinetic energy are composed additively of two parts, the first of which contains only quantities with reference to the second, only quantities with reference to Then, as is known, , where depends only on the coordinates of depends only on the coordi- nates of In this case it must be required that the motions of the total system be combinations of possible motions of the partial systems.[8]

The indicated schema does not correspond to this condition. Namely, let μ be an index that belongs to one of the coordinates of ν be an index belonging to a coordinate of Then does not vanish. This is consequently connected (comp. 5a) to that the of Σ do not agree with the of and provided each of these systems is subjected to observation as an isolated system.

Mr. Grommer[9] has pointed out that this objection could be addressed by a modification of the presented reasoning, in which not the scalar ψ itself is applied to the definition of the principal directions, but the scalar.

The implementation poses no difficulty, but will be given when it has been confirmed by examples.

qμ · h 2𦱩 ------ λα ψ -----· -– ¹ § A(μα) α = λα- ψ ----- Σ1 Σ2 Σ1, Σ2. ψ ψ1 ψ2 ⋅ = ψ1 Σ1 ψ2 Σ2. Σ1, Σ2. ψμν λ(α)’s λ(α)’s Σ1 Σ2, χ lgψ