Energy conservation

6 Energy conservation

In the last section [section (5)] we derived a second differential equation (26) for our dynamical variables $ρ$ and $S$ . This equation has some terms in common with the Hamilton-Jacobi equation of classical mechanics but contains an unknown function $Q$ depending on $ρ$ and $S$ ; in principle it could also depend on $x$ and $t$ but this would contradict the homogeneity of space-time. We need further physical condition(s) to determine those functions $Q$ which are appropriate for a description of quantum mechanical reality or its classical counterpart.

A rather obvious requirement is conservation of energy. In deterministic theories conservation laws - and in particular the energy conservation law which will be considered exclusively here - are a logical consequence of the basic equations; there is no need for separate postulates in this case. In statistical theories energy conservation with regard to time-dependence of single events is of course meaningless. However, a statistical analog of this conservation law may be formulated as follows: “The statistical average of the random variable energy is time-independent”. In the present framework it is expressed by the relation
$[ ] ∫ ∫ d ∞ p2 ∞ ---- dpw (p, t) ------+ dx ρ (x, t )V (x ) = 0. dt - ∞ 2m - ∞$ (33)

We will use the abbreviation $---- --- ---- E = T + V$ for the bracket where $--- T$ denotes the first and $---- V$ denotes the second term respectively. Here, in contrast to the deterministic case, the fundamental laws [namely (2), (3), (7)] do not guarantee the validity of (33). It has to be implemented as a separate statistical condition. In fact, Eq. (33) is very simple and convincing; it seems reasonable to keep only those statistical theories which obey the statistical version of the fundamental energy conservation law.

In writing down Eq. (33) a second tacit assumption, besides the postulate of energy conservation, has been made, namely that a standard random variable “kinetic energy” exists; this assumption has already been formulated and partly justified in the last section. This means, in particular, that the probability density $w (p, t)$ , which has been introduced in the statistical conditions (2), (3) to obtain the expectation value of $p$ may also be used to calculate the expectation value of $2 p$ . This second assumption is - like the requirement of energy conservation - not a consequence of the basic equations (26), (7). The latter may be used to calculate the probability density $ρ$ but says nothing about the calculation of expectation values of $p -$ dependent quantities. Thus, Eq. (33) is an additional assumption, as may also be seen by the fact that two unknown functions, namely $h$ and $Q$ occur in (33).

Eq. (33) defines a relation between $Q$ and $h$ . More precisely, we consider variables $ρ$ and $S$ which are solutions of the two basic equations (26) and (7), where $Q$ may be an arbitrary function of $ρ$ , $S$ . Using these solutions we look which (differential) relations between $Q$ and $h$ are compatible with the requirement (33). Postulating the validity of (33) implies certain relations (yet to be found in explicit form) between the equations determining the probabilities and the equations defining the expectation values of $p$ -dependent quantities (like the kinetic energy).

In a first step we rewrite the statistical average of $p2$ in (33) using (21). The result is
$∫ ∞ ∫ ∞ ( )2 ∫ ∞ 2 ⋆ s ∂ 2 dpw (p, t )p = dx ψ ------- ψ+ dp h (p, t )p , i ∂ x - ∞ - ∞ - ∞$ (34)

as may be verified with the help of (20). Using (34), transforming to $ρ, S$ , and performing an integration by parts, the first term of (33) takes the form
$--- ∫ { [ ( ) ( ) ] ∞ 2 2 2 2 2 d-T-- --1--- -s---- ∂-ρ-- s--- ∂--ρ--- ∂-S-- ∂-ρ-- = dx 2 - 2 + dt 2m - ∞ 4ρ ∂ x 2 ρ ∂ x ∂ x ∂ t } [ 2 ] ∫ ∞ 2 ∂ ρ ∂ S ∂ S ∂ S ∂ h (p, t) p - 2 ---------- + ρ ------- ----- + dp ------------------. ∂ x ∂ x ∂ x2 ∂ t ∂ t 2m - ∞$ (35)

If we add the time derivative of $---- V$ to (35) we obtain the time derivative of $---- E$ , as defined by the left hand side of (33). In the integrand of the latter expression the following term occurs
$[ ] [ ] ( )2 2 ∂ S ∂ ρ ∂ ρ ∂ S ∂ S ∂ S ----- + 2mV ----- - 2 ---------- + ρ ------- ----- . ∂ x ∂ t ∂ x ∂ x 2 ∂ t ∂ x$ (36)

The two brackets in (36) may be rewritten with the help of (7) and (26). Then, the term (36) takes the much simpler form
$( ) - 1 ∂ ρ ∂ Q ∂ ρ 2m ----- ----------. ∂ x ∂ x ∂ t$ (37)

Using (35) and (37) we find that the statistical condition (33) implies the following integral relation between $Q$ and $h$ .
$∫ [ ( ) √ --] ∞ ∂ ρ - 1 ∂ Q s2 1 ∂ 2 ρ ∂ ρ ----- ------ --------------------- ----- dx - √ -- 2 - ∞ ∂ x ∂ x 2m ρ ∂ x ∂ t ∫ ∞ ∂ h (p, t ) p2 + dp -------------------= 0. - ∞ ∂ t 2m$ (38)

Let us first investigate the classical solution. We may either insert the classical, “hybrid” solution (10) for $w (p, t )$ directly into Eq.(33) or insert $h (p, t )$ according to (21) with $w (p, t)$ as given by (10) in (38), to obtain
$∫ ∞ ( ) - 1 ∂ ρ ∂ Q ∂ ρ dx ----- ----------- = 0, ∂ x ∂ x ∂ t - ∞$ (39)

which implies $∂ Q ∕ ∂ x = 0$ . Thus, the hybrid probability density (10) leads, as expected, to a classical (the equation for $S$ does not contain terms dependent on $ρ$ ) statistical theory, given by the Hamilton-Jacobi equation and the continuity equation. These equations constitute the classical limit of quantum mechanics which is a statistical theory (of type 2 according to the above classification) and not a deterministic (type 1) theory like classical mechanics. This difference is very important and should be borne in mind. The various ambiguities [27] one encounters in the conventional particle picture both in the transitions from classical physics to quantum mechanics and back to classical physics, do not exist in the present approach.

If we insert the quantum-mechanical result (29) with properly adjusted constant in (38), we obtain
$∫ ∞ p2 ------ dp h (p, t) = T0, - ∞ 2m$ (40)

where $T 0$ is an arbitrary time-independent constant. This constant reflects the possibility to fix a zero point of a (kinetic energy) scale. An analogous arbitrary constant $V0$ occurs for the potential energy. Since kinetic energy occurs always (in all physically meaningful contexts) together with potential energy, the constant $T 0$ may be eliminated with the help of a properly adjusted $V0$ . Therefore, we see that - as far as the calculation of the expectation value of the kinetic energy is concerned - it is allowed to set $h = 0$ . Combined with previous results, we see that $h$ may be set equal to $0$ as far as the calculation of the expectation values of $n p$ , for $n = 0, 1, 2$ is concerned. These cases include all cases of practical importance. A universal rule for the calculation of averages of arbitrary powers of $p$ is not available in the present theory. The same is true for arbitrary powers of $x$ and $p$ . Fortunately, this is not really a problem since the above powers cover all cases of physical interest, as far as powers of $p$ are concerned (combinations of powers of $x$ and $p$ do not occur in the present theory and will be dealt with in a future work).

It is informative to compare the present theory with the corresponding situation in the established formulations of quantum mechanics. In the conventional quantization procedure, which is ideologically dominated by the structure of particle mechanics, it is postulated that all classical observables (arbitrary functions of $x$ and $p$ ) be represented by operators in Hilbert space. The explicit construction of these operators runs into considerable difficulties [42] for all except the simplest combinations of $x$ and $p$ . But, typically, this does not cause any real problems since all simple combinations (of physical interest) can be represented in a unique way by corresponding operators. Thus, what is wrong - or rather ill-posed - is obviously the postulate itself, which creates an artificial problem. This is one example, among several others, for an artificial problem created by choosing the wrong (deterministic) starting point for quantization.

If we start from the r.h.s. of (38) and postulate $h = 0$ , then we obtain agreement with the standard formalism of quantum mechanics, both with regard to the time evolution equation and the rules for calculating expectation values of $p -$ dependent quantities. Thus, $h = 0$ is a rather strong condition. Unfortunately, there seems be no intuitive interpretation at all for this condition. It is even less understandable than our previous formal postulate leading to Schrödinger’s equation, the requirement of a complex state variable. Thus, while we gained in this section important insight in the relation between energy conservation, time-evolution equation and rules for calculating expectation values, still other methods are required if we want to derive quantum mechanics from a set of physically interpretable postulates.

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