The Long Journey to the Schrödinger Equation

Abstract

This chapter marks the beginning of an itinerary that will take us to the quantum theory of matter. All along this exciting journey we will be accompanied by the zero-point radiation field introduced in Chap. 3, considered as a real, fluctuating field in permanent interaction with matter. Our point of departure is the (nonrelativistic) Abraham-Lorentz!equationAbraham-Lorentz equation governing the particle motion under the action of this field plus an arbitrary external (binding, conservative) force. A statistical treatment leads to a generalized phase-space Fokker-Planck equation!and laws of evolutionFokker-Planck equation. In the transition to configuration space through a partial averaging, a hierarchy of equations is obtained for the local moments of the momentum. When a balance is eventually reached in the mean between the energy lost by the particle through radiation reaction Radiation reactionand the energy gained by it from the background field, any remaining effect of the radiation terms becomes negligible. In this (time-asymptotic) limit, theFokker-Planck equation!generalized generalized Fokker-Planck equation!and laws of evolutionFokker-Planck equation transforms into a true Fokker-Planck equationFokker-Planck equation, describing a Markov processMarkov process. Further, in the Approximation!radiationlessradiationless approximationRadiationless approximation Linear sed!radiationless approximation the first two equations of the hierarchy decouple from the rest and are shown to be equivalent to Schrödinger’s equation. The main lessons and implications of these results are discussed. In particular, an explanation is given for the impossibility to go back from the Schrödinger description Quantum regime!and Schrödinger descriptionand retrieve a fullDetailed energy balance!and Schrödinger equation phase-space description that is consistent with quantum mechanics.

I am, in fact, rather firmly convinced that the Essentially statistical theory essentially statistical character of contemporary quantum theory is solely to be ascribed to the fact that this [theory] operates with an incomplete description of physical systems.

[In] a complete physical description, the statistical quantum theory would $\ldots$ take an approximately analogous position to the statistical mechanics within the framework of classical mechanics $\ldots$

A. Einstein (1949)

$\ldots$ I think that we cannot afford to neglect any possible point of view for looking at Quantum Mechanics and in particular its relation to Classical Mechanics. Any point of view which gives us any interesting feature and any novel idea should be closely examined to see whether they suggest any modification or any way of developing the theory along new lines.

P. A. M. Dirac (1951)

Appendices

Appendix A: Derivation of the Generalized Fokker-Planck Equation

In this appendix the generalized Fokker-Planck equation!and laws of evolutionFokker-Planck equation associated with the stochastic Eq. (4.3) is derived, borrowing from de laDe la Peña, L. Cetto, A. M. Peña and Cetto (1977a). For the sake of simplicity the derivation is presented in one dimension. The Fokker-Planck equation!generalizedgeneralization to three dimensions is made at the end of the appendix.

As discussed in the text, for any given realization of the field the density $R(x,p,t)$ of points in the phase space of the particle satisfies the continuity Eq. (4.12),

$$\begin{aligned} \frac{\partial R}{\partial t}+\frac{1}{m}\frac{\partial }{\partial x}pR+\frac{\partial }{\partial p}\left( f+m\tau \dddot{x}+eE\right) R=0. \end{aligned}$$

(A.1)

The differential equation for $Q,$ the mean value of $R$ over the field realizations $\left\{ (i)\right\}$, can be constructed by means of the smoothing method (see e.g. Frisch 1968)Bharucha-Reid, A. T.,Frisch, U. as follows. A smoothing operator $\hat{P}$ is introduced, which acts on any phase function $A(x,p,t)$ by giving its local (in the particle phase space) average,

$$\begin{aligned} \hat{P}A=\overline{A}^{(i)}, \quad \text {so} \quad A=\overline{A}^{(i)}+(1-\hat{P})A. \end{aligned}$$

(A.2)

Clearly $\delta A=(1-\hat{P})A$ is the random component of $A,$ which means that the second Equation in (A.2) is a decomposition of $A$ into its average $\overline{A}$ plus its fluctuating part. Further, $\hat{P}=\hat{P}^{2},$ so $\hat{P}$ is a projection (idempotent) operator. The application of this smoothing operator to the density $R$ separates it into its average and its random parts $Q$ and $\delta Q$, respectively,

$$\begin{aligned} R=Q+\delta Q,\mathrm \quad Q=\hat{P}R,\quad \delta Q=(1-\hat{P})R. \end{aligned}$$

(A.3)

We are interested in constructing the differential equation for $Q$. For this purpose we rewrite Eq. (A.1) in the form

$$\begin{aligned} \frac{\partial }{\partial t}\left( Q+\delta Q\right) +\hat{L}\left( Q+\delta Q\right) =-e\frac{\partial }{\partial p}E\left( Q+\delta Q\right) , \end{aligned}$$

(A.4)

where $\hat{L}$ stands for the (nonrandom)Liouville operator for the particle, including the radiation-reaction force $m\tau \dddot{x}$—strictly speaking, the operator $\hat{L}$ differs from a true Liouville operator due to the (small) radiation reaction term—,

$$\begin{aligned} \hat{L}=\frac{1}{m}\frac{\partial }{\partial x}p+\frac{\partial }{\partial p}\left( f+m\tau \dddot{x}\right) . \end{aligned}$$

(A.5)

Equation (A.4) becomes separated into its nonstochastic and fluctuating parts by applying to it the projection operators $\hat{P}$ and $1-\hat{P}$ in succession. Using that $\hat{P}E=0$ [see Eq. (4.4)], one thus obtains the couple of equations

$$\begin{aligned} \left( \frac{\partial }{\partial t}+\hat{L}\right) Q=-e\frac{\partial }{\partial p}\hat{P}E\delta Q, \end{aligned}$$

(A.6)

$$\begin{aligned} \left( \frac{\partial }{\partial t}+\hat{L}\right) \delta Q=-e\frac{\partial }{\partial p}EQ-e\frac{\partial }{\partial p}\left( 1-\hat{P}\right) E\delta Q. \end{aligned}$$

(A.7)

The next step is to eliminate $\delta Q$ from these equations. This can be achieved by introducing the operator $\hat{G}=(\partial /\partial t+\hat{L})^{-1},$ which corresponds to the Green function of the differential operator $\partial /\partial t+\hat{L},$ so for any phase function $A(x,p,t)$ one has

$$\begin{aligned} \hat{G}A(x,p,t)=\int \nolimits _{-\infty }^{t}e^{-\hat{L}\left( t-t^{\prime }\right) }A(x,p,t^{\prime })dt^{\prime }. \end{aligned}$$

(A.8)

The operator $\hat{G}$ is now used to invert Eq. (A.7),

$$\begin{aligned} \delta Q=-e\hat{G}\frac{\partial }{\partial p}EQ-e\hat{G}\frac{\partial }{\partial p}\left( 1-\hat{P}\right) E\delta Q, \end{aligned}$$

(A.9)

or, even better,

$$\begin{aligned} \left[ 1+e\hat{G}\frac{\partial }{\partial p}\left( 1-\hat{P}\right) E\right] \delta Q=-e\hat{G}\frac{\partial }{\partial p}EQ. \end{aligned}$$

(A.10)

Applying from the left the inverse of the operator in square brackets gives an expression for $\delta Q,$ which combined with Eq. (A.6) gives a complicated integro-differential equation for $Q,$

$$\begin{aligned} \left( \frac{\partial }{\partial t}+\hat{L}\right) Q=e^{2}\frac{\partial }{\partial p}\hat{P}E\left[ 1+e\hat{G}\frac{\partial }{\partial p}\left( 1-\hat{P}\right) E\right] ^{-1}\hat{G}\frac{\partial }{\partial p}EQ. \end{aligned}$$

(A.11)

This is the gfpe for the problem. However, this is a formal expression in which the random field and the operator $\hat{G}$ appear in a form which makes it quite impractical to use. A more manageable form is obtained by formally expanding the expression within square brackets into a power series,

$$\begin{aligned} \left( \frac{\partial }{\partial t}+\hat{L}\right) Q&=e^{2}\frac{\partial }{\partial p}\hat{P}E\sum _{k=0}^{\infty }\left[ -e\hat{G}\frac{\partial }{\partial p}\left( 1-\hat{P}\right) E\right] ^{k}\hat{G}\frac{\partial }{\partial p}EQ \\&=-e\frac{\partial }{\partial p}\hat{P}E\sum _{k=0}^{\infty }\left[ -e\hat{G}\frac{\partial }{\partial p}\left( 1-\hat{P}\right) E\right] ^{k+1}Q \end{aligned}$$

$$\begin{aligned}&=-e\frac{\partial }{\partial p}\hat{P}E\sum _{k=1}^{\infty }\left[ -e\hat{G}\frac{\partial }{\partial p}\left( 1-\hat{P}\right) E\right] ^{k}Q. \end{aligned}$$

(A.12)

To write the second equality we used the fact that $EQ$ can also be written as $(1-\hat{P})EQ.$

To somewhat simplify Eq. (A.12) we take into consideration some statistical properties of the random field. First of all, for the zpf the distribution is symmetric and centered around zero, so that the average of products of an odd number of factors vanishes,

$$\begin{aligned} \hat{P}E(t_{1})E(t_{2})\ldots E(t_{2n+1})\hat{P}A=0. \end{aligned}$$

(A.13a)

Assuming the distribution to be Gaussian, the average of products with an even number of factors take the form Gardiner, C. W.(Gardiner 1983, Sect. 2.8; Wang and Uhlenbeck Uhlenbeck, G. E. Wax, N.in Wax 1954/1985, Sect. 9a)²⁶

$$\begin{aligned} \hat{P}E(t_{1})E(t_{2})\ldots E(t_{2n})\hat{P}A=\sum \overline{E(t_{i})E(t_{j})}\ldots \overline{E(t_{r})E(t_{s})}^{(i)}\,\overline{A}^{(i)}, \end{aligned}$$

(A.13b)

where the sum is to be effected over all possible different pairs of factors.

Now from Eq. (A.13a) we note that all terms on the right-hand side of (A.12) with even $k$ vanish, so the equation simplifies into

$$\begin{aligned} \left( \frac{\partial }{\partial t}+\hat{L}\right) Q=e\frac{\partial }{\partial p}\hat{P}E\sum _{k=0}^{\infty }\left[ e\hat{G}\frac{\partial }{\partial p}\left( 1-\hat{P}\right) E\right] ^{2k+1}Q, \end{aligned}$$

(A.14)

which is equivalent to

$$\begin{aligned} \left( \frac{\partial }{\partial t}+\hat{L}\right) Q=e^{2}\frac{\partial }{\partial p}\hat{P}E\hat{G}\frac{\partial }{\partial p}E\sum _{k=0}^{\infty }\left[ e\hat{G}\frac{\partial }{\partial p}\left( 1-\hat{P}\right) E\right] ^{2k}Q. \end{aligned}$$

(A.15)

This is the generalized Fokker-Planck equation!and laws of evolutionFokker-Planck equation (gfpe) that we use here. The generalization to three dimensions is straightforward and gives (summation over repeated indices is to be understood)

$$\begin{aligned} \left( \frac{\partial }{\partial t}+\hat{L}\right) Q=e^{2}\frac{\partial }{\partial p_{i}}\hat{P}E_{i}\hat{G}\frac{\partial }{\partial p_{j}} E_{j}\sum _{k=0}^{\infty }\left[ e\hat{G}\frac{\partial }{\partial p_{l}}\left( 1-\hat{P}\right) E_{l}\right] ^{2k}Q, \end{aligned}$$

(A.16)

where

$$\begin{aligned} \hat{L}=\frac{1}{m}\frac{\partial }{\partial x_{i}}p_{i}+\frac{\partial }{\partial p_{i}}\left( f_{i}+m\tau \dddot{x}_{i}\right) \end{aligned}$$

(A.17)

is the Liouville operator with the radiation reactionRadiation reaction force added as an ‘external’ force. Substitution of this expression into (A.16) gives Eq. (4.13).

The above derivations have been made in terms of operators; a more common form involves the Green function, as follows. The differential equation for the Green function $\mathcal {G}$ of the Liouville equationLiouville equation is

$$\begin{aligned} \left( \frac{\partial }{\partial t}+L\right) \mathcal {G}=\delta (\varvec{x,x}^{\prime })\delta (\varvec{p,p}^{\prime }), \end{aligned}$$

(A.18)

in terms of which the evolution of a dynamical variable $A(\varvec{x},\varvec{p},t),$ described above with the aid of the operator $e^{-\hat{L}\left( t-t^{\prime }\right) },$ is given by the expression

$$\begin{aligned} e^{-\hat{L}(t-t^{\prime })}A(\varvec{x},\varvec{p},t^{\prime })=\int dx^{\prime }dp^{\prime }\mathcal {G}(\varvec{x},\varvec{p};\varvec{x}^{\prime },\varvec{p}^{\prime };t-t^{\prime })A(\varvec{x}^{\prime },\varvec{p}^{\prime },t^{\prime }), \end{aligned}$$

(A.19)

where the prime refers to the values of the dynamical variables at $t^{\prime }<t,$ subject to the condition $\left. A(\varvec{x}^{\prime },\varvec{p}^{\prime },t^{\prime })\right| _{t}=A(\varvec{x},\varvec{p},t).$ The evolution from ($\varvec{x}^{\prime },\varvec{p}^{\prime }$) to ($\varvec{x},\varvec{p}$) is deterministic, as follows from the (modified) Liouville operator (A.17).

Appendix B: Diffusion Coefficients in the Markovian Approximation

We now proceed to the derivation of a simplerDiffusion coefficient!Markovian approximation version of Eq. (4.13), by considering the approximation to second order of the gfpe, known as Approximation!MarkovianMarkovian approximation. The results obtained are suitable to develop the sed theory further, once the system has reached a reversible condition, which is called quantum regime Quantum regimefor the reasons mentioned in Sect. (4.4.4). It is important to remark that between the gfpe with an infinity of terms and the fpe written to order $n=2$ in the derivatives, there is no intermediate approximation for a positive probability density. In fact, any truncation of the gfpe above $n=2$ will automatically revert the expansion to second order for a nonnegative $Q$ (all higher-order coefficients vanish). Thus, there are only three nontrivial possibilities: truncation at $n=1,$ which corresponds to deterministic (Newtonian) processes (described by the Liouville equationLiouville equation); truncation at $n=2,$ which corresponds to diffusions (Markovian) processes (described by a true fpe, which may be only approximate); and, finally, no truncation at all, which corresponds to the gfpe of infinite order (A.16). This is the essential content of Pawula’s theorem, a detailed discussion of which can be seen in Risken, H.Risken (1984).

To get the (true, but approximate) fpe (4.19),

$$\begin{aligned}&\frac{\partial Q}{\partial t}+\frac{1}{m}\frac{\partial }{\partial x_{i}}p_{i}Q+\frac{\partial }{\partial p_{i}}\left( f_{i}+m\tau \dddot{x}_{i}\right) Q \end{aligned}$$

$$\begin{aligned}&\quad =\frac{\partial }{\partial p_{i}}D_{ij}^{pp}\frac{\partial Q}{\partial p_{j}}+\frac{\partial }{\partial p_{i}}D_{ij}^{px}\frac{\partial Q}{\partial x_{j}}, \end{aligned}$$

(B.1)

we start from the gfpe (4.13). The first step is to write Eq. (4.14) to first order in $e^{2}$ (i.e., to take only the term of the sum with $k=0$), which is

$$\begin{aligned} \left. \mathcal {\hat{D}}_{i}Q\right| _{k=0}=\hat{P}E_{i}\hat{G}\frac{\partial }{\partial p_{j}}E_{j}Q=\hat{P}E_{i}(t)\int \nolimits _{-\infty }^{t}dt^{\prime }e^{-\hat{L}\left( t-t^{\prime }\right) }\frac{\partial }{\partial p_{j}}(E_{j}Q)(t^{\prime }), \end{aligned}$$

(B.2)

where Eq. (4.15) has been used. Since $E_{j}$ depends on time only, this can be written in the form

$$\begin{aligned} \left. \mathcal {\hat{D}}_{i}Q\right| _{k=0}=\mathop {\displaystyle \int }\nolimits _{-\infty }^{t}dt^{\prime }\overline{E_{i}(t)E_{j}(t^{\prime })}^{(i)}e^{-\hat{L}\left( t-t^{\prime }\right) }\frac{\partial }{\partial p_{j}}Q(t^{\prime }). \end{aligned}$$

(B.3)

Now, the evolution law

$$\begin{aligned} Q(t)=e^{-\hat{L}\left( t-t^{\prime }\right) }Q(t^{\prime }) \end{aligned}$$

(B.4)

allows us to write, inserting the identity operator $e^{\hat{L}\left( t-t^{\prime }\right) }e^{-\hat{L}\left( t-t^{\prime }\right) },$

$$\begin{aligned} e^{-\hat{L}\left( t-t^{\prime }\right) }\frac{\partial }{\partial p_{j}}Q(t^{\prime })&=\left( e^{-\hat{L}\left( t-t^{\prime }\right) }\frac{\partial }{\partial p_{j}}e^{\hat{L}\left( t-t^{\prime }\right) }\right) \left( e^{-\hat{L}\left( t-t^{\prime }\right) }Q(t^{\prime })\right) \end{aligned}$$

$$\begin{aligned}&=\frac{\partial }{\partial p_{j}^{\prime }}Q(t), \end{aligned}$$

(B.5)

where, as explained in Appendix A, the prime refers to the values of the dynamical variables at $t^{\prime }\le t,$ from where they follow a deterministic evolution to their end values at time $t$. Substitution into Eq. (B.3) using Eq. (4.9) gives

$$\begin{aligned} \left. \mathcal {\hat{D}}_{i}Q\right| _{k=0}=\mathop {\displaystyle \int }\nolimits _{-\infty }^{t}dt^{\prime }\varphi (t-t^{\prime })\delta _{ij}\frac{\partial }{\partial p_{j}^{\prime }}Q(x,p,t). \end{aligned}$$

(B.6)

From the chain rule for the derivation,

$$\begin{aligned} \frac{\partial Q}{\partial p_{j}^{\prime }}=\frac{\partial p_{k}}{\partial p_{j}^{\prime }}\frac{\partial Q}{\partial p_{k}}+\frac{\partial x_{k}}{\partial p_{j}^{\prime }}\frac{\partial Q}{\partial x_{k}}, \end{aligned}$$

(B.7)

it follows that

$$\begin{aligned} \left. \mathcal {\hat{D}}_{i}Q\right| _{k=0}=\left( \mathop {\displaystyle \int }\nolimits _{-\infty }^{t}dt^{\prime }\varphi (t-t^{\prime })\frac{\partial p_{j}}{\partial p_{i}^{\prime }}\right) \frac{\partial Q}{\partial p_{j}}+\left( \mathop {\displaystyle \int }\nolimits _{-\infty }^{t}dt^{\prime }\varphi (t-t^{\prime })\frac{\partial x_{j}}{\partial p_{i}^{\prime }}\right) \frac{\partial Q}{\partial x_{j}}. \end{aligned}$$

(B.8)

Direct substitution into (4.13) gives the (approximate) equation

$$\begin{aligned} \frac{\partial Q}{\partial t}+\frac{1}{m}\frac{\partial }{\partial x_{i}}p_{i}Q+\frac{\partial }{\partial p_{i}}f_{i}Q+m\tau \frac{\partial }{\partial p_{i}}\dddot{x}_{i}Q=\frac{\partial }{\partial p_{i}}\left( D_{ij}^{pp}\frac{\partial Q}{\partial p_{j}}+D_{ij}^{px}\frac{\partial Q}{\partial x_{j}}\right) , \end{aligned}$$

(B.9)

with the diffusion coefficients in the Markovian approximation Markovian approximationgiven by

$$\begin{aligned} D_{ij}^{pp}&=e^{2}\mathop {\displaystyle \int }\nolimits _{-\infty }^{t}dt^{\prime }\varphi (t-t^{\prime })\frac{\partial p_{j}}{\partial p_{i}^{\prime }},\end{aligned}$$

(B.10a)

$$\begin{aligned} D_{ij}^{px}&=e^{2}\mathop {\displaystyle \int }\nolimits _{-\infty }^{t}dt^{\prime }\varphi (t-t^{\prime })\frac{\partial x_{j}}{\partial p_{i}^{\prime }}. \end{aligned}$$

(B.10b)

Now, the dominant contribution to these time integrals comes from times $t^{\prime }$ close to $t$, when the system is already in the regime in which Eq. (4.78) controls the dynamics. This means that the dynamical variables are now represented by operators (see Sect. 4.4.2), and the factors $\partial p_{j}/\partial p_{i}^{\prime },$ $\partial x_{j}/\partial p_{i}^{\prime }$ should be properly expressed in terms of them. By noticing that for the corresponding classical variables the equalities

$$\begin{aligned} \frac{\partial p_{j}}{\partial p_{i}^{\prime }}&=\left[ x_{i}^{\prime },p_{j}\right] _{\text {PB}},\end{aligned}$$

(B.11a)

$$\begin{aligned} \frac{\partial x_{j}}{\partial p_{i}^{\prime }}&=\left[ x_{i}^{\prime },x_{j}\right] _{\text {PB}} \end{aligned}$$

(B.11b)

apply, in terms of operators the following substitutions must be made,

$$\begin{aligned} \frac{\partial p_{j}}{\partial p_{i}^{\prime }}&\Rightarrow \frac{1}{2i\eta }\left[ \hat{x}_{i}^{\prime },\hat{p}_{j}\right] ,\end{aligned}$$

(B.12a)

$$\begin{aligned} \frac{\partial x_{j}}{\partial p_{i}^{\prime }}&\Rightarrow \frac{1}{2i\eta }\left[ \hat{x}_{i}^{\prime },\hat{x}_{j}\right] . \end{aligned}$$

(B.12b)

These can be obtained from the general rules

$$\begin{aligned} \frac{\partial }{\partial p_{j}}\Rightarrow \frac{1}{2i\eta }\left[ \hat{x}_{j},\quad \right] ,\quad \frac{\partial }{\partial x_{j}}\Rightarrow \frac{1}{2i\eta }\left[ \quad ,\hat{p}_{j}\right] . \end{aligned}$$

(B.13)

With (B.12a, B.12b), Eqs. (B.10a), (B.10b) take the form

$$\begin{aligned} D_{ij}^{pp}&=\frac{e^{2}}{2i\eta }\mathop {\displaystyle \int }\nolimits _{-\infty }^{t}dt^{\prime }\varphi (t-t^{\prime })\left[ \hat{x}_{i}^{\prime },\hat{p}_{j}\right] ,\end{aligned}$$

(B.14a)

$$\begin{aligned} D_{ij}^{px}&=\frac{e^{2}}{2i\eta }\mathop {\displaystyle \int }\nolimits _{-\infty }^{t}dt^{\prime }\varphi (t-t^{\prime })\left[ \hat{x}_{i}^{\prime },\hat{x}_{j}\right] . \end{aligned}$$

(B.14b)

It is now straightforward to show that the diffusion coefficients [Eqs. (B.10a), (B.10b)] comply with the relation

$$\begin{aligned} \frac{\partial D_{ij}^{pp}}{\partial p_{j}}+\frac{\partial D_{ij}^{px}}{\partial x_{j}}=0. \end{aligned}$$

(B.15)

For this purpose we apply Eqs. (B.13) to (Eqs. B.14a), (B.14b), thus obtaining

$$\begin{aligned} \frac{\partial D_{ij}^{pp}}{\partial p_{j}}+\frac{\partial D_{ij}^{px}}{\partial x_{j}}=\frac{e^{2}}{(2i\eta )^{2}}\mathop {\displaystyle \int }\nolimits _{-\infty }^{t}dt^{\prime }\varphi (t-t^{\prime })\left\{ \left[ \hat{x}_{j},\left[ \hat{x}_{i}^{\prime },\hat{p}_{j}\right] \right] +\left[ \left[ \hat{x}_{i}^{\prime },\hat{x}_{j}\right] ,\hat{p}_{j}\right] \right\} . \end{aligned}$$

(B.16)

Resorting to the Jacobi identity (which is valid both for Poisson brackets and for commutators)

$$\begin{aligned} \left[ x_{j},\left[ x_{i}^{\prime },p_{j}\right] \right] +\left[ x_{i}^{\prime },\left[ p_{j},x_{j}\right] \right] +\left[ p_{j},\left[ x_{j},x_{i}^{\prime }\right] \right] =0, \end{aligned}$$

(B.17)

and noticing that the second term vanishes, we find

$$\begin{aligned} \left[ x_{j},\left[ x_{i}^{\prime },p_{j}\right] \right] =-\left[ p_{j},\left[ x_{j},x_{i}^{\prime }\right] \right] =-\left[ \left[ x_{i}^{\prime },x_{j}\right] ,p_{j}\right] , \end{aligned}$$

(B.18)

and therefore the right-hand side of (B.16) is null, which proves Eq. (B.15). A direct consequence of this result is that Eq. (4.44) reduces to (4.45).

Appendix C: Detailed Derivation of the ‘Generalized’ Schrödinger Equation

The starting point for the present derivation is the couple of Eq. (4.68)

$$\begin{aligned}&\frac{\partial \rho }{\partial t}+\frac{\partial }{\partial x_{j}}\left( v_{j}\rho \right) =0,\end{aligned}$$

(C.1a)

$$\begin{aligned}&m\frac{\partial }{\partial t}\!\left( v_{i}\rho \right) +m\frac{\partial }{\partial x_{j}}\!\left( v_{i}v_{j}\rho \right) \!-\!\frac{\eta ^{2}}{m}\frac{\partial }{\partial x_{j}}\!\left( \!\rho \frac{\partial ^{2}}{\partial x_{i}\partial x_{j}}\ln \rho \!\!\right) \!+\frac{1}{m}\frac{\partial }{\partial x_{j}}\Sigma _{ij}\rho -\!f_{i}\rho \\&=\tau v_{j}\frac{\partial f_{i}}{\partial x_{j}}\rho -e^{2}\!\left. (\widetilde{\mathcal {\hat{D}}Q})_{i}\right| _{\varvec{z}=0}. \end{aligned}$$

(C.1b)

Inserting the continuity equation into (C.1b) and multiplying by $\rho ^{-1}$ results in

$$\begin{aligned} m\frac{\partial \varvec{v}}{\partial t}\!\!+\!\frac{m}{2}\varvec{\nabla v}^{2}-m\varvec{v}\times \left( \varvec{\nabla }\times \varvec{v}\right) +\varvec{\nabla }\left( -\frac{2\eta ^{2}}{m}\frac{\varvec{\nabla }^{2}\sqrt{\rho }}{\sqrt{\rho }}+V\right) =\varvec{F}_{\text {rad}}+\varvec{F}_{\Sigma }, \end{aligned}$$

(C.2)

since $\left( \varvec{v}\cdot \varvec{\nabla }\right) \varvec{v}=\tfrac{1}{2}\varvec{\nabla v}^{2}-\varvec{v}\times \left( \varvec{\nabla }\times \varvec{v}\right)$. The $i$ components of the vectors $\varvec{F}_{\text {rad}}$ and $\varvec{F}_{\Sigma }$ are, respectively,

$$\begin{aligned} F_{i\text {rad}}&=\tau v_{j}\frac{\partial f_{i}}{\partial x_{j}}-\frac{e^{2}\!}{\rho }\left. (\widetilde{\mathcal {\hat{D}}Q})_{i}\right| _{\varvec{z}=0},\end{aligned}$$

(C.3)

$$\begin{aligned} F_{i\Sigma }&=-\frac{1}{m\rho }\frac{\partial }{\partial x_{j}}\!\Sigma _{ji}\rho . \end{aligned}$$

(C.4)

Now, according to Eq. (4.66a) $m\varvec{v}$ decomposes as

$$\begin{aligned} m\varvec{v}=-i\eta \varvec{\nabla }\ln \frac{q(\varvec{x},t)}{q^{*}(\varvec{x},t)}+\varvec{g}=2\eta \varvec{\nabla }S+\varvec{g}, \end{aligned}$$

(C.5)

where the last equality follows from writing $q(\varvec{x},t)$ in its polar form

$$\begin{aligned} q(\varvec{x},t)=\sqrt{\rho }e^{iS(\varvec{x},t)}. \end{aligned}$$

(C.6)

Substitution of Eqs. (C.5) into (C.2) gives (using $\varvec{\nabla }\times m\varvec{v}=\varvec{\nabla }\times \varvec{g})$

$$\begin{aligned} \!\!\varvec{\nabla }M=\varvec{F}_{\text {rad}}+\varvec{F}_{\Sigma }-\frac{\partial \varvec{g}}{\partial t}+\varvec{v}\times \left( \varvec{\nabla }\times \varvec{g}\right) , \end{aligned}$$

(C.7)

with

$$\begin{aligned} M=2\eta \frac{\partial S}{\partial t}+\frac{1}{2}m\varvec{v}^{2}-\frac{2\eta ^{2}}{m}\frac{\varvec{\nabla }^{2}\sqrt{\rho }}{\sqrt{\rho }}+V. \end{aligned}$$

(C.8)

This is basically the Hamilton-Jacobi-type equation of Bohm’s theory [see Sect. (4.4.1)], when $M=0$ (and $\eta =\hbar /2$), which means that it should be possible to arrive at the Schrödinger equation from the above expressions. For this purpose we proceed as follows.

From Eq. (C.6) it follows $\ln q=(1/2)\ln \rho +iS$, which combined with the continuity Eq. (C.1a) gives

$$\begin{aligned} \frac{\partial S}{\partial t}=-\frac{i}{q}\frac{\partial q}{\partial t}-\frac{i}{2}\nabla \cdot \varvec{v}-i\varvec{v}\cdot \frac{\varvec{\nabla }q}{q}-\varvec{v}\cdot \varvec{\nabla }S. \end{aligned}$$

(C.9)

Using here Eq. (C.5) leads to

$$\begin{aligned} \frac{\partial S}{\partial t}&=-\frac{i}{q}\frac{\partial q}{\partial t}-\frac{i\eta }{m}\varvec{\nabla }^{2}S-\frac{i}{2m}\nabla \cdot \varvec{g}-\frac{2i\eta }{m}\varvec{\nabla }S\cdot \frac{\varvec{\nabla }q}{q}- \end{aligned}$$

$$\begin{aligned}&\quad -\frac{i}{m}\varvec{g}\cdot \frac{\varvec{\nabla }q}{q}-\frac{2\eta }{m}\left( \varvec{\nabla }S\right) ^{2}-\frac{1}{m}\varvec{g}\cdot \varvec{\nabla }S. \end{aligned}$$

(C.10)

This expression, together with $m\varvec{v}^{2}=(1/m)\left( 2\eta \varvec{\nabla }S+\varvec{g}\right) ^{2}$, allows to recast Eq. (C.8) in the form

$$\begin{aligned} M&=\frac{1}{q}\left[ -2i\eta \frac{\partial q}{\partial t}+\frac{1}{2m}\left( -2i\eta \varvec{\nabla }+\varvec{g}\right) ^{2}q+Vq\right] -\\&\quad -\frac{2\eta ^{2}}{m}\left[ i\varvec{\nabla }^{2}S+\frac{\varvec{\nabla }^{2}\sqrt{\rho }}{\sqrt{\rho }}+\left( \varvec{\nabla }S\right) ^{2}+2i\varvec{\nabla }S\cdot \frac{\varvec{\nabla }q}{q}-\frac{\varvec{\nabla }^{2}q}{q}\right] . \end{aligned}$$

(C.11)

Now we notice that as a consequence of Eq. (C.6),

$$\begin{aligned} \frac{\varvec{\nabla }q}{q}=\frac{1}{2}\frac{\varvec{\nabla }\rho }{\rho }+i\varvec{\nabla }S, \end{aligned}$$

(C.12)

whence

$$\begin{aligned} \frac{\varvec{\nabla }^{2}q}{q}=\frac{\varvec{\nabla }^{2}\sqrt{\rho }}{\sqrt{\rho }}+i\varvec{\nabla }^{2}S+i\varvec{\nabla }S\cdot \frac{\varvec{\nabla }\rho }{\rho }-\left( \varvec{\nabla }S\right) ^{2}. \end{aligned}$$

(C.13)

From here it follows that the term in the second line of Eq. (C.11) vanishes. Consequently $M$ reduces to

$$\begin{aligned} M=\frac{1}{q}\left[ -2i\eta \frac{\partial q}{\partial t}+\frac{1}{2m}\left( -2i\eta \varvec{\nabla }+\varvec{g}\right) ^{2}q+Vq\right] =\frac{1}{ q}\hat{M}q, \end{aligned}$$

(C.14)

with $\hat{M}$ the linear operator

$$\begin{aligned} \hat{M}=-2i\eta \frac{\partial }{\partial t}+\frac{1}{2m}\left( -2i\eta \varvec{\nabla }+\varvec{g}\right) ^{2}+V. \end{aligned}$$

(C.15)

Finally, inserting Eq. (C.14) into (C.7) we arrive at

$$\begin{aligned} \!\!\varvec{\nabla }\left( \frac{1}{q}\hat{M}q\right) =\varvec{F}_{\text {rad}}+\varvec{F}_{\Sigma }-\frac{\partial \varvec{g}}{\partial t}+\varvec{v}\times \left( \varvec{\nabla }\times \varvec{g}\right) . \end{aligned}$$

(C.16)

We call this the generalized Schrödinger equation, since as is shown in Sect. 4.4.1, it reduces to the Schrödinger equation in the radiationless approximationRadiationless approximation. Equation (C.16) and its adjoint are equivalent to the first two equations of the hierarchy, (C.1a, C.1b).

Appendix D: Diffusive Contribution to the Energy Balance

This appendix is devoted to the calculation of theDetailed balance!frequency mixing right-hand sideDetailed balance!condition of the energy-balance condition (4.37). The particle is considered in equilibrium with the zpf, which means that it must be in its ground state, represented by the solution $\psi _{0}(x)$ of the Schrödinger equation (4.78) (still in terms of $\eta$). We recall that in the time-asymptotic limit, the Markovian approximation holds, described by the fpe (B.1). This means that one may use the simpler Eq. (4.103), which in one dimension reads

$$\begin{aligned} \tau \left\langle \dddot{x}p\right\rangle _{0}=-\frac{1}{m}\left\langle D^{pp}\right\rangle _{0}. \end{aligned}$$

(D.1)

Introducing the diffusion coefficient $D^{pp}$ given by Eq. (B.14a), one obtains

$$\begin{aligned} -\frac{1}{m}\left\langle D^{pp}\right\rangle _{0}=\frac{ie^{2}}{2\eta m}\mathop {\displaystyle \int }\nolimits _{-\infty }^{t}dt^{\prime }\varphi (t-t^{\prime })\left\langle \left[ \hat{x}_{i}^{\prime },\hat{p}_{j}\right] \right\rangle _{0}, \end{aligned}$$

(D.2)

With $\varphi (t-t^{\prime })$ given by (4.10) and $\rho =\rho _{0}$ given by (4.101), this becomes (recall that $\tau =2e^{2}/3mc^{3}$)

$$\begin{aligned} -\frac{1}{m}\left\langle D^{pp}\right\rangle _{0}=\frac{i\hbar \tau }{2\eta \pi }\mathop {\displaystyle \int }_{0}^{\infty }d\omega \ \omega ^{3}\int \nolimits _{-\infty }^{t}dt^{\prime }\cos \omega (t-t^{\prime })\ \left\langle \left[ \hat{x}_{i}^{\prime },\hat{p}_{j}\right] \right\rangle _{0}, \end{aligned}$$

(D.3)

with

$$\begin{aligned} \left\langle \left[ \hat{x}^{\prime },\hat{p}\right] \right\rangle _{0}=\sum _{k}\left( \hat{x}_{0k}^{\prime }\hat{p}_{k0}-\hat{p}_{0k}\hat{x}_{k0}^{\prime }\right) , \end{aligned}$$

(D.4)

Equations (4.104) and (4.105) can be used to write $x_{kn}(t)=e^{i\omega _{kn}t}x_{kn}$ and $p_{nm}(t)=im\omega _{nm}e^{i\omega _{nm}t}x_{nm},$ which gives

$$\begin{aligned} \left\langle \left[ \hat{x}^{\prime },\hat{p}\right] \right\rangle _{0}=2im\sum _{k}\omega _{k0}\left| x_{0k}\right| ^{2}\cos \omega _{k0}(t-t^{\prime }). \end{aligned}$$

(D.5)

Introducing this into Eq. (D.3) leads to

$$\begin{aligned} -\frac{1}{m}\left\langle D^{pp}\right\rangle _{0}&=-\frac{\hbar m\tau }{\pi \eta }\sum _{k}\omega _{k0}\left| x_{0k}\right| ^{2}\\&\times \mathop {\displaystyle \int }_{0}^{\infty }d\omega \ \omega ^{3}\int \nolimits _{-\infty }^{t}dt^{\prime }\cos \omega (t-t^{\prime })\cos \omega _{k0}(t-t^{\prime }). \end{aligned}$$

(D.6)

The integral over time can be calculated with the formula

$$\begin{aligned} \int \nolimits _{-\infty }^{\infty }dke^{ikx}=\int \nolimits _{-\infty }^{\infty }dk\cos kx=2\pi \delta (x). \end{aligned}$$

(D.7)

This gives

$$\begin{aligned} \int \nolimits _{-\infty }^{t}dt^{\prime }\cos \omega (t-t^{\prime })\cos \omega _{k0}(t-t^{\prime })=\frac{\pi }{2}[\delta (\omega -\omega _{k0})+\delta (\omega +\omega _{k0})]. \end{aligned}$$

(D.8)

For the ground state there are no negative frequencies, i.e. all $\omega _{k0}>0,$ whence the second term in the right-hand side does not contribute to (D.6), and therefore

$$\begin{aligned} -\frac{1}{m}\left\langle D^{pp}\right\rangle _{0}=-\frac{\hbar m\tau }{2\eta }\sum _{k}\omega _{k0}^{4}\left| x_{0k}\right| ^{2}. \end{aligned}$$

(D.9)