The Road to Heisenberg Quantum Mechanics

Abstract

This chapter takes us through an alternative itinerary to quantum mechanics, on this occasion to the Heisenberg formulation of the theory. Our starting point is again the stochastic Abraham-Lorentz equationAbraham-Lorentz!equation for the particle embedded in the zero-point field. A detailed analysis of the stationary solutions of this equation exhibits the resonant response Linear sed!resonant responseof the particle to certain modes of the field, determined in each case by the problem itself. The condition of ergodicity Matrix mechanics!and ergodicity Energy-balance condition!and ergodicity of these stationary states has far-reaching consequences, both for the physical behavior of the system and for the formalism used to describe it. In particular, the response of the particle to the field turns out to be linear, regardless of the nonlinearities of the external force. Further, the dynamical variables become represented by matrices, which satisfy the Linear sed!Heisenberg equationsHeisenberg equation. The statistical nature of the description becomes evident. The ensuing fundamental commutator $[x,p]=i\hbar$ represents a direct measure of the intensity of the Commutator!and correlationfluctuations impressed by the zero-point field upon the particle. The path followed in the derivations throws thus new light on the physical meaning of the Linear sed!Heisenberg descriptionquantities involved in the Heisenberg description.Harmonic oscillator!Heisenberg description

... quantum phenomena do not occur in a Hilbert space, they occur in a laboratory.

A. Peres (1995, p. XI)

Appendices

Appendix A: The Ergodic Principle and the Algebraic Description

This appendix is devoted to a detailed analysis of the implications of imposing the ergodic condition in the form (5.46), for the different elements that enter into the description.

We start by recalling from Sect. 5.1.1 that to distinguish the solutions $x(t)$ that correspond to different accessible stationary states $\alpha$, the substitution

$$\begin{aligned} x(t)=\sum _{k}\tilde{x}_{k}a_{k}e^{i\omega _{k}t}\rightarrow x_{\alpha }(t)=\sum _{\beta }\tilde{x}_{\alpha \beta }a_{\alpha \beta }e^{i\omega _{\alpha \beta }t} \end{aligned}$$

(A.1)

has been made, where $\tilde{x}_{\alpha \beta }=\tilde{x}(\omega _{\alpha \beta })$ stands for the amplitude corresponding to the frequency $\omega _{\alpha \beta }$ and similarly for $a_{\alpha \beta }$. The purpose of the first part of the appendix is to establish the correspondence for higher powers of $x,$ $x^{n}\rightarrow (x^{n})_{\alpha }$ $,$ that is consistent with Eq. (5.46). The problem is equivalent to determining the appropriate coefficients $\tilde{A}_{\alpha \beta }$ for any dynamical variable $A_{\alpha }(t)$ that can be expanded as a power series of $x,$ with $A_{\alpha }$ given by Eq. (5.34), namely

$$\begin{aligned} A_{\alpha }(t)=\sum _{\beta }\tilde{A}_{\alpha \beta }a_{\alpha \beta }e^{i\omega _{\alpha \beta }t}. \end{aligned}$$

(A.2)

Let us first analyze the quadratic case, $A_{\alpha }=( x^{2}) _{\alpha }.$ The introduction of the index $\alpha$ must be such as to guarantee that the expansion for $( x^{2}) _{\alpha }$ is consistent with the demands imposed by the theory. We therefore start by writing

$$\begin{aligned} x^{2}(t)=\sum _{k,k^{\prime }}\tilde{x}_{k}\tilde{x}_{k^{\prime }}a_{k}a_{k^{\prime }}e^{i\left( \omega _{k}+\omega _{k^{\prime }}\right) t} \end{aligned}$$

(A.3)

and define

$$\begin{aligned} \omega _{K}\equiv \omega _{k}+\omega _{k^{\prime }}, \end{aligned}$$

(A.4)

so that Eq. (A.3) rewrites as

$$\begin{aligned} x^{2}(t)=\sum _{k,K}\tilde{x}(\omega _{k})\tilde{x}(\omega _{K}-\omega _{k})a(\omega _{k})a(\omega _{K}-\omega _{k})e^{i\omega _{K}t}. \end{aligned}$$

(A.5)

As explained in Sect. 5.1.2, passing from $x^{2}(t)$ to $\left( x^{2}\right) _{\alpha }(t)$ requires focusing not on the complete set $\{ \omega _{K}\}$ but rather on the subset $\{ \omega _{\alpha \beta }\}$, so that the sum over $K$ becomes a sum over $\beta$. Similarly, we make the substitution $\omega _{k}\rightarrow \omega _{\alpha \beta ^{\prime }}$. Thus,

$$\begin{aligned} ( x^{2}) _{\alpha }(t)=\sum _{\beta ,\beta ^{\prime }}\tilde{x} (\omega _{\alpha \beta ^{\prime }})\tilde{x}(\omega _{\alpha \beta }-\omega _{\alpha \beta ^{\prime }})a(\omega _{\alpha \beta ^{\prime }})a(\omega _{\alpha \beta }-\omega _{\alpha \beta ^{\prime }})e^{i\omega _{\alpha \beta }t}. \end{aligned}$$

(A.6)

On the other hand, according to Eq. (A.2), $( x^{2}) _{\alpha }$ has the form

$$\begin{aligned} ( x^{2}) _{\alpha }(t)=\sum _{\beta }\widetilde{x^{2}}(\omega _{\alpha \beta })a(\omega _{\alpha \beta })e^{i\omega _{\alpha \beta }t}. \end{aligned}$$

(A.7)

Comparison of these two equations leads to

$$\begin{aligned} \widetilde{x^{2}}(\omega _{\alpha \beta })a^{(i)}(\omega _{\alpha \beta })=\sum _{\beta ^{\prime }}\tilde{x}(\omega _{\alpha \beta ^{\prime }})\tilde{x}(\omega _{\alpha \beta }\,-\,\omega _{\alpha \beta ^{\prime }})a^{(i)}(\omega _{\alpha \beta ^{\prime }})a^{(i)}(\omega _{\alpha \beta }\,-\,\omega _{\alpha \beta ^{\prime }}), \end{aligned}$$

(A.8)

where the index $(i)$ denotes the dependence of the $a$’s on the field realization. Since according to (5.46) $\widetilde{x^{2}}(\omega _{\alpha \beta })$ is a sure quantity, the sum

$$\begin{aligned} \sum _{\beta ^{\prime }}\tilde{x}(\omega _{\alpha \beta ^{\prime }})\tilde{x} (\omega _{\alpha \beta }-\omega _{\alpha \beta ^{\prime }})\frac{a^{(i)}(\omega _{\alpha \beta ^{\prime }})a^{(i)}(\omega _{\alpha \beta }-\omega _{\alpha \beta ^{\prime }})}{a^{(i)}(\omega _{\alpha \beta })} \end{aligned}$$

(A.9)

must be a sure quantity as well, for any $\alpha$, $\beta$. The only way to ensure that (A.9) is an $i$-independent quantity is by taking

$$\begin{aligned} a^{(i)}(\omega _{\alpha \beta ^{\prime }})a^{(i)}(\omega _{\alpha \beta }-\omega _{\alpha \beta ^{\prime }})=a^{(i)}(\omega _{\alpha \beta }). \end{aligned}$$

(A.10)

Equation (A.8) reduces thus to

$$\begin{aligned} \widetilde{x^{2}}(\omega _{\alpha \beta })=\sum _{\beta ^{\prime }}\tilde{x} (\omega _{\alpha \beta ^{\prime }})\tilde{x}(\omega _{\alpha \beta }-\omega _{\alpha \beta ^{\prime }}). \end{aligned}$$

(A.11)

Together with Eq. (A.7), this gives the expression for $\left( x^{2}\right) _{\alpha }$ that is consistent with the ergodic demand:

$$\begin{aligned} ( x^{2}) _{\alpha }(t)=\sum _{\beta \beta ^{\prime }}\tilde{x} (\omega _{\alpha \beta ^{\prime }})\tilde{x}(\omega _{\alpha \beta }-\omega _{\alpha \beta ^{\prime }})a(\omega _{\alpha \beta })e^{i\omega _{\alpha \beta }t}. \end{aligned}$$

(A.12)

We now focus on the frequency

$$\begin{aligned} \Omega _{\beta ^{\prime }\beta }^{(\alpha )}\equiv \omega _{\alpha \beta }-\omega _{\alpha \beta ^{\prime }}, \end{aligned}$$

(A.13)

which is a difference between two relevant frequencies of the subensemble $\alpha$. For a given $\alpha$, this frequency depends on the indices $\beta ,\beta ^{\prime }$ only. By its definition it has the following properties:

$$\begin{aligned} \Omega _{\beta ^{\prime }\beta }^{(\alpha )} =-\Omega _{\beta \beta ^{\prime }}^{(\alpha )}, \end{aligned}$$

(A.14a)

$$\begin{aligned} \Omega _{\beta \beta ^{\prime }}^{(\alpha )}+\Omega _{\beta ^{\prime }\beta ^{\prime \prime }}^{(\alpha )}+\cdots +\Omega _{\beta ^{\left( n-1\right) }\beta ^{(n)}}^{(\alpha )}=\Omega _{\beta \beta ^{(n)}}^{(\alpha )}. \end{aligned}$$

(A.14b)

The second equation involves an arbitrary number of terms, with each $\beta ^{(m)}$ an element of the set $\{ \beta \}$ of indices introduced to enumerate the different relevant frequencies in $\{ \omega _{\alpha \beta }\} .$ In particular, for $\beta ^{\prime }=\beta _{0},$ $\omega _{\alpha \beta ^{\prime }}$ vanishes [see Eq. (5.23)], and $\Omega _{\beta ^{\prime }\beta }^{(\alpha )}$ reduces to

$$\begin{aligned} \Omega _{\beta _{0}\beta }^{(\alpha )}=\omega _{\alpha \beta }. \end{aligned}$$

(A.15)

In terms of the new frequencies, and writing

$$\begin{aligned} \tilde{x}(\Omega _{\beta \beta ^{\prime }}^{(\alpha )})=\tilde{x}_{\beta \beta ^{\prime }}^{(\alpha )}, \end{aligned}$$

(A.16)

Equation (A.11) reads

$$\begin{aligned} \widetilde{x^{2}}(\Omega _{\beta _{0}\beta }^{(\alpha )})=\sum _{\beta ^{\prime }}\tilde{x}_{\beta _{0}\beta ^{\prime }}^{(\alpha )}\tilde{x}_{\beta ^{\prime }\beta }^{(\alpha )}. \end{aligned}$$

(A.17)

Now we resort to the fact that $\beta ^{\prime }$ and $\beta$ run over the same domain ($\{ \beta \}$) to define a square matrix $\Omega (\alpha )$ with elements $\Omega _{\beta ^{\prime }\beta }^{(\alpha )}.$ Thus, using the (provisional) notation

$$\begin{aligned} \{ \beta \} =\{ 0,1,2,{\ldots },\beta _{0}-1,\beta _{0},\beta _{0}+1,{\cdots }\} , \end{aligned}$$

(A.18)

we write

$$\begin{aligned} \Omega (\alpha )=\left( \begin{array}{lllll} \Omega _{00}^{(\alpha )}=0 &{} \Omega _{01}^{(\alpha )} &{} {\ldots ~} &{} \Omega _{0\beta _{0}}^{(\alpha )} &{} {\ldots } \\ \Omega _{10}^{(\alpha )} &{} \Omega _{11}^{(\alpha )}=0 &{} {\ldots } &{} &{} \\ {\ldots } &{} &{} &{} &{} \\ \omega _{\alpha 0} &{} \omega _{\alpha 1} &{} {\ldots ~} &{} \omega _{\alpha \beta _{0}}=0 &{} {\ldots } \\ {\ldots } &{} &{} &{} &{} \end{array} \right) , \end{aligned}$$

(A.19)

where Eq. (A.15) was used in the $\beta _{0}$-th row. This shows that the set of relevant frequencies $\{ \omega _{\alpha \beta }\}$ is a row in the wider set of frequencies $\Omega (\alpha )$. This, together with the fact that all the elements $\Omega _{\beta ^{\prime }\beta }^{(\alpha )}$ should be taken into consideration on an equal footing when calculating $\left( x^{2}\right) _{\alpha }$ [see Eq. (A.17)], leads us to extend the analysis to the entire matrix $\Omega (\alpha ).$

The order of the rows in $\Omega (\alpha )$ is physically irrelevant, since it was established via the arbitrary ordering (A.18). This means that the rows in $\Omega (\alpha )$ are physically equivalent. Thus, if the $\beta _{0}$-th row gives the set $\{ \omega _{\alpha \beta }\}$ for the subensemble $\alpha ,$ another row, say $\beta ^{\prime }=\eta \ne \beta _{0},$ gives the set $\{\Omega _{\eta \beta }^{(\alpha )}\}$ for the subensemble labeled with $\eta ,$ whose relevant frequencies are $\{\Omega _{\eta \beta }^{(\alpha )}=\omega _{\alpha \beta }-\omega _{\alpha \eta }\}$ . When the system is in the corresponding stationary state $\eta ,$ the expansion of the dynamical variable $A$ reads

$$\begin{aligned} A_{\eta }(t)\equiv \sum _{\beta }\tilde{A}(\Omega _{\eta \beta }^{(\alpha )})a(\Omega _{\eta \beta }^{(\alpha )})e^{i\Omega _{\eta \beta }^{(\alpha )}t}, \end{aligned}$$

(A.20)

which is a generalization of

$$\begin{aligned} A_{\alpha }(t)=\sum _{\beta }\tilde{A}(\omega _{\alpha \beta })a(\omega _{\alpha \beta })e^{i\omega _{\alpha \beta }t} \end{aligned}$$

(A.21)

for a variable $A$ in the stationary state $\alpha$, in the same way that $\Omega (\alpha )$ generalized the set $\{ \omega _{\alpha \beta }\} .$ Clearly (A.21) can be obtained from (A.20) taking $\eta =\beta _{0}.$ A first conclusion that derives from here is that the row-index $\beta ^{\prime }$ is in direct correspondence with a stationary state index, so that the indices in $\{ \beta \}$, originally introduced as labels that distinguished each of the relevant frequencies, denote subensembles corresponding to new stationary states. It follows that the indices $\alpha$ and $\beta$ have the same domain and the same physical interpretation. In particular, this allows us to write

$$\begin{aligned} \beta _{0}=\alpha , \end{aligned}$$

(A.22)

so that Eq. (A.18) rewrites as

$$\begin{aligned} \{ \beta \} =\{ \alpha -\alpha ,\alpha -(\alpha +1),\cdots ,\alpha -1,\alpha ,\alpha +1,\cdots \} . \end{aligned}$$

(A.23)

According to the discussion preceding Eq. (A.20), the (arbitrary) state $\eta$ $(\ne \alpha )$ is related to $\alpha$ via their relevant frequencies, an observation that discloses a relation among all stationary states, and ultimately connects them all. Of course, the origin of such relation goes back to the $\alpha$-dependence of $\Omega (\alpha )$ (and hence of all its rows). However, even though $\Omega (\alpha )$ was constructed taking $\alpha$ as a privileged ensemble, this matrix does not depend on $\alpha$ since the same matrix is obtained when considering the difference between two relevant frequencies of any subensemble $\eta .$ In order to see this we construct the matrix $\Omega (\eta )$ with elements $\Omega _{\beta ^{\prime }\beta }^{(\eta )}$ defined, in analogy with Eq. (A.13), as the difference between two relevant frequencies of the subensemble $\eta$:

$$\begin{aligned} \Omega _{\beta ^{\prime }\beta }^{(\eta )}\equiv \Omega _{\eta \beta }^{(\alpha )}-\Omega _{\eta \beta ^{\prime }}^{(\alpha )}. \end{aligned}$$

(A.24)

Resorting now to Eq. (A.14b) we obtain

$$\begin{aligned} \Omega _{\beta ^{\prime }\beta }^{(\eta )}=\Omega _{\beta ^{\prime }\beta }^{(\alpha )}, \end{aligned}$$

(A.25)

hence $\Omega (\eta )=\Omega (\alpha ).$ Since $\eta$ is arbitrary, it follows that the matrix (A.19) is indeed $\alpha$-independent so that $\Omega (\alpha )=\Omega$ and the superindex $(\alpha )$ in $\Omega _{\beta ^{\prime }\beta }^{(\alpha )}$ may be dropped. The matrix $\Omega$ can then be understood as an array (in row form) of all the relevant frequencies corresponding to all the accessible stationary states of the mechanical system. Because of equations (A.14), its elements can be written in the general form

$$\begin{aligned} \Omega _{\beta ^{\prime }\beta }=\Omega _{\beta ^{\prime }}-\Omega _{\beta }. \end{aligned}$$

(A.26)

The physical meaning of the parameters $\Omega _{\beta }$ is determined in Sect. (5.4.1). Together with Eq. (A.15) (with $\beta _{0}=\alpha$), (A.26) gives

$$\begin{aligned} \Omega _{\alpha \beta }=\omega _{\alpha \beta }=\Omega _{\alpha }-\Omega _{\beta }, \end{aligned}$$

(A.27)

hence

$$\begin{aligned} \omega _{\beta \alpha }=-\omega _{\alpha \beta } \end{aligned}$$

(A.28)

and

$$\begin{aligned} \omega _{\alpha \beta ^{\prime }}+\omega _{\beta ^{\prime }\beta ^{\prime \prime }}+...+\omega _{\beta ^{\left( n-1\right) }\beta }=\omega _{\alpha \beta }. \end{aligned}$$

(A.29)

With these results we may now go back to Eq. (A.10) and write

$$\begin{aligned} a^{(i)}(\omega _{\alpha \beta ^{\prime }})a^{(i)}(\omega _{\beta ^{\prime }\beta })=a^{(i)}(\omega _{\alpha \beta }). \end{aligned}$$

(A.30)

Using the short notation $a_{\beta ^{\prime }\beta }=a(\omega _{\beta ^{\prime }\beta }),$ this relation can be easily generalized to any number of factors by a successive (chained) application of it,

$$\begin{aligned} a_{\alpha \beta ^{\prime }}a_{\beta ^{\prime }\beta ^{\prime \prime }}a_{\beta ^{\prime \prime }\beta ^{\prime \prime \prime }}\ldots a_{\beta ^{\left( n-1\right) }\beta }&=\left( a_{\alpha \beta ^{\prime }}a_{\beta ^{\prime }\beta ^{\prime \prime }}\right) a_{\beta ^{\prime \prime }\beta ^{\prime \prime \prime }}\ldots a_{\beta ^{\left( n-1\right) }\beta }\\&=\left[ \left( a_{\alpha \beta ^{\prime \prime }}\right) a_{\beta ^{\prime \prime }\beta ^{\prime \prime \prime }}\right] \ldots a_{\beta ^{\left( n-1\right) }\beta } \\&=\left[ a_{\alpha \beta ^{\prime \prime \prime }}\right] \cdots a_{\beta ^{\left( n-1\right) }\beta } \end{aligned}$$

$$\begin{aligned}&=a_{\alpha \beta }. \end{aligned}$$

(A.31)

With each $a_{\beta ^{(n)}\beta ^{(m)}}^{(i)}$ written in polar form according to (5.21), this implies that also the stochastic phases must satisfy the relation

$$\begin{aligned} \varphi _{\alpha \beta ^{\prime }}^{(i)}+\varphi _{\beta ^{\prime }\beta ^{\prime \prime }}^{(i)}+\cdots + \varphi _{\beta ^{\left( n-1\right) }\beta }^{(i)}=\varphi _{\alpha \beta }^{(i)}, \end{aligned}$$

(A.32a)

and can therefore be expressed as a difference of terms,

$$\begin{aligned} \varphi _{\alpha \beta }^{(i)}=\phi _{\alpha }^{(i)}-\phi _{\beta }^{(i)}, \end{aligned}$$

(A.33)

where each of the $\phi _{\lambda }$ represents a random phase. Equation (5.21) becomes thus

$$\begin{aligned} a_{\alpha \beta }=e^{i\varphi _{\alpha \beta }}=e^{i(\phi _{\alpha }-\phi _{\beta })}, \end{aligned}$$

(A.34)

from where it follows, in particular, that

$$\begin{aligned} a_{\beta \alpha }=a_{\alpha \beta }^{*}. \end{aligned}$$

(A.35)

The relations (A.29) and (A.31), which are fundamental for the theory, constitute what is called the chain rule . The frequencies that enter in the chain rule can refer to relevant frequencies of any stationary state (they are elements of the matrix $\Omega )$, and can be either resonance frequencies or linear (chained) combinations of them.

The result $\left| a_{\alpha \beta }\right| =1$ requires a comment. In Chap. 3 it was found that the energy of the oscillators of the zpf Coherence!zpf modes have an important dispersion. Here we got what seems to be a contradictory result, namely that the $a_{\alpha \beta }$’s have a fix amplitude. Consistency is recovered by considering the discussion following Eq. (5.4). Further, the strict meaning of (A.34) is that only the modes of the field that have an important role in the dynamics of the mechanical subsystem in the ergodic regime, are those that have amplitudes with a Gaussian distribution with a negligible dispersion around an average value 1. The remaining modes simply contribute to the background noise.

Finally, with the results obtained above, Eq. (A.12) becomes

$$\begin{aligned} ( x^{2}) _{\alpha }(t)=\sum _{\beta \beta ^{\prime }}\tilde{x}_{\alpha \beta ^{\prime }}\tilde{x}_{\beta ^{\prime }\beta }a_{\alpha \beta }e^{i\omega _{\alpha \beta }t}, \end{aligned}$$

(A.36)

which, when compared with (A.7), gives

$$\begin{aligned} (\widetilde{x^{2}})_{\alpha \beta }=\sum _{\beta ^{\prime }}\tilde{x}_{\alpha \beta ^{\prime }}\tilde{x}_{\beta ^{\prime }\beta }. \end{aligned}$$

(A.37)

Iteration of the procedure presented above leads to the following expression for the $n$-th power of the variable $x^{n}$ in state $\alpha$

$$\begin{aligned} ( x^{n}) _{\alpha }(t)=\sum _{\beta }(\widetilde{x^{n}})_{\alpha \beta }a_{\alpha \beta }e^{i\omega _{\alpha \beta }t}, \end{aligned}$$

(A.38)

with

$$\begin{aligned} (\widetilde{x^{n}})_{\alpha \beta }=\sum _{\beta ^{\prime }...\beta ^{(n-1)}}\tilde{x}_{\alpha \beta ^{\prime }}\tilde{x}_{\beta ^{\prime }\beta ^{\prime \prime }}...\tilde{x}_{\beta ^{(n-1)}\beta }. \end{aligned}$$

(A.39)

Appendix B: A Simple Example: The Harmonic Oscillator

The aim of this appendix is to analyze a simple system by applying some of the methods presented in the body of the chapter, in order to clarify the meaning of the resonance and relevant frequencies, respectively. For this purpose let us consider the case of a harmonic oscillator Detailed balance!oscillatorof natural frequency $\omega _{0}.$ The force is $f=-m\omega _{0}^{2}x,$ so that in the state $\alpha$

$$\begin{aligned} f_{\alpha }=-m\omega _{0}^{2}x_{\alpha }. \end{aligned}$$

(B.1)

Equation (5.29) becomes then

$$\begin{aligned} \omega _{\alpha \beta }^{2}\approx \omega _{0}^{2}, \end{aligned}$$

(B.2)

with solutions $\pm \omega _{0}.$ The independence of the resonance frequencies from the state is characteristic of the harmonic oscillator; it does not hold in general for arbitrary forces. There exist therefore only two resonance frequencies for the state $\alpha$, which will be labeled with the indices $\beta _{+}=\alpha -1$ and $\beta _{-}=\alpha +1$ [see Eq. (A.23)] in such a way that

$$\begin{aligned} \omega _{\alpha ,\alpha -1}=\omega _{0},\quad \omega _{\alpha ,\alpha +1}=-\omega _{0}. \end{aligned}$$

(B.3)

Since these are the frequencies that contribute significantly to the expansion

$$\begin{aligned} x_{\alpha }(t)=\sum _{\beta }\tilde{x}_{\alpha \beta }a_{\alpha \beta }e^{i\omega _{\alpha \beta }t}, \end{aligned}$$

(B.4)

we conclude that when the noisy terms are neglected in Eq. (B.4), the coefficients $\tilde{x}_{\alpha \beta }$ are

$$\begin{aligned} \tilde{x}_{\alpha \beta }=\tilde{x}_{\alpha ,\alpha -1}\delta _{\beta ,\alpha -1}+\tilde{x}_{\alpha ,\alpha +1}\delta _{\beta ,\alpha +1}, \end{aligned}$$

(B.5)

hence

$$\begin{aligned} x_{\alpha }(t)=\tilde{x}_{\alpha ,\alpha -1}a_{\alpha ,\alpha -1}e^{i\omega _{0}t}+\tilde{x}_{\alpha ,\alpha +1}a_{\alpha ,\alpha +1}e^{-i\omega _{0}t}. \end{aligned}$$

(B.6)

To determine the relevant frequencies of the system we resort to the chain rule Relevant frequencies!and chain rule(Eq. A.29) and to the antisymmetry $\omega _{\alpha \beta }=-\omega _{\beta \alpha }$ to combine the resonance frequencies, thus obtaining

$$\begin{aligned} \omega _{\alpha +1,\alpha }+\omega _{\alpha ,\alpha -1}=\omega _{\alpha +1,\alpha -1}=2\omega _{0}. \end{aligned}$$

(B.7)

This defines a new (relevant) frequency $\omega _{\alpha +1,\alpha -1}=2\omega _{0}.$ Now, according to the discussion following Eq. (A.21), $\alpha +1$ and $\alpha -1$ represent new stationary states. Their resonance frequencies are again $\pm \omega _{0},$ since as stated above, the solutions of (B.2) are state-independent. As before, the two resonance frequencies for the state $\alpha \pm 1$ will be labeled with the indices $\beta _{+}=(\alpha \pm 1)-1$ and $\beta _{-}=(\alpha \pm 1)+1.$ This gives, in analogy with (B.3),

$$\begin{aligned} \omega _{\alpha +1,\alpha } \quad&= \quad \omega _{0},\quad \omega _{\alpha +1,\alpha +2}=-\omega _{0}, \\ \omega _{\alpha -1,\alpha -2} \quad&= \quad \omega _{0},\quad \omega _{\alpha -1,\alpha }=-\omega _{0}. \end{aligned}$$

(B.8)

A chained combination of these frequencies with $\omega _{\alpha +1,\alpha -1}$ in Eq. (B.7) defines two more relevant frequencies,

$$\begin{aligned} \omega _{\alpha +2,\alpha +1}+\omega _{\alpha +1,\alpha -1} \quad&= \quad \omega _{\alpha +2,\alpha -1}=3\omega _{0}, \\ \omega _{\alpha +2,\alpha -1}+\omega _{\alpha -1,\alpha -2} \quad&= \quad \omega _{\alpha +2,\alpha -2}=4\omega _{0}. \end{aligned}$$

(B.9)

It is clear that the procedure can be applied iteratively, so that the relevant frequencies for the harmonic oscillator take the form

$$\begin{aligned} \omega _{\alpha +n,\alpha +m}=\omega _{nm}=\omega _{0}(n-m), \end{aligned}$$

(B.10)

with $n,m=0,$ $\pm 1,\pm 2,\ldots$

In particular, the resonance frequencies ($\pm \omega _{0}$) correspond to $m=n\pm 1;$ the remaining relevant frequencies (those in Eq. (B.10) with $m\ne n\pm 1$) will be important in expansions of higher powers of $x(t)$ (or $p(t)$). For example, for $\widetilde{x^{2}}_{\alpha \beta }(t)$ one finds, using Eqs. (5.57) and (B.5), that the only terms that represent an important contribution to $\widetilde{x^{2}}_{\alpha \beta }(t)$ are

$$\begin{aligned} \widetilde{x^{2}}_{\alpha \alpha }&= \tilde{x}_{\alpha ,\alpha -1}\tilde{x} _{\alpha -1,\alpha }+\tilde{x}_{\alpha ,\alpha +1}\tilde{x}_{\alpha +1,\alpha }, \\ \widetilde{x^{2}}_{\alpha ,\alpha -2}(t)&= \tilde{x}_{\alpha ,\alpha -1}\tilde{x}_{\alpha -1,\alpha -2}e^{i\omega _{\alpha ,\alpha -2}t}= \widetilde{x^{2}}_{\alpha ,\alpha -2}e^{i2\omega _{0}t}, \\ \widetilde{x^{2}}_{\alpha ,\alpha +2}(t)&= \tilde{x}_{\alpha ,\alpha +1}\tilde{x}_{\alpha +1,\alpha +2}e^{i\omega _{\alpha ,\alpha +2}t}= \widetilde{x^{2}}_{\alpha ,\alpha +2}e^{-i2\omega _{0}t}. \end{aligned}$$

(B.11)

The frequencies of oscillation of $( x^{2}) _{\alpha }$ are thus $0$ (for $\left\langle (x^{2})_{\alpha }\right\rangle =\widetilde{x^{2}}_{\alpha \alpha },$ see Eq. (5.73)) and $\pm 2\omega _{0}.$ The extension of this exercise to other powers allows to recover the remaining relevant frequencies (B.10).

This example serves to show that the relevant frequencies $\omega _{\alpha \beta }$ are combinations of resonance frequencies, as dictated by the chain rule (A.29). This latter is the quantum counterpart of the classical rule to construct the combination frequencies intervening in nonlinear systems, $\omega _{j}=\sum _{i}\pm \,n_{ji}\omega _{i},$ where $\omega _{i}$ is a fundamental frequency of oscillation and $n_{ji}$ are integer multiples; these frequencies correspond to the different harmonic and intermodulation frequencies.