Critical look at the time-energy uncertainty relations

TL;DR

This paper critically examines the Heisenberg and Mandelstam-Tamm time-energy uncertainty relations, arguing they lack the rigorous status of position-momentum uncertainty relations and are not universally valid within standard quantum mechanics.

Contribution

It provides a detailed analysis showing that time-energy uncertainty relations are not equivalent to position-momentum relations and are not universally valid in quantum mechanics.

Findings

Time-energy uncertainty relations are not as rigorous as position-momentum relations.

These relations cannot be considered universally valid within Schrödinger and von Neumann quantum mechanics.

The analysis challenges common interpretations of the time-energy uncertainty principle.

Abstract

The Heisenberg and Mandelstam-Tamm time-energy uncertainty relations are analyzed. The conlusion resulting from this analysis is that within the Quantum Mechanics of Schr\"{o}dinger and von Neumann, the status of these relations can not be considered as the same as the status of the position-momentum uncertainty relations, which are rigorous. The conclusion is that the time--energy uncertainty relations can not be considered as universally valid.

Equations

$$\Delta x\cdot \Delta p_x\ge \frac{\hslash }{2},$$

$${\Delta }_{\varphi }x\cdot {\Delta }_{\varphi }{p}_x\ge \frac{\hslash }{2},$$

$${\Delta }_{\varphi }F=\Vert \delta F|\varphi ⟩\Vert ,$$

$$\delta F\stackrel{\mathrm{def}}{=}F-⟨F{⟩}_{\varphi }\mathbb{I},$$

$${\Delta }_{\varphi }A\cdot {\Delta }_{\varphi }B\ge \frac{1}{2}\left|⟨[A,B]{⟩}_{\varphi }\right|.$$

$$[A,B]\equiv [\delta A,\delta B]\ne 0.$$

$${\left|⟨\varphi |[A,B]|\varphi ⟩\right|}^2$$

$${\Vert \delta A|\varphi ⟩\Vert }^2{\Vert \delta B|\varphi ⟩\Vert }^2\ge {\left|⟨\varphi |\delta A\delta B|\varphi ⟩\right|}^2.$$

$$\delta A\delta B=\frac{\delta A\delta B+\delta B\delta A}{2}+i\frac{(-i)(\delta A\delta B-\delta B\delta A)}{2}.$$

$$\{(-i)(\delta A\delta B-\delta B\delta A){\}}^{+}=[(-i)(\delta A\delta B-\delta B\delta A)]\equiv (-i)[A,B]$$

$${\left|⟨\varphi |\delta A\delta B|\varphi ⟩\right|}^2=\frac{1}{4}{\left|⟨\varphi |(\delta A\delta B+\delta B\delta A)|\varphi ⟩\right|}^2+\frac{1}{4}{\left|⟨\varphi |[A,B]|\varphi ⟩\right|}^2.$$

$${\left|⟨\varphi |\delta A\delta B|\varphi ⟩\right|}^2\ge \frac{1}{4}{\left|⟨\varphi |[A,B]|\varphi ⟩\right|}^2$$

$${\Vert \delta A|\varphi ⟩\Vert }^2{\Vert \delta B|\varphi ⟩\Vert }^2\ge {\left|⟨\varphi |\delta A\delta B|\varphi ⟩\right|}^2\ge \frac{1}{4}{\left|⟨\varphi |[A,B]|\varphi ⟩\right|}^2,$$

$${\left|⟨\varphi |\delta A\delta B|\varphi ⟩\right|}^2={\left[Re.(⟨\varphi |\delta A\delta B|\varphi ⟩)\right]}^2+{\left[Im.(⟨\varphi |\delta A\delta B|\varphi ⟩)\right]}^2.$$

$${\left|⟨\varphi |\delta A\delta B|\varphi ⟩\right|}^2\ge {\left[Im.(⟨\varphi |\delta A\delta B|\varphi ⟩)\right]}^2$$

$$Im.(⟨\varphi |\delta A\delta B|\varphi ⟩)$$

$${\left|⟨\varphi |\delta A\delta B|\varphi ⟩\right|}^2\ge \frac{1}{4}{\left|⟨\varphi |[A,B]|\varphi ⟩\right|}^2.$$

$${\Vert \delta A|\varphi ⟩\Vert }^2{\Vert \delta B|\varphi ⟩\Vert }^2\ge \frac{1}{4}{\left|⟨\varphi |[A,B]|\varphi ⟩\right|}^2.$$

$$L_{\lambda }=\delta A+i\lambda \delta B\ne L_{\lambda }^{+}$$

$${\Vert L_{\lambda }|\varphi ⟩\Vert }^2$$

$${\left(⟨\varphi |(i[A,B])|\varphi ⟩\right)}^2-4{\Vert \delta A|\varphi ⟩\Vert }^2{\Vert \delta B|\varphi ⟩\Vert }^2\le 0,$$

$$\delta B|\varphi ⟩=i\kappa \delta A|\varphi ⟩,$$

$${\Delta }_{\varphi }t\cdot {\Delta }_{\varphi }E\ge \frac{\hslash }{2}.$$

$${\Delta }_{\varphi }A\cdot {\Delta }_{\varphi }E\ge \frac{1}{2}\left|⟨[A,H]{⟩}_{\varphi }\right|,$$

$$⟨[A,H]{⟩}_{\varphi }\equiv i\hslash \frac{d}{dt}⟨A{⟩}_{\varphi }.$$

$${\Delta }_{\varphi }A\cdot {\Delta }_{\varphi }E\ge \frac{\hslash }{2}\left|\frac{d}{dt}⟨A{⟩}_{\varphi }\right|.$$

$$\frac{{\Delta }_{\varphi }A}{\left|\frac{d}{dt}⟨A{⟩}_{\varphi }\right|}\cdot {\Delta }_{\varphi }E\ge \frac{\hslash }{2},$$

$${\tau }_A\stackrel{\mathrm{def}}{=}\frac{{\Delta }_{\varphi }A}{\left|\frac{d}{dt}⟨A{⟩}_{\varphi }\right|},$$

$${\tau }_A\cdot {\Delta }_{\varphi }E\ge \frac{\hslash }{2},$$

$$[H,T^n]=i\hslash n{T}^{n-1}\equiv i\hslash \frac{\partial }{\partial T}{T}^n.$$

Full text

PACS: 03.65.-w - Quantum mechanics; 03.65.Ta - Foundations of quantum mechanics; 01.55.+b - General physics;

Key words: time–energy uncertainty relations

1 Introduction

Before the emergence of quantum mechanics, physicists were convinced that always two different physical quantities can be measured at the same time with any accuracy. Heisenberg analyzing such quantities as a position and a momentum of the moving electron found that such a belief is wrong on the quantum level that is in all cases when a particle manifests its quantum properties [1, 2]. Results of this Heisenberg’s analysis was formulated in the form of mathematical formulae, which are know as the uncertainty relations. Heisenberg’s uncertainty relations describe connections between uncertainties of the position and momentum and also between uncertainties of time and the energy [1]. These relations play an important and significant role in the understanding of the quantum world and in explanations of its properties. We have a mathematically rigorous derivation of the position–momentum uncertainty relation but so far within the Schroödinger and von Neumann quantum mechanics there does not exist a rigorous derivation of the time–energy uncertainty relation. Nonetheless the time–energy uncertainty relation is considered by many authors as having the same status as the position–momentum uncertainty relation and it is often used as the basis for drawing far–reaching conclusions regarding the prediction of the behavior of some physical systems in certain situations in various areas of physics and astrophysics and from time to time such conclusions were considered as the crucial. So, the time–uncertainty relation still requires its analysis and checking whether it is correct and well motivated by postulates of quantum mechanics. We present here an analysis of the Heisenberg and Mandelstam–Tamm time–energy uncertainty relations and show that the time–energy uncertainty relation can not be considered as universally valid. In Sections 2 and 3 the reader finds theory and calculations. Discussion is presented in Sec. 4. Sec. 5 contains conclusions.

2 Preliminaries: Uncertainty principle

One of characteristic and the most important consequences of the quantum mechanics is the uncertainty principle. The most known form of this principle is the Heisenberg uncertainty principle for the position and momentum [1, 2],

[TABLE]

where $\Delta x$ and $\Delta p_{x}$ are Hesienberg’s "uncertainties". Unfortunately there is no precise definitions of these "uncertainties" in [1]. The rigorous definition of uncertainties was proposed in [3, 4]. Following [3, 4] the uncertainty relation can be written as follows (see e.g. [5]),

[TABLE]

where $\Delta_{\phi}x$ and $\Delta_{\phi}p_{x}$ are the standard (root–mean–square) deviations: In the general case for an observable $F$ the standard deviation is defined as follows

[TABLE]

where

[TABLE]

and $\langle F\rangle_{\phi}\stackrel{{\scriptstyle\rm def}}{{=}}\langle\phi|F|\phi\rangle$ is the average (or expected) value of an observable $F$ in a system whose state is represented by the normalized vector $|\phi\rangle\in{\cal H}$, provided that $|\langle\phi|F|\phi\rangle|<\infty$. Equivalently: $\Delta_{\phi}F\equiv\sqrt{\langle F^{2}\rangle_{\phi}-\langle F\rangle_{\phi}^{2}}$. (In Eq. (2) $F$ stands for position and momentum operators $x$ and $p_{x}$ as well as for their squares). The observable $F$ is represented by a hermitian operator $F$ acting in a Hilbert space ${\cal H}$ of states $|\phi\rangle$. In general, the relation (2) results from basic assumptions of the quantum theory and from the geometry of Hilbert space [6]. Similar relations hold for any two observables, say $A$ and $B$, represented by non–commuting hermitian operators $A$ and $B$ acting in the Hilbert space of states (see [3, 4]), such that $[A,B]$ exists and $|\phi\rangle\in{\cal D}(AB)\bigcap{\cal D}(BA)$, (${\cal D}({\cal O})$ denotes the domain of an operator $\cal O$ or of a product of operators):

[TABLE]

The inequality (5) is rigorous and its derivation simple. Indeed, let us consider two observables represented by noncommuting operators $A$ an $B$. Then if to apply the definition (4) to operators $A$ and $B$, respectively, one finds that

[TABLE]

Hence for all $|\phi\rangle\in{\cal D}(AB)\bigcap{\cal D}(BA)$,

[TABLE]

which reproduces inequality (5). It is because $(\Delta_{\phi}A)^{2}\equiv\left\|\delta A|\phi\rangle\right\|^{2}$ and $(\Delta_{\phi}B)^{2}\equiv\left\|\delta B|\phi\rangle\right\|^{2}$. The above derivation seems to be the simplest one.

One can find in the literature and in many textbooks the other derivations of the Heisenberg inequality. For completeness they are presented below in short. The most commonly used methods of deriving the uncertainty relation (5), which can be found in the literature, are the following: The first one follows the Robertson [3] and Schrödinger [4]. This methods uses the obvious relation resulting from the Schwartz inequality,

[TABLE]

The next step within this method is (see, e.g., [5, 6, 7, 8] and so on) to write the product $\delta A\;\delta B$ as a combination of the hermitian and anti–hermitian parts,

[TABLE]

Here

[TABLE]

is the hermitian operator. Hence

[TABLE]

Now in order to obtain the desired result one should ignore the first component of the right hand side of the above equation which leads to the following inequality

[TABLE]

and finally

[TABLE]

which is the proof of the inequality (5).

The similar simpler version of the above proof can be found eg. in [9, 10]. This proof also makes use of the inequality (8) but the estimation of the right hand side of this inequality is simpler. Namely within this proof expression $\left|\langle\phi|\delta A\;\delta B|\phi\rangle\right|$ is written as follows

[TABLE]

Next ignoring the contribution coming from the real part of the above expression one obtains

[TABLE]

but

[TABLE]

Thus

[TABLE]

This result together with (8) means that

[TABLE]

The another method to proof the Heisenberg inequality one can find eg. in [11, 12]. Within this method using selfadjoint operators $\delta A$ and $\delta B$ one builds a new non–selfadjoint operator

[TABLE]

where $\lambda=\lambda^{\ast}\geq 0$. Then

[TABLE]

Note that $(+i[A,B])=(+i[A,B])^{+}$ therefore the average value of $(+i[A,B])$ is a real number: $\langle\phi|(i[A,B])|\phi\rangle=(\langle\phi|(i[A,B])|\phi\rangle)^{\ast}$. So we have the second–degree polynomial in $\lambda$, which is positive for any $\lambda$. This implies that discriminant of this equation can not be a positive

[TABLE]

which again reproduces the inequality (5).

Summarizing the above part of the analysis of the Heisenberg uncertainty relations, let us note that from (8) if follows that the equality in the uncertainty relation (5) takes place when $|\phi\rangle$ is an eigenvector of $A$ or $B$ or when vectors $\delta A|\phi\rangle$ and $\delta B|\phi\rangle$ are parallel: $\delta A|\phi\rangle\,\parallel\,\delta B|\phi\rangle$ but from (14) it follows that this equality is possible only if additionally ${\rm Re.}\,(\langle\phi|\delta A\;\delta B|\phi\rangle)=0$. From these conditions the following conclusion results: The necessary nad sufficient condition for the minimum uncertainty, that is for the equality in the uncertainty relation (5) is

[TABLE]

where $\kappa\in\mathbb{R}$. In particular for $A=x$ and $B=p$, that is for the position–momentum uncertainty relation, solutions of the criterion (22) are the so–called coherent states, which have the Gaussian form.

The defect, or perhaps the weakness, of methods (8) — (21) of deriving of the uncertainty relation (5) is that using them one should know in advance what a result should be obtained and which components appearing in the intermediate equations during the derivation process should be ignored. For example following the method (8) — (13) in order to obtain the desired result one should know earlier that the hermitian part in (11) should be ignored. Using the approach described by relations (14) — (18) one should know in advance that the result (5) can be obtained by ignoring in (14) the contribution coming from the real part of scalar product $\langle\phi|\delta A\;\delta B|\phi\rangle$. Finally using the method analyzed in (19) — (21) one should guess the form of such an auxiliary operator $L_{\lambda}$ (see (19)), and then one should know in advance that in order to proof the relation (5) the $\left\|L_{\lambda}|\phi\rangle\right\|^{2}$ should be considered. Comparing methods described in relations (8) — (21) with the method (7) one can see that the derivation (7) of the relation (5) is much shorter and simpler than the others and it does not need any hidden assumption making in advance to find a desired result.

3 Analysis of the Heisenberg and Mandelstam–Tamm time–energy uncertainty relations

Heisenberg in [1] postulated also the validity of the analogous relation to (1) for the time and energy (see also [13]). This relation was a result of his heuristic considerations and it is usually written as follows

[TABLE]

The more rigorous derivation of this relation was given by Mandelstm and Tamm [14] and now it is known as the Mandelstam–Tamm time–energy uncertainty relation. Their derivation is reproduced in [5] and goes as follows: In the general relation (5) the operator $B$ is replaced by the selfadjoint non–depending on time Hamiltonian $H$ of the system considered and $\Delta_{\phi}B$ is replaced by $\Delta_{\phi}H$ and then identifying the standard deviation $\Delta_{\phi}H$ with $\Delta_{\phi}E$ one finds that

[TABLE]

where it is assumed that $A$ does not depend upon the time $t$ explicitly, $|\phi\rangle\in{\cal D}(HA)\bigcap{\cal D}(AH)$, and $[A,H]$ exists. The next step is to use the Heisenberg representation and corresponding equation of motion which allows to replace the average value of the commutator standing in the right–hand side of the inequality (24) by the derivative with respect to time $t$ of the expected value of $A$,

[TABLE]

Using this relation one can replace the inequality (24) by the following one,

[TABLE]

(Relations (24) — (26) are rigorous). Next authors [5, 14] and many others divide both sides of the inequality (26) by the term $\left|\frac{d}{dt}\langle A\rangle_{\phi}\right|$, which leads to the following relation

[TABLE]

or, using

[TABLE]

they come to the final result known as the Mandelstam–Tamm time–energy uncertainty relation,

[TABLE]

where $\tau_{A}$ is usually considered as a time characteristic of the evolution of the statistic distribution of $A$ [5]. The time–energy uncertainty relation (29) and the above described derivation of this relation is accepted by many authors analyzing this problem or applying this relation (see, e.g. [15, 16, 17, 18] and many other papers). On the other hand there are some formal controversies regarding the role and importance of the parameter $\tau_{A}$ in (29) or $\Delta t$ in (23). These controversies are caused by the fact that in the quantum mechanics the time $t$ is a parameter. Simply it can not be described by the hermitian operator, say $T$, acting in the Hilbert space of states (that is time can not be an observable) such that $[H,T]=i\hbar\mathbb{I}$ if the Hamiltonian $H$ is bounded from below. This observation was formulated by Pauli [19] and it is know as "Pauli’s Theorem" (see, eg. [15, 20]). Therefore the status of the relations (23) and relations (2), (5) is not the same regarding the basic principles of the quantum theory (see also discussion, e.g., in [21, 22, 23, 24]).

The Pauli’s conclusion follows from the following analysis: If $T=T^{+}$ and $[H,T]=i\hbar\mathbb{I}$ then

[TABLE]

From this last relation it follows that

[TABLE]

The consequence of the above commutation relations is that for all $\lambda\in\mathbb{R}$

[TABLE]

From the Stone theorem we know that if $T$ is a selfadjoint operstor, $T=T^{+}$, then the operator

[TABLE]

is the unitary operator: $U_{\lambda}\;U_{\lambda}^{+}\;=\;U_{\lambda}^{+}\;U_{\lambda}\;=\;\mathbb{I}$. So, operators $H$ and $H_{\lambda}$ are unitarily equivalent and hence they must have the same spectrum. From (32) it follows that they commute: $[H,H_{\lambda}]=0$ and thus they have common eigenfunctions. The spectrum of $H_{\lambda}$ ranges over the whole real line $\mathbb{R}$ but, by assumption (see, e.g. [20]) the spectrum of $H$ is bounded from below. Therefore $H_{\lambda}$ and $H$ can not be unitarily equivalent. In other words, the operator $U_{\lambda}$ can not be the unitary operator. Hence the conclusion that the operator $T$ defining the operator $U_{\lambda}$ can not be a selfadjoint operator.

The Mandelstam–Tamm uncertainty relation (29) is also not free of controversies. People applying and using the above described derivation of (29) in their discussions of the time-energy uncertainty relation made use (consciously or not) of a hidden assumption that right hand sides of Es. (24), (26) are non–zero, that is that there does not exist any vector $|\phi_{\beta}\rangle\in{\cal H}$ such that $\langle[A,H]\rangle_{\phi_{\beta}}=0$, or $d/dt\langle A\rangle_{\phi_{\beta}}=0$. Although in the original paper of Mandelstam and Tamm [14] there is a reservation that for the validity of the formula of the type (29) it is necessary that $\Delta_{\phi}H\neq 0$ (see also, e.g. [25, 26]), there are not an analogous reservations in [5] and in many other papers.

Basic principles of mathematics require that before the dividing the both sides of Eq. (26) by $\left|\frac{d}{dt}\langle A\rangle_{\phi}\right|$, one should check whether $\frac{d}{dt}\langle A\rangle_{\phi}$ is different from zero or not. Let us do this now: Let $\Sigma_{H}\subset{\cal H}$ be a set of eigenvectors $|\phi_{\beta}\rangle$ of $H$ for the eigenvalues $E_{\beta}$. We have $H|\phi_{\beta}\rangle=E_{\beta}|\phi_{\beta}\rangle$ for all $|\phi_{\beta}\rangle\in\Sigma_{H}$ and therefore for all $|\phi_{\beta}\rangle\in\Sigma_{H}\bigcap{\cal D}(A)$ (see (25)),

[TABLE]

Similarly,

[TABLE]

for all $|\phi_{\beta}\rangle\in\Sigma_{H}$. This means that in all such cases the non–strict inequality (26) takes the form of the following equality

[TABLE]

In other words, one can not divide the both sides of the inequality (26) by $\left|\frac{d}{dt}\langle A\rangle_{\phi}\right|\equiv 0$ for all $|\phi_{\beta}\rangle\in\Sigma_{H}$, because in all such cases the result is an undefined number and such mathematical operations are unacceptable. It should be noted that although the authors of the publications [5, 25] and many others knew that the property (34) occurs for the vectors from the set $\Sigma_{H}$, it did not prevent them to divide both sides of the inequality (26) by $\left|\frac{d}{dt}\langle A\rangle_{\phi}\right|$, that is by $\left|\frac{d}{dt}\langle A\rangle_{\phi}\right|\equiv 0$, without taking into account (35) and without any explanations. What is more, this shows that there is no reason to think of $\tau_{A}$ as infinity in this case as it was done, e.g, in [5, 25]. In general,the problem is that usually the set $\Sigma_{H}$ of the eigenvectors of the Hamiltonian $H$ is a linearly dense (complete) set in the state space ${\cal H}$.

Similar picture one meets when $|\phi\rangle=|\phi_{\alpha}\rangle$ is an eigenvector for $A$. (This case was also noticed in [25]). Then also for any $|\phi_{\alpha}\rangle\in\Sigma_{A}\bigcap{\cal D}(H)$, (where by $\Sigma_{A}$ we denote the set of eigenvectors $|\phi_{\alpha}\rangle$ for $A$), $\left|\frac{d}{dt}\langle A\rangle_{\phi}\right|\equiv 0$ and $\Delta_{\phi}A\equiv 0$. Thus, instead of (36) one once more has,

[TABLE]

and once again dividing both sides of this inequality by zero has no mathematical sense. Now note that the relations (2), (5) are always satisfied for all $|\phi\rangle\in{\cal H}$ fulfilling the conditions specified before Eq. (5). In contrast to this property, results (36), (37) mean that we have proved that the Mandestam–Tamm relation (27) can not be true not only on the set $\Sigma_{H}\subset{\cal H}$, whose span is usually dense in ${\cal H}$, but also on the set $\Sigma_{A}\subset{\cal H}$.

Hence the conclusion that such relations as (27) and then (29) can not be considered as correct and rigorous seems to be justified. Summing up, we have proved that contrary to the uncertainty relations (2) and (5), the relations of type (23) and (29) can not hold on linearly dense sets in the state space ${\cal H}$ and therefore such relations can not be considered as the universally valid.

4 Discussion

Conclusions presented in the previous Section agrees with the intuitive understanding of stationary states. These states are represented by eigenvectors of the Hamiltonian $H$ of the system under considerations and if one knows that the system is in a stationary state represented, assume, by the state vector $|\phi_{\beta}\rangle$ and then $\langle E\rangle_{\phi}\equiv\langle H\rangle_{\phi}=E_{\beta}$ and $\Delta_{\phi_{\beta}}E=\Delta_{\phi_{\beta}}H=0$ and then one is sure that at any time $t$ (and during any time interval $\Delta t=t_{2}-t_{1}$, where $t_{1}<t_{2}<\infty$) the energy is equal $E_{\beta}$ or that $\Delta E=E_{\beta}(t_{2})-E_{\beta}(t_{1})\equiv 0$.

In addition to the doubts discussed above and relating to validity of the time–energy uncertainty relations a thorough analysis of the relation (23) suggests one more interpretative ambiguity. Namely let us consider the minimal uncertainty version of (23):

[TABLE]

Analyzing the ideas expressed in [1, 2], it can be seen that Heinserberg was sure that the time– energy uncertainty relation is a completely general relation and applies in the quantum world without any exceptions. This means that according to Heisenberg’s ideas this relation should be also valid in the case of photons. Then, let us invoke a much older relation, namely the Planck–Einstein relation:

[TABLE]

(where $h$ is the Planck’s constant nad $\nu_{\phi}$ is the frequency), which constituted one of the foundations enabling the emergence of quantum mechanics. This relation plays still a fundamental role in Quantum Theory and it was verified many times using direct and indirect methods.

The frequency $\nu_{\phi}$ is connected with the period $\mathcal{T}_{\phi}$ by the relation

[TABLE]

Now using the last relation (40) one can rewrite the Planck–Einstein relation (39) as follows:

[TABLE]

which means that there is,

[TABLE]

From the mathematical point of view equations (38) and (41) are identical. (To be more precise: the equation (41) is a re–scaled version of the equation (38) and scaling factor equals $\frac{1}{4\pi}\simeq 0.08$). On the other hand the inequality (42) is the strong case of the Heisenberg inequality (23). The problem is that relations (41) (and (39)) combine exact values of time $t=\mathcal{T}_{\phi}$ and energy $E_{\phi}$ (or $E_{\phi}$ and $\nu_{\phi}$) with each other while the equation (38) combines uncertainties of time $t$ and energy $E$. In the light of this analysis the standard interpretation of the Heisenberg relation (38) (and (23)) may not be obvious and correct. Simply: Equation (41) says that if one find that the exact value of the energy of the photon in the state $|\phi\rangle$ is $E_{\phi}$ , then one is sure that the period is $\mathcal{T}_{\phi}$ (or that the frequency is $\nu_{\phi}\,=\,1/\mathcal{T}_{\phi}$) and, of course because the value of $E$ is exact then in this case there must be $\Delta_{\phi}E=0$. At the same time, equation (38) and inequality (23) state that if the value of $E$ is exact and thus $\Delta_{\phi}E=0$ then simultaneously there must be $\Delta_{\phi}t=\infty$, which means that it should be impossible to determine the exact value of the period $\mathcal{T}_{\phi}$ or frequency $\nu_{\phi}$.

It was signaled earlier there is a reservation in [14] that derivation of (29) does not go for eigenvectors of $H$ (Then $\Delta H=0$): it can be only applied for eigenvectors corresponding to the continuous part of the spectrum of $H$. As an example of possible applications of the relation (29) unstable states modeled by wave–packets of such eigenvectors of $H$ are considered in [14], where using (29) the relation connecting half–time $\tau_{1/2}$ of the unstable state, say $|\varphi\rangle$, with the uncertainty $\Delta_{\varphi}H$ was found: $\tau_{1/2}\,\cdot\,\Delta_{\varphi}H\geq\frac{\pi}{4}\,h$. In general, when one considers unstable states such a relation and the similar one appear naturally [32, 33, 34] but this is quite another situation then that described by the relations (2), (5). The other example is a relation between a life–time $\tau_{\varphi}$ of the system in the unstable state, $|\varphi\rangle$, and the decay width $\Gamma_{\varphi}$: In such cases we have $\tau_{\varphi}\cdot\Gamma_{\varphi}=\hbar$ but there are not any uncertainties of the type $\Delta E$ and $\Delta t$ in this relation (see, e.g., [32]). Note that in all such cases the vector $|\varphi\rangle$ representing the unstable state can not be the eigenvector of the Hamiltonian $H$. In general relations described in this paragraph are rather examples of relations analogous to the relation (41) and can not be interpreted as an realization of the Heisenberg’s ideas, i.e. as relation connecting uncertainties of the time and energy.

5 Conclusions

The analysis of the discussion of relations (23) and (29) in previous Sections together with the conclusions presented in [35] show that these time–energy uncertainty relations are not well founded and can not be considered as universally valid. This means that using these relations as the basis for predictions of the properties and of a behavior of some systems in physics or astrophysics (including cosmology — see, e.g., [18, 36]) one should be very careful interpreting and applying results obtained. In general in some problems the use of the relation (29) may be reasonable (see, e.g. the case of unstable states) but then they should not be interpreted analogously to the relations (2), (5) (i.e., as the relations connecting the uncertainties) but as the relations of the type (41) connecting the exact values of quantities corresponding to the time and energy.