The paper [4] was the first one to talk about possible uncertainty relationships. The whole article did only qualitative discussion with no rigorous mathematical derivation. Heisenberg, based on his established QM in matrix form, found that the commutator of the two matrices q and p had the following result: p q − q p = i ℏ . (2.1)

The nonzero result meant that the two matrices could not exchange the order in their product. From this, he guessed that when a particle’s position and momentum were measured, both had some uncertainties.

“Let q 1 be the precision with which the value q is known ( q 1 is, say, the mean error of q); therefore, here, it is the wavelength of the light. Let p 1 be the precision with which the value p is determinable; that is, here, it is the discontinuous change of p in the Compton effect. Then, according to the elementary laws of the Compton effect, p 1 and q 1 stand in the relation q 1 p 1 ∼ ℏ . ” [ 4 ] . (2.2)

Here, the definition of the uncertainty of q was obvious: “ q 1 be the precision with which the value q is known ( q 1 is, say, the mean error of q)”. q 1 is also the measurement precision of q. Then, its value must rely on the measurement equipment and measurement process. For instance, he mentioned a special case where photons were used to impinge a particle. Then, the uncertainty of the particle’s position was the photon’s wave length. This example showed that the measurement precision was indeed closely related to the measurement instrument chosen.

According to the basic viewpoint I.1 in Supplementary Appendix SA, every physical concept should have an explicit mathematical expression, or people would not clearly understand the conception.

First, in QM, a particle is described by its wave function. The wave function is the function of the spatial coordinate q, that is to say, q is an argument in a function, e.g., Eqs. 3, 12–(14) in [4]. In Heisenberg’s words, “Let q 1 be the precision with which the value q is known”. What is the meaning of the q in this sentence was not clear. In Eq. 2.1, q and p are matrices. It seems that Heisenberg unknowingly regarded them as numbers.

Next, we discuss the contents in the QM field. Following the viewpoint II.2, a wave function must be the solution of a fundamental QM equation. Heisenberg put down functions for discussion, but some of them were not the solutions of the Schrödinger equation for the system under consideration.

“If, for any definite state variable η , we determine the position q of the electron as q ′ with an uncertainty q 1 , then we can express this fact by a probability amplitude S ( η , q ) , which differs appreciably from zero only in a region of spread q 1 near q ′ . For example, one can write S ( η , q ) ∝ exp [ − ( q − q ′ ) 2 / 2 q 1 2 − i p ′ ( q − q ′ ) / ℏ ] ” . [ 4 ] . (2.3)

This should be a wave function in QM. Such a function was called a Gaussian wave packet and used in the literature [6, 13–16]. It is time independent. Only the stationary eigenfunctions of a harmonic oscillator are of the form of ψ ( x ) ∝ e − α x 2 , and the possible parameter is the location of the center of the oscillator. Otherwise, we do not know what the Hamiltonian of the wave function Eq. (2.3) is, while describing a moving particle we are talking about. When the Hamiltonian of a free particle is acted on this function, − ℏ 2 2 m ∂ 2 ∂ q 2 ψ ( q ) = ℏ 2 2 m [ 1 2 q 1 2 − ( q 2 q 1 2 + i p ′ ℏ ) 2 ] ψ ( q ) . (2.4)

It is seen that the function (Eq. 2.3) is not a free particle. We do not know what a potential V ( q ) enables us to put down the stationary equation for the wave function (2.3). [ − ℏ 2 2 m ∂ 2 ∂ q 2 + V ( q ) ] ψ ( q ) = E ψ ( q ) . (2.5)

We are unable to write down a V ( q ) other than the oscillator potential, and neither the expression of the corresponding Hamiltonian H.

In Eq. 2.3, q is the spatial coordinate of the wave function, but not the position of the electron. Heisenberg confused the concepts.

Eq. 18 in [4] was S ( t , p ) = e − α t ψ ( E 2 , p ) e − i E 2 t / ℏ + ( 1 − e − 2 α t ) 1 / 2 ψ ( E 1 , p ) e − i E 1 t / ℏ . (2.6)

This was a time-dependent function. When it is substituted into the left-hand side of Eq. 1.1, the result is i ℏ ∂ ∂ t S ( t , p ) = ( − α − i E 2 t / ℏ ) e − α t ψ ( E 2 , p ) e − i E 2 t / ℏ + ( α e − 2 α t 1 − e − 2 α t − i E 1 / ℏ ) ( 1 − e − 2 α t ) 1 / 2 ψ ( E 1 , p ) e − i E 1 t / ℏ . (2.7)

We are unable to find a time-dependent Hamiltonian H ( t ) such that the function Eq. 2.6 satisfies Schrödinger Eq. 1.1: i ℏ ∂ ∂ t S ( t , p ) = H ( t ) S ( t , p ) . (2.8)

Therefore Eq. 2.6 is not a wave function in QM.

For the wave function (22) in [4], one was unable to write a corresponding Hamiltonian as well, and so it was not a wave function in QM.

When writing down a function, one should first prove that it is the solution of a fundamental QM equation or has its corresponding Hamiltonian. Otherwise, it cannot be treated as a wave function in QM.

Third, Heisenberg did not present explicit expressions or rigorous mathematical derivations when he mentioned some physical conceptions.

For example, he mentioned “statistical error” more than once, but we do not know what he meant by it.

In Eq. 2.6, a concept of “radiation damping” was used. However, Heisenberg did not provide the mathematical derivation for the form in Eq. 2.6. In QM, a single particle does not have the concept of “radiation damping.” This concept must belong to a many-body system.

In [4], the argument below Eq. 8 was questionable. A beam of electrons was arranged to run through two fields successively in two different manners. In the second manner, no derivation was presented. Therefore, one could not know how the result Z n l = ∑ m c n m c ¯ n m d m l d ¯ m l was reached.

Fourth, according to viewpoint I.3, when a gedanken experiment leads to a positive conclusion, it cannot explain anything. Such a conclusion could neither be proved nor be disproved.

In short, Heisenberg’s primary paper [4] lacked rigorous mathematics, was not quantitatively related to real experiments, and was not very clear in some physical concepts.

Heisenberg’s paper [4] just considered the measurement precisions of the position and the momentum of a single particle.

In 1929, Robertson [5] derived the famous mathematical inequality, see Supplementary Appendix SB. The conclusion was that the mean square errors of coordinate and momentum obeyed the following inequality: Δ x Δ p ≥ ℏ / 2 . (2.9)

This inequality was believed to be the mathematical expression of the uncertainties of coordinate and momentum that Heisenberg guessed. So, (2.9) was called the Heisenberg uncertainty relation.

Here, we distinguish the concepts of the position of a particle and coordinate. Heisenberg discussed the uncertainties of the measured position and momentum of a particle, so that his assumed relation was called the position–momentum uncertainty relation. However, in QM, a particle at a state is described by a wave function, which is a function of coordinates. The Δ x in (2.9) does not involve the meaning of a particle’s position. Hereafter, (2.9) is called the coordinate–momentum uncertainty relation.

According to the current understanding, the Δ x and Δ p in (2.9) are two quantities related to measurement. We point out in Supplementary Appendix SB that (2.9) is irrespective to both measurement and the content in Ref. [4].

Now, we explain that it is not right to understand the Δ x and Δ p in (2.9) as measurement uncertainties of coordinate and momentum, and as a matter of fact, it is impossible to implement measurement in the way of (2.9).

Usually, the recognition of (2.9) is that if one measures the position x and momentum p of a particle, they cannot be precisely simultaneously measured, and the smaller the measuring deviation of one quantity is, the greater the other. This recognition is incorrect.

Since (2.9) is regarded as the relation between the ncertainties of position and momentum, it ought to be related to the statistics of measured quantities. This prompts us to explain the implication of the inequality and the way of statistics of measured quantities.

We have pointed out above that in [4], there was no rigorous argument and mathematical derivation, and some concepts were confused. However, people a priori believed that the content in this paper was right and the mathematical expression was what Robertson [5] provided. Some textbooks presented (2.9) without explanation [8, 17, 18]. Some others tried various devices in order to explain that this uncertainty relation was correct.

Because the narration in [4] was not clear, when later people talked about the uncertainty relation, they did not have fixed rules, but depended on their own imaginative development. Different people had different explanations. Each explanation was unable to overturn others. Therefore, according to the basic viewpoint I.2, none of the explanations were right. The so-called examples, that were believed to support the uncertainty relation, were just farfetched ex-post explanations.

There are three typical examples giving farfetched explanations: finite-length wave train or a piece of truncated plane wave, single-slit diffraction, and the ground state of hydrogen atom. Before analyzing these examples, we distinguish between two concepts: a particle’s dimension and the uncertainty of its position.

Until now, when discussing the uncertainty relation, often idealized experiments have been concerned, which are irrelative to the experiment of measurement.

Actually, there is no such experiment in which the position and momentum of a particle are measured simultaneously, and their uncertainties are estimated from the measured information, so as to meet Eq. 2.9.

To gain the uncertainty of a particle’s position, one first has to measure its position. Nevertheless, in QM, a particle is described by a wave function, and has a dimension as having been defined in Section 2.3.1. In QM, the concept of a particle’s position is not clearly defined. Because of this fact, in experiments, no measurement of the so-called position of a particle is carried out.

People did measure a particle’s momentum, and estimate the uncertainty from the information of the experiment. However, they did not measure the particle’s position in the same experiment simultaneously. There are two examples [14].

One is that the momentum of a charged particle is measured by deflection in a constant magnetic field, the strength of which is denoted by B. An electron with a charge e enters the magnet after passing through a diaphragm with width d 1 , and leaves it through another diaphragm with width d 2 after having suffered a 180-degree deflection. The trajectory of the electron in the magnet is a semicircle, the radius R of which is equal to half of the distance between the two diaphragms. When the momentum of the electron is measured to be p, the precision is estimated to be Δ p = p R ( d 1 + d 2 ) . (2.28)

At the instant at which the electron enters the magnet through the first diaphragm, it moves along the y axis. The uncertainty is estimated as Δ y ≈ 4 π ℏ e B d 1 . (2.29)

Thus, it seems that Δ y Δ p ≈ 4 π ℏ ( 1 + d 2 d 1 ) , satisfying the coordinate–momentum uncertainty relation.

However, the right-hand side of Eq. 2.29 has been already known before the measurement. In the experiment, the position of the electron is not detected, and consequently, one is unable to estimate the precision of the position of the electron from measurement. This experiment does not need the knowledge of QM.

Another experiment measuring a particle’s momentum is to let a photon collide with the particle. Before the collision, the photon’s frequency ν is precisely known and the momentum of the particle is p. Let us suppose that before and after the collision, the photon and the particle move along a line, set as y axis. After the collision, the measured momentum and frequency are p ′ and ν ′ , respectively. The precision Δ p ′ of the p ′ relies on the Δ ν ′ of ν ′ : Δ p ′ ≈ m c Δ ν ′ ν ′ + ν . (2.30)

Assuming that the uncertainty at the time of the collision is Δ t , Δ ν ′ Δ t ≥ 1 . (2.31)

Within the time, the particle can go a distance of Δ y = | p − p ′ | m Δ t . (2.32)

Substitution of (2.31) into (2.32) leads to the uncertainty of the position of the particle. Δ y ≥ | p − p ′ | m Δ ν ′ = 2 π ℏ m c ν + ν ′ Δ ν ′ . (2.33)

It seems that Δ y Δ p ′ ≥ ℏ , meeting the uncertainty relation.

This explanation is problematic. The relation 2.31 has neither rigorous mathematical derivation nor experimental verification. Eq. 2.32 assumes that within the time uncertainty Δ t , the particle goes with the velocity of about p / m . Unfortunately, Δ t is a part of the time period within which the collision occurs. In this period, the particle’s momentum changes drastically, such that the particle’s velocity cannot be estimated in the way of 2.32. This experiment does not involve the particle’s wave function in QM, i.e., the particle is treated as a classical one.

The common features of these two examples are as follows: the measurement of a particle’s position is out of question; the measured particles are actually treated as classical ones. Though the momentum is measured, the estimated Δ p based on the measurements has nothing to do with Eq. 2.14. In conclusion, these two experiments do not embody the uncertainty relation.

In Ref. [4], having discussed the possible uncertainty relation between the position and momentum, Heisenberg noticed that the product of the coordinate and momentum was of the dimension of angular momentum, and the result was proportional to the Planck constant. He associated the commutator of time and energy, and thus postulated the following commutator: [ t , E ] = i ℏ . (3.1)

Then, imitating the discussion of position and momentum, he thought that the uncertainties of time and energy obeyed, similarly to (2.2), the relation t E ∼ ℏ . (3.2)

Later, people accepted his postulation. Furthermore, imitating Δ x Δ p ≥ ℏ / 2 , people guessed that there was a similar inequality, Δ t Δ E ≥ ℏ / 2 . (3.3)

Since Δ t and Δ E in 3.3 are uncertainties, they should be quantities related to measurement.

Eq. 3.3 is the so-called time–energy uncertainty, but it is even worse than the coordinate–momentum uncertainty relation.

Heisenberg put forth Eq. 3.1 without any derivation and proof. He did not even present the concrete form of the operator E. Hilgevoord [32] thought that “a relation like” (3.1) “does not occur in quantum mechanics”. His reason was that “there is no Poisson bracket defined between t and H. Consequently, in quantum mechanics, one does not have a relation like” Eq. 3.3. “Accordingly, there is no natural analog for energy and time of the ‘canonical’ uncertainty relations” Eq. 2.9.

At the time when QM was established, Heisenberg himself did not know explicitly what the relation between time and energy was. The mathematical theory of QM had not been accomplished yet. In Heisenberg’s paper [4], there was neither rigorous derivation nor an association with real experiments.

Thus, Heisenberg did not give a convincing conclusion, but people deemed that what he said was right. Later, many people tried to show that there was indeed the inequality (3.3). Everyone raised his own version, without rigorous derivation and experimental correspondence.

Until 1961, “there has been an erroneous interpretation of uncertainty relations of energy and time.” [31] Until 1990, “no general agreement has been reached. One finds physicists claiming that ‘there is no energy–time uncertainty relation at all,’ while others stress, for instance, that the relation is applied quite effectively to the analysis of individual short-lived elementary particles (resonances). Even among those who accept the validity of the relation, there is appreciable disagreement as to meanings of the relation.” [43] Until 1996, “It is generally thought desirable that quantum theory entail an uncertainty relation for time and energy similar to the one for position and momentum. Nevertheless, the existence of such a relation has still remained problematic” [32].

As a matter of fact, up to now, there has been no final verdict with respect to the time–energy uncertainty relation.

It should also be noted that Heisenberg discussed the measurement of individual particles, which can be inferred from his examples about the uncertainty relation (2.9). Subsequently, the discussion of time–energy uncertainty should be for individual particles. However, later, people often discuss many-particle systems.

Now we start to carefully analyze the so-called time–energy uncertainty relation.

First, let the two operators in Eq. (B1) be time and energy, respectively, F = t and G = H . Nevertheless, the commutator [ t , H ] (3.4) does not have a definite result. When the Hamiltonian H is independent of time, the result is zero. [ t , H ] = 0 , w h e n   H   i s   i n d e p e n d e n t   o f   t i m e . (3.5)

The key is that the coordinate and momentum operators on the left-hand side of (2.1) have explicit forms, no matter what function they act on. By contrast, the form of Hamiltonian H depends on the system under investigation. Hilgevoord noticed that “for a system of particles, one should not demand a communication relation between t and H as a complement to the ones between q and p, nor could there be such a commutation relation.” [32].

Time t is not an operator. According to Pauli, “the introduction of an operator t is basically forbidden, and the time t must necessarily be considered as an ordinary number (‘c-number’).” [12] The average of t in any normalized state is still the time itself, t ¯ = ( ψ , t ψ ) = t ( ψ , ψ ) = t , which makes the average meaningless.

Therefore, there is no way to derive a time–energy uncertainty relation starting from (3.4) in the way in Supplementary Appendix SB.

People may think that although the Hamiltonian H in (3.4) depends on systems, the operator i ℏ ∂ ∂ t on the left-hand side of (1.1) corresponds to Hamiltonian and is independent of systems. There is a definite commutator [ t , i ℏ ∂ ∂ t ] = i ℏ . (3.6)

It seems, then, that imitating the procedure of deriving (2.9) can lead to (3.3). It is not so. Obviously, the averages of time t and its square t 2 in any normalized function are still t and t 2 , so that Δ t = 0 . It is easily verified that ( ψ , i ℏ ∂ ∂ t ψ ) 2 = ( ψ , ( i ℏ ∂ ∂ t ) 2 ψ ) . Therefore, along this routine, one is unable to reach Eq. 3.3.

The conclusion is that there is no way to acquire (3.3) through the procedure in Supplementary Appendix SB. In [44], Eq. 3.3 was just a hypothesis.

Here we intend to clarify the implication of the operator i ℏ ∂ ∂ t . Someone thought it to be an energy operator and denoted it by [22] E ^ = i ℏ ∂ ∂ t . (3.7)

Following this definition, (3.1) could be understood as (3.5), but this problematic.

When we put down ∂ ∂ t , it is just an operator taking a derivative with respect to time, without any other physical information. When we put down i ℏ ∂ ∂ t , we just let the dimension of the operator become that of energy and make it a Hermitian one, and no physical information is added yet.

An operator should have its eigenvalues and corresponding eigenfunctions under appropriate boundary conditions, such as a momentum operator. The operator i ℏ ∂ ∂ t is not of this property. In [19], an attempt was made to define a time operator, but no eigenvalue and eigenfunction could be given. One may define an operator in some way, but it is meaningless if the operator does not have eigenvalues and eigenfunctions.

Then, why do people think of i ℏ ∂ ∂ t as an energy operator? The reason is based on the fundamental QM equation. In Eq. 1.1, i ℏ ∂ ∂ t is directly connected to Hamiltonian H. When, and only when, in this equation, the operator i ℏ ∂ ∂ t represents the Hamiltonian on the right-hand side of this equation.

It is stressed that i ℏ ∂ ∂ t should not be regarded as a Hamiltonian operator or energy operator carelessly except in the case of (1.1). The following two points are important: (i) according to Eq. (1.1), the operator i ℏ ∂ ∂ t must be related to a specific Hamiltonian of the system under investigation; (ii) only when the operator i ℏ ∂ ∂ t acts on the wave function ψ satisfying (1.1), can it show the meaning of energy, because this action is just that of this specific Hamiltonian on this wave function ψ .

When these two points are met, the result calculated through ( ψ , i ℏ ∂ ∂ t ψ ) = ( ψ , H ψ ) = E ¯ (3.8) is of the meaning of the energy average in this state.

For any function of φ not satisfying (1.1), ( φ , i ℏ ∂ ∂ t φ ) is not of the meaning of energy.

Since the implication of the operator i ℏ ∂ ∂ t is clarified, we are able to figure out the Pauli difficulty, which is as follows [12, 19].

Let Hamiltonian H be independent of time and its eigenfunction be denoted as ψ E . H ψ E = E ψ E . (3.9)

We construct a wave function e − i α t / ℏ ψ E , where α can be any real number. Let the Hamiltonian act on this function H e − i α t / ℏ ψ E , and make the Taylor expansion of the factor e − i α t / ℏ . It follows that H e − i α t / ℏ ψ E = ( E − α ) e − i α t / ℏ ψ E . (3.10)

This result shows that for any α , E − α is an eigenvalue of the H. That is to say, the eigenvalues of the H compose a continuous spectrum covering the whole real axis. However, the energy spectrum of the Schrödinger equation (1.1) must have a lower limit. This contradiction is the Pauli difficulty.

We explain how this difficulty is yielded. When taking the Taylor expansion of the factor e − i α t / ℏ on the left-hand side of (3.10), the commutator [ t , H ] = i ℏ is employed [19], which results on the right-hand side of (3.10). The reason of the Pauli difficulty is the employment of the relation [ t , H ] = i ℏ . We have pointed out above that there is no such relation. Eq. 3.8 is one for stationary states, i.e., H is independent of time. So, Eq. 3.5 has to be employed. Starting from the left-hand side of (3.10), one obtains H e − i α t / ℏ ψ E = E e − i α t / ℏ ψ E , (3.11) instead of the right-hand side of (3.10). If ψ = e − i α t / ℏ ψ E (3.12) is still an eigenfunction of the H, it must meet (1.1). i ℏ ∂ ∂ t e − i α t / ℏ ψ E = H e − i α t / ℏ ψ E . (3.13)

Since ψ E is independent of time, i ℏ ∂ ∂ t e − i α t / ℏ ψ E = α e − i α t / ℏ ψ E . (3.14)

The right-hand side of (3.11) and (3.14) should be equal. It is seen that α = E ; i.e., α must be an eigenvalue of the H, not an arbitrary number.

In [4], the relation (3.3) between the uncertainties of time and energy was guessed without derivation, and the discussion was vague. People believed that what Heisenberg said was right. Some first assumed (3.3), resembling (3.3), and then, tried to derive it by supposing various scenarios. Different persons present the derivation based on their own understanding of the uncertainties of time and energy. Among different derivations, none of them could overturn the others. Therefore, according to viewpoint I.2, none was correct. In fact, every derivation was apparently right but actually wrong. Although it was noticed [31, 32] that some of the derivations were wrong, a thorough analysis is desired.

In the following, we list several derivations and present our comments. In each case, we extract the concepts of Δ t and Δ E , demonstrating that the concepts differ from person to person. The common features are that almost every proof has concept stealing and that no one made the measurements that could match the formulas.

Before the introduction, we emphasize that in inequality (B14), Δ A Δ E ≥ 1 2 | 〈 [ H , A ] 〉 | , (3.15) and Δ A is defined by Δ A = A 2 ¯ − A ¯ 2 = [ ( ψ , A 2 ψ ) − ( ψ , A ψ ) 2 ] 1 / 2 . (3.16)

It is a definite quantity determined by the known wave function but not a variable. One more point should be stressed that Δ A is finite, meaning that it is neither infinitely small nor infinitely large.

1) Using the concept of wave packet [14, 15, 19]:

This is for a single particle. Suppose that the particle is a wave packet with width Δ x , moving along the x axis with speed v. The time it passes one point in the x axis is not definitely determined, but has an uncertainty Δ t ≈ Δ x v . (3.17)

On the other hand, the wave packet has some extension in momentum space, so that the particle’s energy has an uncertainty Δ E . Δ E ≈ ∂ E ∂ p Δ p = v Δ p . (3.18)

The product of these two equations yields Δ t ⋅ Δ E ≈ Δ x ⋅ Δ p . (3.19)

Then Eq. 2.9 is used to result in (3.3), “which limits the product of the spread Δ E of the energy spectrum of the wave packet and the accuracy Δ t of the prediction of the time of passage at a given point” [19].

Comment:

Since the right-hand side of (3.19) is just (2.9), the Δ x and Δ p ought to be evaluated in the way of (3.16). However, the Δ x from (3.17) is the dimension of the wave packet and the Δ p from (3.18) is the increment of the momentum, both not being the meanings of uncertainties. The concept stealing is obvious.

The use of (3.19) means that Eqs. (2.9) and (3.3) ought to be compatible. However, the two relations were thought to express two different and incompatible viewpoints in [19].

Here, Δ t is the time a wave packet needs to pass through a distance. The Δ E is the increment of energy irrespective to measurement.

2) Making use of the formulas in Supplementary Appendix SB [3, 14, 19, 21, 23]:

When a quantity A varies, the time it needs to change Δ A is Δ t = Δ A | d 〈 A 〉 / d t | . (3.20)

We make use of the formula d 〈 A 〉 d t = 1 ℏ | 〈 [ H , A ] 〉 | . (3.21)

Then, by (3.15) Δ A Δ E ≥ 1 2 | 〈 [ H , A ] 〉 | = ℏ 2 d 〈 A 〉 d t . (3.22)

The combination of these three equations results in Δ t Δ E ≥ ℏ 2 . (3.23)

Comment:

We point out that Eq. 3.20 is strange. In the denominate and following equations, A is regarded as an operator, and the numerator should be written as Δ 〈 A 〉 . It is not clear from which fundamental formula (3.20) was derived.

In Eq. 3.20, the Δ t is variable and is the time increment when A has an increment Δ A . In (3.22), both the Δ A and Δ E should be calculated by (3.16), and are not variable. The Δ A in (3.20) is different form that in (3.22). The concept confusion happens from (3.20) to (3.22). Furthermore, both the Δ t and Δ E do not have the meanings of uncertainty.

The authors of [45] recast (3.22) to be the form Δ t Δ E ≥ ℏ 2 d 〈 A 〉 d A by replacement of the Δ A by d A and of the d t by Δ t , which seemed to be a smart way to obtain (3.23), but this was wrong. The d A was a variable and could be taken as infinitesimal, but the Δ A on the left-hand side of (3.22) is calculated by (3.16), so it is finite and not a variable. This distinction stands also for the d t and Δ t . Therefore, the replacements were illegal. The authors of [45] addressed that the quantity A in (3.20) could be arbitrary: “its physical meaning depending thus on the choice of this quantity.” However, the A could not be t.

Here, the Δ t is a time increment when a quantity A changes Δ A , and the Δ E is calculated by (3.16), both being irrespective to measurement.

3) Making use of the difference of two energy levels [3, 14]:

Suppose that a particle had two energy levels, E 1 and E 2 . Their difference is Δ E = | E 1 − E 2 | . (3.24)

When the two states superpose, the particle oscillates between the two states and the oscillation period is τ = ℏ Δ E . (3.25)

Then, Δ E is explained as the uncertainty of the energy level, and τ is explained as the time one needs to observe the system’s variation. Eq. 3.25 is recast to be τ Δ E ≈ ℏ , (3.26) which is explained as the time–energy uncertainty relation.

Comment:

Here, the τ is the oscillation period between two energies of the system, and the Δ E is the difference of the two energies, both being not of the meaning of uncertainty.

Eq. 3.26 is simply the copy of (3.25), but the concepts are endowed different connotations, i.e., concept stealing. In Eq. 3.26, the Δ E and τ are respectively said to be the uncertainties of energy and time, which are created out of nothing.

4) Using the concept of “time packet” [6]:

It was assumed that a particle’s behavior was a pulse or ‘time packet’.

“We consider the case such that ψ ( t ) is a pulse or ‘time packet’, which is negligible except in a time interval Δ t .” This time packet can be expressed as a superposition of monochromatic waves of angular frequency ω by the Fourier integral ψ ( t ) = ∫ − ∞ ∞ d ω G ( ω ) e − i ω t , (3.27) where the function G ( ω ) is given by G ( ω ) = 1 2 π ∫ − ∞ ∞ d t ψ ( t ) e i ω t . (3.28)

As the ψ ( t ) takes only significant values for a duration Δ t , it follows from the general properties of Fourier transformations that G ( ω ) is only significant for a range of angular frequencies such that Δ ω Δ t ≥ 1 . (3.29)

Since E = ℏ ω , (3.30) the width of the distribution in energy, Δ E , satisfies the time–energy uncertainty relation Δ E Δ t ≥ ℏ . ” [ 6 ] . (3.31)

In this way, it seems that the uncertainty relation can be proved.

Comment:

Here, the Δ t is the time the particle exists—from its appearance to its vanishing, and the Δ E is the width of the distribution of the particle’s energy. Both the concepts are strange, because we cannot image what the system is and how its Hamiltonian is written.

It is well known that a light pulse can be produced experimentally. But what about a massive particle? This imagined pulse or ‘time packet’ of a massive particle is not possible. No one is able to find a Hamiltonian H such that the solution ψ ( t ) of Eq. 1.1 is a ‘time packet’. If someone knew such a Hamiltonian, he would have put it down.

5) Making use of the interaction between the measured system and measuring device [10, 24, 46]:

The measured system and measuring device are combined to become a larger system. In other words, the whole system is divided into two parts, measured system and measuring devices, the energies of which are E and ε, respectively. “We suppose that it is known that at some instant these parts have definite values of the energy, which we denote by E and ε, respectively.” “The energies E, ε, on the other hand, can be measured to any degree of accuracy at any instant” [10].

Because of the interaction between the two parts, each time the measurement would cause the energy E to change, say, to be E ′ . It was treated as the transition between the energies E and E ′ . The transition probability for a system subject to a periodic perturbation was given by the formula (43.2) in [10]. By taking ω = 0 in this formula, the transition probability was sin ⁡   2 [ ( E ′ − E ) t / 2 ℏ ] ( E ′ − E ) 2 . (3.32)

According to this formula, “The most probable value of E ′ − E is of the order of the magnitude of ℏ / t .” [10]. Subsequently, it was believed that | E + ε − E ′ − ε ′ | Δ t ∼ ℏ . (3.33)

This was called the “uncertainty relation for energy.”

Comment:

According to this result, the energy conservation in QM was understood in an alternative way. “It shows that, in quantum mechanics, the law of conservation of energy can be verified by the means of two measurements only to an accuracy of the order of ℏ / Δ t , where Δ t is the time interval between the measurements.” “The quantity ( E + ε ) − ( E ′ + ε ′ ) in (44.1) is the difference between two exactly measured values of the energy E + ε at two different instants, and not the uncertainty in the value of the energy at a given instant” [10]. However, “the statement that the conservation law of energy may be violated by an amount δ E during a time δ t ≅ ℏ / δ E …… confuses the energy of the actual system with the energy of the unperturbed system” [32].

It was believed that the relation Δ t Δ E > ℏ “does not signify that the energy cannot be known exactly at a given time (for in that case the concept of energy would have no meaning), nor does it means that the energy cannot be measured with arbitrary accuracy within a short time” [10].

This scenario is totally different from the above ones. Here the energy of a system can be measured in any accuracy, which contradicts the uncertainty of energy.

According to [10, 46], because the measured system is interacted by the measuring device, its energy shifts after the measurement. The amount of the shift and the time interval between adjacent measurements form the time–energy uncertainty relation: “the smaller the time interval Δ t , the greater the energy change that is observed” [10]. This brings a question that why the shorter the time interval, the greater the energy shift.

In the transition probability formula, the difference of two energy levels is used, which is not the energy uncertainty. Furthermore, in [10], Eq. (42.3) was valid under a condition of (42.1) which required that the frequency ω should not be zero. It is hard to understand the transition expressed by (3.32) without releasing or absorbing photons.

Equations (3.32) and (3.33) contradict each other. Eq. 3.32 means that there were two energy levels E and E ′ in the system. The existence of the two energy levels was determined by the Hamiltonian of the system, independent of whether the transition happened or not. What is more, it was thought that Δ ( E − E ′ ) > ℏ / Δ t [46], which meant both E and E ′ had some uncertainties. However, Eq. 3.33 reflected that there was only one energy level E, which, after the measurement, shifted to E ′ . That is to say, before the measurement, there was no energy level E ′ , and after the measurement, there was no energy level E.

Here, the Δ t is the lifetime of an energy level. The Δ E is the energy shift caused by the measurement, and meanwhile, it is the difference of two energy levels.

This scenario was criticized in [31].

In [10, 46], following the above content, momentum variation was discussed by collision as an example. Nevertheless, Eq. 3.32 was obtained by perturbation theory, while collision could not be treated by the perturbation theory.

The last part of Section 44 in [10] related the difference E − E ′ to the lifetime of energy level. This recognition, also seen in other textbooks [6, 7], is going to be expounded in next subsection.

All in all, it is seen from the entries 1) – 5) that people presumed the relation (3.3) and then designed certain ideas to scrape it together. None of the above scenarios were connected to a real measurement. All of them are incorrect.

Different people had different explanations of the Δ t and Δ E in (3.3). In one textbook, more than one explanation arose [3, 10, 14, 18]. This fully revealed the serious confusion regarding these concepts. Some other narrations are as follows.

In [13], the relation (3.3) was given without derivation. The explanation was that “an energy determination that has an accuracy Δ E must occupy at least a time interval Δ t ∼ ℏ / Δ E ; thus if a system maintains a particular state of motion not longer than a time Δ t , the energy of the system in that state is uncertain by at least the amount Δ E ∼ ℏ / Δ t , since Δ t is the longest time interval available for the energy determination.” Here the Δ t had the implication of the lifetime of the state of the system.

In [14], the Δ E was regarded as the uncertainty of the measured energy or the change of the energy of the system, but meanwhile, the Δ E is the difference of two energy levels, which could be both measured precisely.

In [19], there were contradictory statements. One was that “the energy of a system can be determined with arbitrary precision at any time.” The other was that Eq. 3.3, written as τ Δ E ∼ ℏ , “is the lifetime–width relation for unstable systems, i.e., systems which are not stationary and do not correspond to a well-defined value of the energy but rather to an energy spectrum with a certain spread Δ E , called the level width. The mean lifetime τ of the stable (or metastable) state here plays the role of the characteristic time considered above”. “In this case, the accuracy Δ E of the energy measurement is connected with the time Δ t required for the measurement itself.”

In [24], the Δ t and Δ E were replaced by δ t and δ E , respectively. “The time–energy uncertainty relation relates the rate at which the state of a system changes to the uncertainty δ E of its energy. If the state of the system changes appreciably during a time interval δ t , then the time–energy uncertainty relation states that δ t δ E ≥ ℏ . I have written δ t and δ E , rather than Δ t and Δ E , to emphasize that these are not standard deviations.” Here, the time interval δ t was regarded as the lifetime of the system.

One explanation of the Δ t in (3.3) was the lifetime of an energy level [16, 31, 32], and correspondingly, one of explanations of the Δ E was the width of the energy level [14, 19, 31, 32]. It is necessary to clarify the concept of the lifetime of an energy level [47, 48].

In Eq. 3.3, the Δ E , as mentioned in Section 3.3, at least has three explanations: the energy shift, energy width, and difference of two energy levels. All the explanations concern the real part of the energy. Unfortunately, the lifetime of a state is irrelative to the real part of the energy of the state. Let us recall the definition of the lifetime of a state.

In a wave function, there is a factor containing energy E and time t, e − i E t . If the energy is a complex number, E = ε − i γ , (3.34) then e − i E t / ℏ = e − i ε t / ℏ e − γ t / ℏ . (3.35)

The wave function decays with time exponentially. After a time period of about τ ∼ ℏ / γ , (3.36) the wave function almost disappears. Due to this fact, we say that the lifetime of this state is about ℏ / γ .

Here, we emphasize the following points: (1) The lifetime τ of a state is determined by the imaginary part, not the real part, of the state’s energy. (2) The lifetime is defined by (3.36). It is not the case that we have first the two quantities Δ E and τ, which then meet Eq. 3.3. (3) The lifetime of the state is determined by the state itself, irrespective of the measurement process. (4) The lifetime reflects the decay of the state wave function, but is not related to the shift of the energy. From the lifetime, one is unable to gain the information of energy shift. (5) The lifetime does not involve energy level broadening, which is a property of the real part of the energy. From the lifetime, one is unable to gain the information of the energy level broadening. (6) The energy (3.34) has an imaginary part. This fact shows that this is a many-particle system.

In a many-particle system, there are interactions between particles, such as electron–phonon interaction, collision, and so on. Due to the interactions, elementary excitations are formed, and they are of finite lifetimes. An elementary excitation’s lifetime is determined by the imaginary part of its energy [47, 48].

A detailed analysis was given in [25]. The interactions inside a system result in transitions between energy levels. The transitions in turn cause an energy shift and broadening. Now we introduce the analysis.

Suppose that in a system there are two states denoted by a and b, respectively. When there is no interaction, both are stationary states, and their energies are E a and E b , respectively. When there are interactions, the transition between them can occur. Suppose that a and b are, respectively, the initial and final states of the transition. In the course of the transition, the initial state a will vary. The state after the change is denoted by a ′ with energy E a ′ − i Γ , i.e., energy changes, E a → E a ′ − i Γ . (3.37)

The change yields not only a shift of the energy but also an imaginary part of the energy, the latter being determined by transition probability. This imaginary part determines the lifetime of the state a ′ . This is because the amplitude of the state a ′ contains a factor e − i ( E a ′ − i Γ ) t / ℏ = e − i E a ′ t / ℏ e − Γ t / ℏ . (3.38)

It is seen that in about time Δ t a , where Γ Δ t a ∼ ℏ , (3.39) the state a ′ will almost vanish. Therefore, Δ t a ∼ ℏ / Γ is the lifetime of the state a ′ .

Meanwhile, the energy of the final state b has a broadening with a Lorentz line shape, the half height width of which happens to be Γ, too. We denote this half height width by Δ E b . Δ E b = Γ . (3.40)

Then, Δ E b Δ t a ∼ ℏ . (3.41)

Equations (3.39) and (3.41) seem to be the form of the time–energy uncertainty relation, but they are not. The former is the definition of the Δ t a , the lifetime of state a ′ . The latter relates the lifetime of state a ′ and the half height width of the energy of state b. Among these quantities, none has the physical meaning of uncertainty, and none is to be determined by measurement.