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3 - The Subjectivity of Heisenberg's Uncertainty Relationship

"The uncertainty principle of Heisenberg is one of the most important aspects of the Copenhagen interpretation. It is also an interpretational aspect of quantum mechanics."

3.1

3-2 Heisenberg's Wave Packet Description.
        The physical aspect of the problem is illustrated in the following way. In the case of Heisenberg's model, we consider an initial wave with an unlimited coherence time. Within a wave packet, coherence is maintained at one frequency as long as the wave maintains its phase relationship. The amplitude of the signal varies according to a time-dependent sine function, as illustrated on Fig 3-A.
        Heisenberg's calculations are applied to the wave packet formed when the unlimited sine wave is cut up by a time shutter, between t₁ and t₂ as seen in Fig. 3-A. Let us examine that wave packet in more detail.
        Experimentally, there are many ways to take measurements of a wave packet but at the time Heisenberg developed his model, only one method was used. It was standard spectroscopy. With the use of spectroscopy, one can easily obtain the information about the frequency and the bandwidth of the signal as illustrated on Fig. 3-B. However, a spectroscope does not reveal all the existing information contained in a wave.

3-3 Static Analysis.
        Let us consider the following example, illustrated on Fig 3-C. Let us generate a sine wave at frequency f₁, having amplitude A₁, during one cycle and a quarter (between 0 and 450 degrees).
        This is identical to a continuous wave packet, with a cut such that only one and a quarter cycles are saved. Such a description is perfectly clear and complete, and anybody can reproduce such a wave packet from that description without losing information.
        In order to study the wave packet, let us consider taking numerous samplings of the amplitude at different times of the signal presented on Figure 3-C. From these data the original waveform can be faithfully reproduced. Furthermore, the exact original frequency can be calculated by fitting (mathematical fitting by computer) the observed data between t₁ and t₂ with a sine wave. That mathematical fitting must be able to take into account three parameters: a) the frequency, b) the phase and g) the amplitude. The fitting is made on the original undisturbed wave and not on the sharp cut induced by the experimenter. It is that sharp cut that widens the spectral lines in spectroscopy.

        The result of such a fitting leads to the exact frequency of the wave packet and the accuracy achieved can be as good as noise and technology will permit. The fitting will not give a wide bandwidth as it does in the case of the Fourier transform. Using the frequency, the phase and the amplitude obtained by the fitting, the wave packet can be reproduced without Heisenberg's uncertainty.

3-4 Characteristics of Some Phase-Sensitive Frequency Meters.
        Other than the static analysis mentioned in section 3-3, to explain basic principles, fast dynamic frequency meters using phase-sensitive detection have been developed in different frequency ranges. For example, an instrument called an automatic wave meter [3.3], uses phase-sensitive detection to measure accurate frequencies during short periods of integration. The instrument uses the fringe-counting Michelson interferometer technique and can count fringes while taking phases into account. The instrument typically counts electronically one million fringes in one second and can distinguish 1/50^th of a fringe. Consequently, the instrument has an accuracy of 2×10^-8 while counting only 10⁶ fringes. This accuracy (±0.02 fringe) is superior to the one set by the uncertainty principle. The uncertainty principle allows only the detection of 1/2p of a fringe.
        Another method involves the use of phased-locked loops [3.4]. To describe the principle involved, we give here a description of the more fundamental phased-locked loops method to measure frequencies. Phased-locked loops circuits are commonly used in all color TVs, to adjust the phase controlling the color on every sweep of the electronic beam on the TV screen.

3-5 Basic Principles Involved in Phase-Locked Detectors.
        The principle involved in phase-locked detectors corresponds to the following simplified procedure. Let us consider the wave packet described in Fig. 3-A. As soon as the electromagnetic signal enters the system at time t₁, the incoming wave is detected and coupled with a reference signal generator at about the same frequency. When the leading part of the wave-packet arrives at the detector, the phase is, of course, completely random with respect to the reference local oscillator. However, during the first few cycles of the wave-packet, an active electronic circuit changes the phase of the local oscillator until it matches the phase and the frequency of the incoming wave packet. The electronic circuit maintains the matching of the phase until the wave packet ends at t₂. The counting of cycles starts automatically, at the first zero crossing, after time t₁, when the phase lock action between the two signals has been achieved. Then the phase difference between the wave packet and the local oscillator is about zero. The counting measured is typically one million fringes. The counting stops, just before time t₂ at the moment another signal indicates the coming end of the wave packet (through a delay line technique). The electronic control stops the counting, at the moment that the last zero phase is crossed, just before t₂. An atomic clock can measure very accurately the time elapsed during the integer number of cycles.
        In practice, there are a few cycles that are wasted during the initial period of electronic phase adjustment but this is negligible compared with the total of one million cycles usually counted. That knowledge of the number of cycles, with an accuracy of ±0.02 cycle, provides much more accurate information about the wave parameters than can be detected by a spectroscope, because the phase relationship is now known and the detector is not handicapped by a simple spectroscope. One of those instruments [3.3] could give reproducible results within ±0.02 cycle. This is smaller than 1/2p, contrary the Heisenberg uncertainty principle. It is smaller than 1/15p.

3-6 Confusion between Poor Instrumentation and a Basic Phenomenon.
        Heisenberg was misled because he assumed that the only way to measure a wave packet was by measuring its spectrum. He even dared to claim that no other information than that obtained with a spectroscope can even exist. Heisenberg's interpretation is compatible with the Berkeley-Copenhagen interpretation on non-realism. According to the Berkeley-Copenhagen interpretation, the wave packet (or the photon) is created at the collapse of the wave function and does not exist independently of the observer. Heisenberg never tried to measure nor suggested measuring the frequency by using a more sophisticated method than the spectroscope.
        The limit suggested by Heisenberg can now be interpreted either as an instrumental limit due to the particular use of a spectroscope or as a mathematical limit due to the information lost while using the Fourier transform without the imaginary part. In the case of a phase-sensitive detector, there is no fundamental limit. With a phase-sensitive detector, the limit of resolution depends only on the limit due to noise. It is unfortunate that Heisenberg could not distinguish between the loss of information given by a poor instrument and a fundamental principle.

3-7 Hidden Criteria.
        We have seen that the Heisenberg uncertainty relationship is not fundamental in physics. We will now see that the amount of information lost, described as h/2p, is not absolute but quite arbitrary. It depends on the units chosen.
        Let us go back to Heisenberg's model to describe the wave packet, as illustrated on Fig. 3-A. We will show how one can obtain three different limits of resolution, using a method compatible with the one used by Heisenberg. The results will be discussed afterwards.
        From Fig. 3-A, let us consider that we are counting the number of cycles, in the wave packet starting at time t₁. Let us assume that the phase is zero when we start counting. When the observing time Dt has elapsed, we have counted x cycles. A fraction of a cycle cannot be counted. However, after counting the last cycle, an unknown fraction of another cycle has started before the end of the time interval Dt. Therefore the exact number of cycles is a fraction between number x and x+1. Then, the maximum possible frequency is:

3.2

        The minimum frequency is:

3.3

        The error in frequency due to the random phase at cutting time t₂ is:

3.4

        Therefore

DnDt = 1 cycle

3.5

        Multiplying both sides by h and using the Planck relation E = hn, we find that:

DEDt = h cycles·h

3.6

        where cycles·h means the unit cycle, multiplied by the units of h.

        Eq. 3.6 is a form of uncertainty when cycles are counted. Before concluding, let us examine the case of radians.
        In the experiment described, instead of counting cycles, the detector could monitor the phase of the incoming wave, counting an integer number of radians. The radians are recorded and counted until the time Dt has elapsed. It is found that y radians are counted at the end of the time interval Dt. Again, after the last radian has been counted, an unknown fraction of a radian has started before the end of Dt. Therefore, the number of radians in the wave packet is between number y and y+1.
        Since one cycle equals 2p radians, the maximum possible frequency is:

3.7

        The minimum frequency is:

3.8

        The error in frequency due to the random phase at the cutting time t₂is:

3.9

        Therefore

3.10

        Multiplying both sides by h and using the Planck relation E = hn, we find that:

3.11

        Although this might appear repetitive, let us finally examine what happens in the case of counting individual degrees. Using the same method, we can find the maximum possible frequency when we count z degrees during the time interval Dt:

3.12

        The minimum frequency is:

3.13

        The error in frequency due to the random phase at the cutting time t₂ is:

3.14

        Therefore

3.15

        Multiplying both sides by h and using Planck relation E = hn, we find that:

3.16

        Three different equations have been obtained from that method. They are Eq. 3.6, 3.11 and 3.16. How can they be different? Since the equations are different, one must search for a hidden parameter. The difference between these three equations comes from that we have arbitrarily set up an experimental error of respectively one cycle, one radian and one degree when measuring the wave.
        With respect to the resolution, the method of counting an integer number of radians is equivalent to the Fourier transform. The Fourier transform however gives supplementary information because one can get the shape of the distribution while the method of counting the integer number of radians (or any determined angle) gives an average value of DE. Other instruments such as the lock-in amplifier can show the frequency distribution experimentally. Of course, the mathematical distribution given by the Fourier transform has an absolute shape (Fig. 3-B). It is the absolute shape of the resolution given by a spectroscope.
        Since Heisenberg's resolution is mathematically identical to the result obtained when we are counting an integer number of radians (Eq. 3.11), we must conclude that the resolution of one radian corresponds to Heisenberg's criteria of resolution. This resolution of one radian is exactly equal to the loss of information that is intrinsic to the use the Fourier transform when the phase is neglected. In other words, Heisenberg's criteria of resolution has been set to be mathematically equal to the loss of information when we use a spectroscope. It is certainly arbitrary in physics to decide that the exact amount of information loss due to the use of a particular instrument (a spectroscope) should become a fundamental principle. Furthermore, we know that such an information loss does not exist when we use a different detecting method like the phase-lock detector.
        The angle of 1 radian, 1/2p cycle or 57.3 degrees, used by Heisenberg is certainly not an absolute or physical criterion. He could have decided to count cycles (or any other unit). In that case, the so-called fundamental uncertainty relationship would be different as shown in equations 3.6 and 3.16 In fact, it was neither rational nor coherent for Heisenberg to use the unit radian since in the same equation, he uses E = hn, in which n is expressed in cycles and not in radians.
        Whatever the interpretation is, the relation should be written:

DEDt = kh

3.17

        where k is any arbitrary criterion of resolution corresponding to many cycles, one cycle or a fraction of a cycle that the detector can resolve. The units of k are cycles. Heisenberg (consciously or not) assumed that k = 1/2p to fit the resolution of a spectroscope. The units of k are compatible with the dimensional analysis of E = hn that shows that the units of h are equal to Js/cycle. Using a phase-sensitive detector, the coefficient k can be experimentally as small as permitted by technology. In fact, there is no fundamental limit.
        The uncertainty relationship claimed is uncertain for another point of view. Some scientists like Schiff, [3.5] state that the relation is good "within an order of magnitude". Furthermore, Schiff writes the equation as an approximation using the symbol ~ giving:

3.18

        More recently, Van Name [3.6] gives a different amount of uncertainty using the relation:

3.19

        It is clear that all those values correspond to a rough estimate of a practical limit when using the Fourier transform or a spectroscope to analyze data. Clearly, the uncertainty relationship has no fundamental meaning.

3-8 Irrelevance of the Planck Constant h to Photon Energy.
        It is believed that following Eq. 3.1, the Planck constant determines the amount of uncertainty of energy of the photon. On the contrary, one can show that the uncertainty is not a function of the Planck constant. Let us divide both sides of Eq. 3.1 by h and let us substitute E by its value hn. Of course, such a division on both sides of the equation does not change the nature of the equation. This gives:

3.20

3.21

        Eqs. 3.20 and 3.21 show that the uncertainty relationship does not change at all even if the Planck constant h has a different value. We must conclude from the above that the Planck constant is fundamentally irrelevant to the uncertainty relationship, even if we are accustomed to believing that h appears to determine the amount of uncertainty.

3-9 Hidden Units.
        Another observation is related to hidden units. In practice, the unit cycle is overlooked. The absence of the unit cycle is confusing because, unfortunately, as seen in the case of Heisenberg's uncertainty relationship, it makes it impossible to detect mixed units, such as cycles and radians, when they are used in the same equation.
        Finally, from Eq. 3.11, we can see now that the units of DEDt = h/2p are: "radian·Joule·s." and not simply "Joule·s." To avoid confusion, it is essential to indicate clearly that the units are "radian·Joule·s".
        So, there is a hidden and arbitrary criterion of resolution in Heisenberg's uncertainty relationship which means that there is no fundamental physical principle involved. Heisenberg's relationship (Eq. 3.11) which corresponds to a criterion of one radian, is just as arbitrary as Eq. 3.16, which corresponds to a criterion of one degree (or 1/57.3 radian), as Eq. 3.6, which corresponds to a criterion of one cycle. Since the Heisenberg uncertainty relationship is an arbitrary criterion that is one of the most important aspects of the Copenhagen Interpretation [3.1]we must realize that it is necessary to reevaluate the significance of Quantum Mechanics [3.1].

3-10 The Relationship E = hn
        In physics, we have the relationship:

E = hn

3.22

        We will show the units in this relationship are not coherent. We know that the units of E are Joules. The units of h are Joules-second. The units of frequency n are "cycles per second". Therefore E = hn gives:

3.23

        We divide both sides of 3.23 by [Cycle]. This gives the energy of the photon per cycle. After simplification, we get:

3.24

        Eq. 3.24 gives: (Joule/cycle) = Joule. That equation does not make sense because the units are not coherent. "Energy per Cycle" cannot be equal to "Energy". This shows the incoherence of the units in the relationship E = hn. The missing "cycle" corresponds to quantizing the energy per cycle. This will be discussed later.

 

References

 

Chapter2 Contents  Chapter 4

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