Written by | Lu Jun (Institute of Physics, Chinese Academy of Sciences)
Source | This article is from Physics, No. 7, 2020
01
introduction
Everything is numbered, the evolution, life, death and interconnection of things at different levels in nature, and the choice of plurals for description is not only concise but also natural. For example, in an AC circuit containing both resistor R and inductor L or capacitor C (the driving angle frequency is set to ω), the conversion between the current waveform and the voltage waveform does not need to perform complex calculus operations for L or C, just through the imaginary unit j (the habit in electrical engineering, to distinguish the current identification, in fact, it is equivalent to the well-known imaginary unit i in mathematics), you can add or subtract jωL and R composite into a complex impedance or add and subtract jωC and 1/R composite into a complex conduction, and the impedance and admittance are calculated, in addition to the pluralization of the operation rules. As with resistance and conductance operations, there is no difference in the analysis and calculation using rules such as series parallel connection. Why did the introduction of imaginary units so magically simplify complex circuit problems? How did the simplicity and convenience we enjoy developed (Note 1)? How is such a wonderful correspondence between pure mathematics and the real physical world established? Is there any revelatory connection between such a miraculous tool for depicting the world and understanding the world? These questions are actually the object of my years of professional work and life related to complex measurement. It happened that on Children's Day 2019, to celebrate the 70th birthday of a respected instrument maintenance engineer, the author made an oral summary of his knowledge of complex and plural measurements. This article is a text summary. The following chapters analyze and discuss these issues separately. Because the main content is the original text of the author's reflection and summary, the relationship between this article and the plural textbook hopes to be: after reading this article, I want to learn the plural again, and I want to read this article after reading the plural.
02
What is a plural?
Fig. 1 Illustration of the stepwise extension process from natural numbers to complex numbers
After the introduction of zero, negative numbers and irrational numbers, human understanding of numbers has gradually expanded from natural numbers to real numbers, basically completing the complete and non-repetitive correspondence between the points on the number axis and the results of the finite or infinite addition, subtraction, multiplication and division of natural numbers, however, on the road to solving problems such as the root of the quadratic or cubic equation, with the new situation that the square root of the negative number does not correspond to a certain real number, it is initially discarded as an unnatural thing. It now appears that the real numbers have the defect that the open square operation cannot be self-enclosed, and later through the redefinition of the -1 square root (usually recorded as i) as the unit of imaginary numbers, the concept of numbers has been extended from the basic real numbers to the complex numbers composed of one real and one virtual pairs of numbers, and finally completed the closed definition of the addition, subtraction, multiplication, and division of the square square and its finite and infinite mixed operation results. The concept of numbers is gradually expanded, summarized in Figure 1, which lists several special numbers that are landmarks in human understanding of the concept of logarithms, including 1, 0, -1, π, e, and just mentioned i. The π pi has been in contact with many people since elementary school, so it is familiar to more than 6 decimal places without explanation. And the natural base e = 2.71828... Why special, a little mention, this lies in the exponential function y based on it
Fig. 2 Algebraic and geometric diagrams of complex numbers
The equivalent conversion relationship between the addition of complex numbers and the exponential expression is shown in Figure 3, for more comprehensive complex number calculations and complex variable functions, please refer to the special complex number textbook [1], it should be pointed out here that some authors may be figure-saving, and simply but incorrectly expressed as the arctangent of b/a when the real imaginary part seeks the amplitude angle, this simplicity is not only wrong when the value of a is 0, but also that the range of functions of the arctangent is only in the first and fourth quadrants. When the plural is in the second or third quadrant, the problem of folding towards the first fourth quadrant will inevitably arise. The correct approach is to classify the symbols of the real and imaginary parts, that is, the need for two-parameter input functions, there are already ready-made function modules available for call in a variety of programming languages, such as the scripting language generally supports the atan2 function, which supports two-parameter input and can output a full four-quadrant angle, graphical languages such as LabView directly have real virtual to amplitude amplitude conversion functions. It is also worth mentioning that although the real imaginary part and the polar coordinate representation can be converted to each other, strictly speaking, the two are not completely equivalent. The key is at the origin, the real imaginary part of the definition of 0 + i0 is natural, but the polar coordinate representation does not have a definite phase angle, although the programming language usually sets the phase angle of the origin to zero, the user should be aware of this subtle but essential difference. In any case, if the reader encounters the arctangent of b/a as the phase angle of the complex number directly in the article or textbook after reading this article, there is every reason to directly doubt the author's understanding and cognitive level of the complex number.
Figure 3 Equivalence conversion between two basic representations of a complex number
The extension of real number operations to complex number operations can not be completely bound by the rules of real number operations because of the new phenomenon, some common sense in real numbers, may not be applicable to the processing of complex numbers to attract attention, here is an example of a counterintuitive point in complex number multiplier operations: complex number multiplier operations will cause a paradoxical paradox of self-contradiction, as shown in Figure 4, it is recorded that Originally proposed by Thomas Clausen, who was born as a cowherd in 1827 [2]. Clausen's paradox is not actually a problem with complex numbers, but rather that in complex multiplication operations, the phase of the complex base is multivalued, and the phase multiplied by the exponent of the multiplication of the exponent of the multiplication operation should be the principal value of the phase plus 2πk (k is an integer). The results of the two power operations shown in Figure 5 are cited
Figure 4 Clausen's problem of the complex exponential paradox[2]
Fig. 5 Example of multivaluability of a virtual exponent
At this point, the basic concepts and operation rules of complex numbers have been introduced. However, I have to admit that how the plural was developed, the previous introduction is only an overly simplistic version, and the real human understanding and the application of the history of the plural is relatively large, if I want to give an excuse for this imprecise introduction of history, it is nothing more than to save the reader's precious time, and this introduction does not affect the acquisition of the knowledge of what the plural is. If you only focus on knowledge, regardless of the truth, you can skip this paragraph and go straight to the next chapter. However, any reader who has a little interest in the truth is recommended to patiently read the text of this paragraph and summarize Figure 6 of this paragraph. The author believes that knowing the process of true knowledge generation not only allows us to cherish the knowledge we have easily learned, but also makes us realize that the knowledge we learn is not the end of human knowledge, so that our hearts can be passed on, and we can get the motivation to further innovate and extend human knowledge. As shown in Figure 6, this is a millennia-old time scale through the key of the event nodes (what/when/who) in sequence to describe the process of human understanding and application of the plural, the original data of the time and person involved in the event is partly from Dr. Nahin's book and mathematical manual [2,3], in order to better understand this progression process, the author divides these events into several categories:
Figure 6 Summarized on a millennium scale shows the historical process of human understanding and application of the plural
(1) Foundation laying
The original origins of complex numbers, from algebra and geometric direction, respectively, can be traced back to Arithmetica (Arithmetica), compiled by Diophantus in 300 AD, and Elements (Geometric Original) by Euclid in 888 AD.
(2) Laying the groundwork
In 1545 Cardano systematically discussed negative numbers and operations in Ars Magna, revealing for the first time the concept of square root numbers of negative numbers; in 1637, Descartes disclosed the a+ib analytic method for treating complex numbers in La Geometrie; Wessel in 1797 exposed the idea of a plural processing vectors composed of virtual and real axes; Cauchy In 1814, the differential conditions of complex functions were exposed, which opened the calculus of complex functions; Laurent exposed a set of series ideas and zero and pole processing methods unique to complex functions in 1843, introducing the number of retention (the coefficient of the one-to-reciprocal term of The Laurent series); in 1807 Fourier disclosed the multiplier but amplitude stable sine cosine as the series of the virtual real part for the decomposition of functions; in 1785 Laplace (Laplace) A complex exponential transformation method is introduced, and it is sometimes necessary to study the instability process.
(3) Finishing touches
The previously mentioned Euler identity was publicly proved by Euler in 1748 that without Euler, this identity may have been introduced, but it is not known how many years later; gauss, the mathematical prince gauss, who has always been cautious and low-key, finally publicly established the historical status of the complex plane in 1813, and the full promotion of the complex began; Minkowski announced in 1907 that by multiplying the units of imaginary numbers and the speed of light with time, three-dimensional space is composite, introducing four-dimensional space-time. Proved absolute invariance in relative change in space-time; Schr dinger proposed the world-famous equation for electron wave dynamics in 1926, despite the original text Phys. Rev., 28, 1049 (1926) does not explicitly write that after his death, he inscribed an equation with the appearance of imaginary units on his tombstone (as listed in the 1926 event block in Figure 6), but the appearance of imaginary units and phase ±2πEt/h in part 9 of the original text for the time-containing wave function is sufficient to express the oscillatory properties of the wave function at a specific energy and inspire posterity to understand the mystery of quantum mechanics.
(4) Carry forward
First of all, it is necessary to mention Riemann, although he died young at the age of 40, in his short life, in addition to the aforementioned Riemann conjecture, he made important contributions to complex analysis like Cauchy, and more creatively carried out complex analysis from the perspective of differential geometry than Cauchy, and for the first time introduced non-Euclidean geometry to the human mathematical knowledge base, which provided a good preparation for the birth of general relativity and string theory in the 20th century.
Dirac made a relativistic modification of the wave function of quantum mechanics, the key step was to turn the Hamiltonian operator into a complex matrix of 4 ×4, using the anti-easy nature of the matrix to reconcile the contradiction between the antisum of time and space square antisum and the first-order differential equivalence, so that the fourth component of the modified wave function corresponds very well to the 1/2 spin of the interpreted electron, and predicted the existence of positrons, which greatly promoted the development of quantum mechanics.
In the process of unifying the electromagnetic and gravitational fields and constructing the great gauge invariance theory, Weyl's crucial step is to find that the symmetry corresponding to the conservation of charge is the phase invariance of the complex wave function, and the phase invariance law can be derived from Maxwell's equations in electromagnetism, which is the ultimate cornerstone of the electromagnetic interaction theory.
C. N. Yang (C. N. Yang) after painstaking group theory deduction, using the 2×2 complex matrix to find the iso-spin symmetry between protons and neutrons, and later Murray Gell-Mann through 3×3 complex matrix operation to further understand the construction rules of elementary particles in the microscopic world, so as to accurately explain or predict the types and interactions of all microscopic elementary particles, Mr. Yang and Gellman introduced the theory known as the Standard Model to achieve the great unity of all known fundamental forces except gravity. By the way, in 2000, Mr. Yang reviewed the three main themes of physics in the 20th century: quantum, symmetry, and phase factor.
(5) Regrets
In the development of complex numbers, from binary complex numbers to multivariate complex matrices, algebra has also experienced the attempts of ternary numbers, quaternions, and even octonates, the most famous of which is the quaternion founded by Hamilton, unfortunately, the simplicity and effect of algebraic operations of multivariate complex numbers are not as simple as 2×2 and 4×4 complex matrices, so that in the big stage of the development of physics in the 20th century, Hamilton himself did not get the practicality expected by Hamilton. Although there are also unique applications in smaller discipline branches, such as when the mode of quaternion is set to 1, it can be used to describe three-dimensional spatial rotation, and spatial navigation has smaller errors and higher stability than the usual Euler angle method. Of course, formally replacing lijk with 2×2 complex matrix groups does not affect the substantial similarities, and in this sense, formal unappreciation should not obscure the greatness of Hamilton's supercomplect number ideas and logic.
In addition, in the middle of the 19th century, when the concept of complex numbers had been recognized and applied by the mainstream of mathematics, Boole, the founder of logical algebra, insisted that imaginary units were unexplainable symbols, which now seemed difficult to understand, but binary bit logic had its own advantages and development paths, imaginary numbers were indeed unnecessary in numerical logic, and perhaps in the future when analog calculations and numerical calculations were equally divided, the importance of complex numbers in computer architecture would be self-evident.
Moreover, Maxwell, in his 1865 paper "A Dynamic-al Theory of the Electronmagnetic Field", did not use imaginary numbers, and the number of sub-equations reached as many as 20, and it was thanks to Oliver Heaviside and Willard Gibbs that they were greatly modified by vector analysis to make Maxwell's equations composed into an easy-to-understand form of four sub-equations. Of course, Maxwell's paper is epoch-making in synthesizing and firmly establishing the concept of field, introducing displacement currents, and proving that electromagnetic waves are light through wave velocity.
(6) Interlude
We know that negative numbers are the source of imaginary numbers, the introduction of imaginary numbers must be premised on negative numbers, but for some reason, in ancient China to introduce and use negative numbers is very natural, and the West officially accepted negative numbers about 1700 years later than China, but the West immediately opened the study of imaginary numbers after accepting negative numbers, and Chinese did not see any substantial contribution to the development of imaginary numbers, but by the "Nine Chapters of the Book of Numbers" Qin Jiushao proposed the second, third and higher equation solution algorithms, it can be seen that he is the closest person to the discovery of imaginary numbers in the ancient Chinese. And what is even more valuable is that the Nine Chapters of the Book of Numbers is a mathematical work on the earth for about 600 years between the Original Geometry and the Great Yanshu.
In addition, before the introduction of Euler's identity, the computation of complex numbers usually required superb mathematical skills, such as John Bernoulli's technique of (x+i)(x-i) = 1+x 2 to find the intrinsic relationship between arctangent and logarithmic functions, and applied it to the factorization of imaginary numbers, which Leibniz evaluated as a beautiful magic of extraordinary intelligence.
(7) Can be expected
Like other human knowledge systems, the development of the plural should not be the end point so far, and more important development events in the future can be expected. We know that the quantum mechanics characterized by the complex wave function are all good, have been proven correct many times, and have not yet found events that have been confirmed by experimental results, but they cannot deduce the gravitational equation, which makes people worry about what is wrong, and the breakthrough event of the illusion that the realization of the great unity or the proof that the great unity is impossible to achieve, let us strive to make it happen in our lifetime.
03
How do I measure complex numbers?
Although the plural and its importance are widely known, it is undeniable that the plural measurement has long been a niche subject, the author once listened to the main lecturer in an online course at Nanjing University and said that the plural can not be measured, perhaps the teacher's expression has a small context that the author did not notice, but the side reflects the fact that the group that understands the plural measurement is much smaller than the group that understands the complex number. There are many peers who use instruments that are essentially complex measurements at the signal or wave function level every day, such as spectrum analyzers, impedance analyzers, dielectric analyzers, vector network analyzers, AC susceptibility meters, AC voltmeters, etc., but because the measured physical quantities or instrument names are not easy to see that they are making complex measurements, the complex measurement itself is rarely concerned.
Real-world complex measurements usually do not directly have a pair of abstract coordinate axis planes like Figure 2 for people to measure with a ruler, but hidden in the changing wave function, such as time-varying AC signals or waveforms that fluctuate with spatial changes, the frequency of the time domain or the number of waves in the airspace can vary widely, and the corresponding complex measurement of the real imaginary result may be the same, but the premise for the two complex numbers to be comparable in the physical world must be "in the same boat", which will be discussed in the next chapter.
Similar to other measurement methods that require probes such as photons, electrons, neutrons, etc. to see structural information equivalent to the probe, the complex measurement wave function generally also uses a complex "probe", a reference wave function that is in step with the measured wave function, as shown in Figure 7 The reference signal is carried by the sine signal after frequency stabilization. In order to obtain both the real and imaginary parts, the reference signal is divided into cosine signals and light blue sine signals in the blue line that are locked into 90° by phase difference pairs through the phase stabilization parts, which are actually multiplicative parts of two in and one out, and the measured signals that need to be measured in red are multiplied separately after the output of the measured signal obtained by low-pass filtering is the same as the real and imaginary parts of the reference signal frequency point ω0, respectively, as shown by the green and light green lines, so as to complete the complex measurement process. In the case of insufficient signal strength, of course, you can also choose an auxiliary front or rear amplification. By the Fourier series rule, the sine and cosine signals of the same frequency are orthogonal (orthogonal means that the output of the two after multiplication and low-pass filtering components is zero), so that the real and imaginary measurements remain independent of each other in principle
Fig. 7 Schematic diagram of the measurement principle of the lock-in amplifier[4]
From the measurement principle to see that the stability of the ω0 frequency point is extremely important, in the ordinary oscillation source frequency stability is not enough, the need to increase the phase lock loop (PLL) part, its structure as shown in Figure 8, in the forced external reference mode, the controlled oscillation source in order to be consistent with the external frequency point, the need to measure the phase difference with the external reference signal in real time and self-adjustment, through the phase difference lock to maintain the stability of the measurement frequency point. Therefore, this complex number measurement method and tool is often referred to as lock-in amplifiers.
Fig. 8 Diagram of phase-locked loop structure[4]
It can be seen that the key to the use of phase-locked technology to achieve complex measurement lies in the phase-locked loop tracking of the AC change dimension and the quadrature phase-sensitive detection measurement of the real imaginary part, for more professional introduction, please refer to the literature [4]. The meaning of the lock phase here, in order for the layman to better understand this powerful tool for complex measurement, do a little extended interpretation: lock phase, and the idiom Xiangfu godson as the verb of the "phase" meaning tacit understanding, the relationship between the husband and wife is actually a long-term phase and pace through the lock to achieve a state of happiness, if the lock is more troublesome. In addition, the phase-locked loop, a magical component that automatically maintains a stable pace but has a certain ability to self-regulate with the external environment, is not a primitive machine mind?
04
A new method of complex number measurement
15 years ago, when I first needed to start contacting lock-in amplifiers because of weak magnetoelectric coupling measurements, the laboratory had an old EG&G PARC Model 124A from Princeton Applied Research that manually set paper-and-pencil records, and when I looked at the waveform while scanning the frequency and using the paper-and-pencil boring record lock-in measurement results, somehow an idea appeared, with waveform data (at that time, the domestic digital storage oscilloscope was already available, I happened to buy the now long-discontinued General Source Rigol DS5062C this domestic ashes-level digital storage oscilloscope), I can complete the lock-up ah, completely replace the function of Princeton's EG&G 124A, why bother to spend energy on antique lock-up? When there is an idea, it usually automatically has non-stop motivation, and after a few months, after dll's C call drive interface, self-taught virtual instrument LabView graphic language, vi implementation of phase-locked algorithm, automatic measurement program control all the way through the level, even the originally conceived function was realized, and the phase-locked performance and measurement efficiency have been significantly improved (performance and efficiency comparison was later published in Meas. Sci. Technol. On [5]), I finally scrapped Princeton's EG&G 124A completely, and I was liberated from the boring work of hand-down transcription. Using a low-end oscilloscope of several thousand yuan in China, coupled with a virtual instrument algorithm written by myself, perfectly replaced the lock-in phase of several thousand dollars imported from Princeton, I realized the fourfold realm of "seeing", "measuring", "changing" and "taming" used by the oscilloscope, and shared it publicly on the Internet [6,7]. The code that I have worked so hard to write is also unreservedly exposed on the Internet[8], and the act of exposing code is, in my opinion, to confidently and openly put aside past achievements and actively adopt a posture of continuing to innovate for the future. In addition, this experience was very helpful for training me to develop a strivers' thinking, and in the later work, when faced with the difficulty of insufficient resources, I never wanted to complain, and only blindly took the initiative to motivate myself to take positive actions to take difficulties as opportunities to improve my ability.
Figure 9 Illustration of the oneness relationship between frequency, amplitude and phase
The first paragraph introduces the starting point of the author's career in complex measurement research, this initial experience is only to verify that the lock-in can be virtually instrumented, and there is no principle innovation, and then enter the main topic, bird's eye view of the author's little progress in the innovation of the complex number measurement principle over the years [9-11]. As mentioned in the previous chapter, the two keys to complex measurement with phase-locked phase, phase-locked loop frequency tracking and virtual real phase-sensitive detection, and the traditional method measurement result is a complex number corresponding to the reference signal frequency, and the true alternating frequency of the measured complex number is used as a hidden variable, which is usually regarded as completely consistent with the reference frequency. But can the frequency of the measured signal always be exactly the same as the frequency of the reference signal? This question is the author in the effort to improve the key entry point when the locked phase, the answer is obviously no, so that the two signals to maintain frequency consistency in the consistency requirements are not high (such as 0.1%) is not difficult, but to achieve 1ppm level of inconsistency is not easy, as can be seen from Figure 8, the phase lock only in the reference signal side of the application of the phase lock loop to ensure frequency stability, and the other end, it is just that the measured signal is usually unable to stabilize the frequency, at this time due to the nonlinearity of the medium, transmission dispersion, Doppler effect, etc. Inevitably introduce frequency drift, The existence of frequency drift, resulting in the existence of a frequency difference between the two input signals of the phase-sensitive detection, will definitely lead to the inaccuracy or even error of the real imaginary measurement results, because the signal frequency and amplitude and phase together describe a complete signal, respectively, like the position and direction of the hull and the sail floating on the water, as shown in Figure 9, with the hull as a reference to see the sail relatively fixed, and when the shore as a reference object, the sail is more seen in the waves, so the lock-in work of the fixed reference signal is like a modern version of the carved boat sword.
Fig. 10 Framework of principles for new phase-locked measurement techniques
Figure 11 Comparison of the frequency measurement accuracy of a phase-locked frequency meter with a fast Fourier transform (FFT).
The author's idea to solve this problem is shown in Figure 10, which is actually to cancel the assumption that the frequency of the measured signal is consistent with the frequency of the reference signal, improve the functional of the frequency through the phase-locked algorithm, and solve the two problems of anti-noise frequency measurement and accurate phase locking of the measured signal, the name of the new method is called the frequency locker or the phase-locked frequency meter. The comparison effect of the frequency measurement accuracy of the use of a phase-locked frequency meter and the fast Fourier transform (FFT) is shown in Figure 11, we know that the frequency accuracy of the Fourier transform is limited by the sampling window, it is difficult to improve further, and the new method of the phase-locked frequency meter can perform ultra-fine scanning in the frequency domain, such as the frequency measurement accuracy of the FFT in Figure 11 is only about one ten-thousandth, and the phase-locked frequency meter can be fine to less than one millionth. Of course, the improvement of accuracy is limited in principle, and there is a statistical lower limit for the uncertainty of frequency measurement under a certain length of time and sampling rate, that is, the lower limit of Cramer-Rao [10,12], and it can be seen after numerical simulation that the frequency measurement accuracy of the new method under different signal-to-noise ratio conditions is close to the lower theoretical limit, as shown in Figure 12. Recently, the author theory derived the local function form of the phase-locked spectrum of a single-period signal near the estimated frequency point, proved that the correct functional form is better than the empirical parabola, and based on this use less calculation
Figure 12 Comparison of the frequency measurement accuracy of the new phase-locked frequency meter under different signal-to-noise ratio conditions and the theoretical limit Cramer-Rao Lower Bound (CRLB).
Fig. 13 Measurement of a 977 Hz real signal by a new phase-locked frequency meter (monitoring lasts about 80 minutes)
From the beginning of graduate school to contact complex measurement in the beginning with a virtual instrument in the domestic low-end oscilloscope to replace princeton's antique lock-in phase, to today's custom chip hardware embedded in the principle of innovative phase locking instrument core indicators beyond the Swiss Zurich instrument and the United States Stanford instrument mainstream lock-in phase, the author as a frequency lock-in struggler time has passed 15 years.
05
The future prospect of complex measurement
Fig. 14 Both the microstructure and the ultra-macroscopic event detection are inseparable from complex measurements (the schematic diagrams of microprobe and gravitational wave detection are derived from the network)
With the accelerated transformation of China's intelligent manufacturing, the universal moderation and depth of complex measurement continue, and within the range that human beings can see and touch, the application prospects of intelligent perception, accurate positioning and interconnection, precision manufacturing, and lock-in measurement in trace matter detection in the future are optimistic. In the ultra-microscopic and ultra-macroscopic world that is difficult for the human eye to perceive, as shown in Figure 14, subatomic and deep space exploration is endless, whether vast and microscopic, whether instantaneous or eternal: complex events are endless, and complex measurements are endless.
Fig. 15 Image and geometric representation of qubits[13]
Earlier, when discussing Boolean's regrets in the development of complex numbers, it has been mentioned that the future computer should be analog computing and numerical computing equally divided, each with its own strengths. The most remarkable analog computing should be quantum computing, which has an unparalleled parallel acceleration efficiency advantage over problems such as near-natural search and large number decomposition. Compared with the current mainstream binary numerical bits, the essence of qubits is complex bits[13], as shown in Figure 15, taking the example of the |0> and |1 > of the bistatal orbitals, the possibility of bimorphic aliasing is on the sphere, and the projection relationship between the sphere and the complex plane satisfies the mutual equivalent (see Appendix 1 for details). Although a large number of lock-in amplifiers must be used in quantum computers to prepare the coherence of quantum states and monitor quantum states, the direct measurement of complex bits is still a serious problem and cannot be achieved in the short term, because the phase in the quantum complex bit is the phase of the inner space, which is completely different from the phase in the macroscopic wave function, and cannot be rigidly set, just like Mr. Dirac Principles of Quantum In the world of small, the conceptual rules of the big world must fail unless technological advances can move the boundary between big and small down the boundary between big and small, such as deconstructing the inner degrees of freedom of qubits to a certain extent and preparing corresponding complex bit probes. From this point of view, quantum computers are still very far from practical, but the attraction that this day will eventually come is so beautiful.
06
Ending revelation
As can be seen from the above, in human history, the plural was developed very late, and even until the 20th century was found to be very useful, and we are currently in an era when human beings use the plural to further accelerate productivity, and lock-in technology will inevitably be more deeply integrated and more widely used. The author has reason to believe that with the more comprehensive penetration of electromagnetics, relativity and quantum information in the development of human scientific and technological civilization, people who master and use complex numbers will be able to face future changes and unknown challenges more calmly.
What can the connotation of plurals teach us? There is a song of plurals, the original English lyrics at the end of this chapter (Note 2), the author's surname is even more famous who unfortunately has not been verified, the lyrics and the tune of the "War Song of the Republic" are very in tune, and very accurately the charm of the Dominican number comes from the imaginary unit, the square root of -1, the root of minus one in the lyrics, however, what does the charm of the imaginary unit come from? What's the root of the root of minus one? The author's understanding of the plural through his own understanding and application of the plural is that i is between positive and negative, which is half of the opposite operation, leaving the positive but not the negative, not positive or negative, like a chess game to be determined, as if it is unpredictable. For this independent natural attribute of i, the reader will certainly be thoughtful, and will believe that personal experience and social events in addition to the general and absolute black and white and right and wrong, before the situation is not completely understood, there is no black and white, not a good moderate orientation to choose, appropriately jump out of their own local time and position, analyze the reliable information at hand, and realistically form a corresponding conclusion based on the existing one, right?
From the essence of complex measurement is the principle of accurate measurement of the overall phase, we can know that local isolated events are difficult to measure the phase information of complex numbers, and if we look at the i orientation of not being positive or negative or black and white for a long time or pulling a long enough perspective, we will be able to more accurately grasp the overall trend of the event, and firmly align our thoughts and actions with the real long-term trend. It just so happens that independent, like imagination, begins with the imaginary unit i. As the following four sentences are sent to the reader:
The move is to be determined, half a step free; positive or negative, with ease.
Seemingly infinite, between reality; not left or right, moderate and independent.
Finally, the author sincerely hopes that through the reading of this article, you will have some thoughts, gains, calmer and happier.
Note 1 to entry: Mathematical experts may not necessarily make a clear and uniform analysis of the terms invented vs discovered, as they involve esoteric discussions of the question of possible desperfiability of scientific justification. Many people agree that the plural as a system of explanatory tools was invented; but the universal logic and rationality that have been repeatedly proved by human practice are not wrong to say that they have been discovered. In order to avoid this detail to separate the reader group orientation and thus affect the transmission of the subject information, this article uses "introduction" or "development" to express the superposition effect of the double meaning in relevant places.
注 2：The Complex Number Song(Author: unkown; Tune: Battle Hymn of the Republic)
Mine eyes have seen the glory of the Argand diagram,
They have seen the i's and thetas of De Moivre's mighty plan.
Now I can find the complex roots with consummate elan,
With the root of minus one.
Complex numbers are so easy;
In Cartesian co-ordinates the complex plane is fine,
But the grandeur of the polar form this beauty doth outshine.
You be raising i+40 to the power of 99,
You'll realise your understanding was just second rate,
When you see the power and magic of the complex conjugate.
Drawing vectors corresponding to the roots of minus eight,
bibliography
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[4] Locked Classic Textbook: Meade M L. Lock-In Amplifiers: Principles and Applications. Peter Peregrinus, London, 1983; first lock-in amplifier paper I found: Michels W C, Curtis N L. Rev. Sci. Instrum., 1941, 12:444;Recommended reading online handout: http://people.ee.duke.edu/~mbrooke/defense/Borte.ppt, https://nationalmaglab. org/images/news_events/searchable_docs/ summerschool/2016/lockin_amplifier_kowitt.pdf
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Lu Jun. Modern Instruments, 2012, 18: 78
LU Jun. Multifunctional magnetoelectric spectrometer (including labview source files of the second-generation phase-locked algorithm), uploaded to the instrument information network in February 2008 and published: https://www. instrument.com.cn/download/shtml/62888. shtml
Lu Jun et al. Anti-noise broadband frequency measurement method and phase-locked frequency meter, Chinese invention patent: 201110380805. X;A wide frequency impedance measurement system and wide frequency impedance measurement method, Chinese invention patent: 201210081375. 6; Signal sampling averager and signal sampling averaging method, Chinese invention patent: 201310535320. 2; A grade interlocking phaser and cascade interlocking phase method, Chinese invention patent: 201410725833.4; a time domain signal phase-locked time-frequency measurement method and device, Chinese invention patent: 201510004030.4; tester for testing the lock-in amplifier and its test method, Chinese invention patent: 201510182065. 7; Frequency measurement device and frequency measurement method, Chinese invention patent: 201710707180. 0
[10] Lu J. Lock- in frequency measurement with high precision and efficiency. Rev. Sci. Instrum.，2020，91：075106
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This article is reprinted with permission from the WeChat public account "Chinese Physical Society Journal Network".
Special mention
1. Enter the "Boutique Column" at the bottom menu of the "Return to Simplicity" WeChat public account to view the series of popular science articles on different topics.
2. "Return to Park" provides the function of retrieving articles on a monthly basis. Follow the official account, reply to the four-digit year + month, such as "1903", you can get the index of articles in March 2019, and so on.