The story of quantum mechanics is a fable of wonder and bewilderment. It has elements of science, philosophy, religion, and dare I say magic. It’ll turn your mind upside down, and sometimes it’ll make you question the existence of an all-powerful creator out there. Even though I find its concepts difficult to grasp, I’ve always been fascinated by it. Some of the concepts presented in this chapter are hard to understand; however don’t be troubled. Nobody has been able to fully describe what this all means, not even the titans of physics fully understand quantum mechanics. However that doesn’t mean we can’t be fascinated by it. The great physicist Richard Feynman once said: If somebody tells you he understands quantum mechanics, it means he doesn’t understand quantum mechanics. This chapter is my take on this fascinating fable and how the struggle of two titans of science shaped its past, present, and future.
It all began in the 1930s, after Albert Einstein rose to world fame with the theory of special relativity which built upon Newtonian physics to unify the heavens and the earth. While Einstein was looking to the heavens, a new breed of scientists were looking at the very small. Spearheaded by giants of physics such as Max Planck, Ernest Rutherford, and Niels Bohr, it started a clash of titans and one of the greatest debates of physics in the twentieth century – on one side, Albert Einstein, fresh from winning the Nobel Prize for his groundbreaking discoveries on the nature of light and special relativity and, on the other side, Niels Bohr, whose contributions to the field of quantum mechanics would earn him a Nobel Prize in 1922 and the prestigious Order of the Elephant, a Danish distinction normally reserved for royalty. Let’s take a look how the struggle between these two greats shaped the science masterpiece, that is, quantum mechanics.
The Golden Age of Physics in the Twentieth Century
At the beginning of the twentieth century, British scientist Ernest Rutherford made a startling discovery about the nature of the atom. He postulated that atoms look like tiny solar systems, made of a tiny nucleus with positive charge and electrons negatively charged rotating like tiny planes around it. This was a remarkable insight as it was previously believed that the atom was a simple spherical blob of mass with positive and negative charges.
Bohr arrived at Rutherford’s lab in Cambridge in 1920 and fell in love with Rutherford’s model of the atom, but there was a problem, and a big one. If classical Newtonian physics are applied to Rutherford’s model where negatively charged electrons rotate around a positively charged nucleus, the electron will eventually fall inside and crash against the nucleus creating a catastrophic paradox. Nothing should exist, as electrons will crash in a matter of seconds. Bohr saw this, and with undeterred excitement, he delayed his marriage and canceled his honeymoon in an effort to save Rutherford’s model. Bohr postulates in a paper that electrons move in fixed orbits that cannot change. This goes against the basis of Newtonian physics but draws upon new ideas from the father of quantum mechanics, Max Planck.
Max Planck and the Ultraviolet Catastrophe Started It All
Black-body radiation experiment results
Light Colorization at Different Temperatures
Temperature (°C) | Color |
|---|---|
500 | Dark red |
800 | Cherry red |
900 | Orange |
1000 | Yellow |
1200 | White |
Figure 1-1 shows the black-body radiation experiment along with the results provided by the classical theory of radiation curves collected from experiments in the 1890s. Classical physicist’s experiments predicted infinite intensities for the ultraviolet spectrum. This became known as the ultraviolet catastrophe and was the product of dubious theoretical arguments and experimental results. If true, this would mean, for example, that it will be dangerous to seat anywhere close to a fireplace! Planck sought to find a solution to the ultraviolet catastrophe.
Planck used the second law of thermodynamics also known as entropy to derive a formula for the experimental results derived from the black-body radiation problem.
S = k log W
This is Boltzmann’s entropy (S), where k is known as the Boltzmann’s constant and W is the probability that a particular arrangement of atoms will occur for an element be that a solid, liquid, or gas.
Using Boltzmann’s statistical method to calculate entropy, Planck sought a formula to match the results of the black-body experiment. By dividing the total energy (e) in chunks proportional to the frequency (f), he came up with the equation:
e = hf
where e is a chunk of energy, h is known as the Planck constant, and f is the frequency. Yet, he faced an obstacle; Boltzmann’s statistical method demanded the chunks decrease to zero over time. This will nullify his equation and thus defeat its validity. After much struggle, Planck was forced reluctantly to postulate that the energy quantity must be finite. And here comes Planck’s incredible insight; if this is correct, it meant that is not possible for an oscillator to absorb or emit energy in a continuous range. It must absorb or emit energy in small indivisible chunks of e = hf which he called “energy quanta,” hence the term quantum mechanics.
Bohr’s Quantum Jump
Bohr applied Plank’s groundbreaking idea of energy quanta to the atom, the smallest unit of matter. He provided a bold description of the relationship of the atom and light where the electron which rotates around the nucleus will emit or absorb light causing a quantum jump. A quantum jump was therefore a transition between two states; however Bohr was incapable of fully describing it.
This idea was met with skepticism by other scientists who labelled his theory as nonsense, a cheap excuse for not knowing, or too bold, too fantastic to be true. The result was a rift in the physics community with one camp around Bohr believing in the quantum nature of matter and those supporting the classical view. Einstein will soon join the fight in the classical side of the struggle.
Clash of Titans: Quantum Cats and the Uncertainty Principle
By the mid-1920s the new theory about the quantum nature of matter is in shaky ground facing the real prospect of an early demise. It will take two new groundbreaking discoveries to solidify its foundation.
Duality of the nature of the photon. It behaves as both a particle and wave.
de Broglie used both Einstein’s famous equation for energy E = mc2 and Planck’s energy quanta e = hf to find a relation between the wavelength (λ) and the momentum (P) of a photon:
E = mc2 = (m c) * c
Given that (mc) is the momentum (P) of the photon and c (speed) = f (frequency) * λ (wavelength), the equation becomes:
E = (P) (f λ)
But wait, Planck’s relation states that energy E = (h)(f); thus using basic algebra, de Broglie concluded:
h * f = P * (f λ)
h = P * λ
λ = h / P
de Broglie relation between the wavelength and the momentum of a photon
Schrödinger used de Broglie’s ideas to find an approach that was more acceptable to the status quo, marking a return to the continuous, visualizable world of classical physics. He was right about his wave function but dead wrong about appeasing the status quo.
Enter the Almighty Wave Function
Schrödinger famous wave function sought to describe any physical system with known energy
Bohr and Heisenberg joined forces with Schrödinger given the incredible power of his wave function, but they needed to work out their differences first. It all took place in 1926 at a newly formed institute in Copenhagen where the three giants met to discuss.
Schrödinger rejected the Bohr/Heisenberg concept of discontinuous quantum jumps in the atom structure. He wanted to use his new discovery as a pathway back to the continuous process of physics undisturbed by sudden transitions. He was in fact proposing a classical theory of matter based entirely on waves, even to the point of doubting the existence of particles. Schrödinger proposed that particles are in fact a superposition of waves, a claim that was later proved wrong by Hendrik Lorentz who brought him to his senses, proving that you can’t win them all after all. Schrödinger will later waver in his conviction on the importance of wave motion as the source of all physical reality.
Bohr, Heisenberg, and Schrödinger argued relentlessly until the point of exhaustion. Bohr demanded absolute clarity in all arguments, trying to force Schrödinger to admit that his interpretation was incomplete, Schrödinger clinging to his classical view, sometimes bemoaning his work on atomic theory and quantum jumps (something that he probably didn’t mean).
Schrödinger loathed Bohr interpretation of the atomic structure. A final piece was required before these two could come to terms on a solid quantum theory.
Probabilistic Interpretation of ψ: The Wave Function Was Meant to Defeat Quantum Mechanics Not Become Its Foundation
Just like when the great rock guitarist Jimi Hendrix heard the tune Hey Joe, released a cover, and made it his own, thus creating arguably one of the greatest tune covers, so did the fathers of quantum mechanics. They realized the tremendous power of the wave function and made it their own. A little factoid about this story is that Schrödinger detested Planck’s noncontinuous interpretation of energy and heat. He wanted to use his smooth and continuous wave function to defeat Planck’s energy quanta. It is hard to believe, but in the 1930s, Planck’s discovery was so revolutionary that most physicists thought he was nuts. Nevertheless, just as Hendrix did with that tune, the founders of quantum mechanics will make the wave function theirs.
Bohr vs. Max Born probabilistic view of the wave function
The Quantum Cat Attempts to Crash Born’s Probabilistic Party
As Born’s idea about the probabilistic nature of ψ gained traction, Schrödinger through his wave function was being misused, and that originated the famous thought experiment that will be later known as the quantum cat, a story that you probably heard of. In the experiment, Schrödinger sought to rebuff Born’s probabilistic interpretation of ψ. It goes like this: a live cat is placed in a box with a radioactive source that triggers the release of a hammer that breaks a flask with poison that will kill the cat. Assuming a 50% probability of radioactive decay per hour, after one hour the mechanism will be triggered, thus killing the cat. Schrödinger claimed that according to Born’s interpretation, quantum theory will predict that after one hour, the box would contain a cat that is neither dead nor alive but a mixture of both states, a superposition of both wave functions. Schrödinger thought this was ridiculous and would create a paradox. Yet today, this so-called paradox is used to teach about quantum probabilities and superposition of states.
This is the genius of superposition; as soon as the box is opened, the superimposed wave functions collapse into a single one making the cat dead or alive – thus the act of observation resolves the impasse. Yet another incredible insight will come from Heisenberg pondering about a certain amount of uncertainty about the position of a particle in the atomic structure championed by Bohr.
Uncertainty Principle
Heisenberg pondered about how the position of a particle cannot be known in Bohr’s atom. After much reflection, in a moment of clarity, he realized that to know where a particle is, you have to look at it, and to look at it, you have to shine a photon of light on it. However, when you do this, it disturbs the particle position; thus the act of observing a particle changes its location. Heisenberg called this idea the uncertainty principle.
To study the problem, Heisenberg devised a hypothetical experiment using a microscope firing gamma rays, which carry high momentum and low frequency, toward a passing electron to be observed. With Bohr’s help, the goal was to describe a quantitative relationship by estimating the imprecision on a simultaneous measurement of the position and momentum. The imprecision of the position was found to be close to the wavelength of the radiation being used, ΔX ~ λ.
Similarly, the imprecision of the momentum of the electron is close to the momentum of the photon used to illuminate the particle, ΔP ~ h/ λ. Note that from the de Broglie equation it is known that the momentum of the photon (P) = h (Planck constant)/ λ (wavelength). Heisenberg showed that multiplying both inequalities, the product will always be greater or equal to h.
Δ X * Δ P ≥ λ * h/ λ
Δ X * Δ P ≥ h
This is Heisenberg uncertainty principle (HUP) which formally states: “The uncertainty of a simultaneous measurement of the momentum and position is always greater than a fixed amount and close to Planck’s constant h.”
Single slit experiment used to show the uncertainty principle in action
The uncertainty principle is extremely important because it unifies the rift between Schrödinger and Bohr laying down the foundation of the modern quantum theory. That is, the electron is a particle, as Bohr postulated, but we don’t know exactly where it is, as the uncertainty principle states (Heisenberg). Lastly, the probability of finding it is given by the wave function (Schrödinger/Born). Thus there is a duality in the nature of the electron, both as a particle and wave. With all this, a rock-solid view of quantum mechanics emerges that will later be known as the Copenhagen interpretation.
Interference and the Double Slit Experiment
Interference is another incredible property of quantum mechanics, one that makes you think what in the world is going on behind the scenes of our reality. The great physicist Richard Feynman once said about interference: The essentials of quantum mechanics could be grasped from an exploration of interference and the double slit experiment.
Double slit experiment by Thomas Young
This mind-bending result perplexed physicists who hypothesized that interference is taking place between the waves and particles going through the slits. If the beam of photons is slowed enough to ensure that individual photons are hitting the plate, one might expect there to see two lines of light (a single photon going through one slit or the other and ending up in one of two possible light lines). However that is not true. What happens is that somehow the light is doing the impossible: each photon not only goes through both slits but also simultaneously traverses every possible trajectory en route to the target (a principle called interference).
The fact that events like interference, which seem impossible, can occur at the atomic scale baffled the greatest minds at the time. Yet soon, this new theory will face its biggest challenge from the titan of physics, Albert Einstein.
Einstein to Bohr: God Does Not Throw Dice
If you are involved in science, or even if you aren’t, you probably heard the famous phrase by Einstein “God does not throw dice.” It was coined during a series of letters exchanged with Bohr about the nature of quantum mechanics. Bohr believed the concepts of space-time do not apply at the atomic level. Einstein, on the other hand, was a firm believer in the fabric of space-time and thought this idea could be extended to the atomic scale. This was essentially the root of the disagreement between the two.
Einstein postulated that the properties of an atomic particle could be measured without disturbing it, an idea that goes against the Bohr/Heisenberg interpretation. The two giants faced in a gathering of the greatest physicists of the time in Brussels in 1927 where Einstein sought to prove once and for all that uncertainty does not rule reality.
Einstein’s experimental box to disprove the uncertainty principle
In the thought experiment in Figure 1-7, the box has a light source with a clock designed to measure the precise time a photon is emitted. At the same time, the box hangs from a spring with a weight at the bottom and corresponding measuring device. The idea was simple: weight the box before and after the photon is emitted and at the same time register the precise time using the clock. The energy levels could be easily calculated using Einstein’s own equation E = mc2. Things didn’t look good for the uncertainty principle at that point. If the experiment was correct, the uncertainty principle will be disproven and quantum theory defeated.
Bohr got to work immediately trying to persuade Einstein that if his box works it would mean the end of physics. Bohr prevailed at the end by stating that Einstein forgot to take his own theory into account, as clocks are affected by gravity yielding uncertainty at the time of measurement. He proved the following uncertainty calculation ΔE Δt ≥ h using Einstein’s equation and the red shift formula. Given (Δp) uncertainty of the momentum and (Δq) uncertainty of the position:
Δp Δq ≥ h(1-1)
The uncertainty of the momentum (Δp) is given by Δp ≤ t g Δm; then we have:
t g Δm Δq ≥ h(1-2)
From the redshift formula and principle of time dilation:
Δt = c-2 g t Δq(1-3)
ΔE = c2 Δm(1-4)
Now, multiply (1-3) and (1-4) to obtain (1-5):
ΔE Δt = g t Δm Δq(1-5)
Finally, comparing (1-5) and (1-2), we obtain an inequality for the uncertainty principle ΔE Δt ≥ h. With this result, round one goes to Bohr; however this will not be the end of it. Einstein believed in a complete picture of physical reality, and the uncertainty principle stood in his way. He will come back with a bigger challenge.
Bohr to Einstein: You Should Not Tell God What to Do
God does not throw dice was Einstein unshakable principle. The firm belief that reality exists independent of one’s self. When Einstein wrote to Bohr that god does not throw dice, he replied that he should not tell god what to do. This set the stage for a second struggle between the two while trying to figure out what holds the nucleus together. By the mid-1930s, the time around which this took place, both general relativity and quantum theory are widely accepted as the strongest ideas to explain how the world works. Round two focuses in the most paradoxical aspect of quantum theory – the idea that atomic particles remain connected to one another even at great distances.
Entanglement and the EPR Paradox: Spooky Action at a Distance
In the beginning, light was thought to behave as a wave, but Einstein proved that it also showed particle behavior also known as photons. The same was true about atoms. They behaved both as particles and waves depending on the measuring instrument being used. Furthermore, both conditions were necessary to obtain a complete picture, an idea that Bohr called complementarity.
So how is matter to be understood in the face of these two contradictions? Bohr believed that the atom, as it is, existed outside our perception. This was more than Einstein could accept as he believed on the idea of space-time at the foundation of all physical reality and wanted to extend this concept to the atomic level. Bohr, on the other hand, thought space and time were meaningless and reality was unknowable, and all that we had were phenomena.
It is around this time that Einstein issues a second and final challenge to Bohr. In a paper written with colleagues Podolsky and Rosen, Einstein postulates the question: Does quantum mechanics provide a complete view of physical reality? He proposes a thought experiment where two particles emitted from the same source have common properties and become separated. It should be possible then to measure the first particle and obtain information about the second one without disturbing it. The purpose of the experiment was to demonstrate absurdities in Bohr’s view of particles behaving differently based on measuring device. According to quantum mechanics, a measurement on the first particle will influence the other across time and space.
Now, imagine if the particles were to be separated across very large distances (e.g., from one side of the universe to the other). This will create a paradox by violating a fundamental principle of science: the principle of cause and effect. The idea that all events in reality have a cause and effect, and events cannot be transmitted faster than the speed of light, the ultimate speed limit in the universe. Einstein called this principle local causality or locality for short. This paradox will be known as the Einstein-Podolsky-Rosen or EPR paradox.
As soon as Bohr got news of the paper, all work was abandoned immediately. The challenge had to be answered. Bohr was reluctant at first in his reply, but finally did so by claiming that both particles ought to be considered as a single system. In other words both particles become entangled, with space and time being meaningless in such system. Therefore a picture of the atomic world was unknowable.
Einstein called the effects of entangled particles over large distances “Spooky action at a distance.” The disagreement between the two was never resolved. Nevertheless, a breakthrough to settle things up came in 1965 by physicist John Bell.
Bell’s Inequality: A Test for Entanglement
Bell proposed a set of inequalities to provide experimental proof of the existence of local hidden variables. Formally, Bell’s inequality theorem states: No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics .1 Mathematically, it is given by the formula:
Polarization of light at three angles
Permutation Table for Photon Polarization at Three Angles
Count | A(0) | B(120) | C(240) | [AB] | [BC] | [AC] | Sum | Average |
|---|---|---|---|---|---|---|---|---|
1 | A+ | B+ | C+ | 1(++) | 1(++) | 1(++) | 3 | 1 |
2 | A+ | B+ | C– | 1(++) | 0 | 0 | 1 | 1/3 |
3 | A+ | B– | C+ | 0 | 0 | 1(++) | 1 | 1/3 |
4 | A+ | B– | C− | 0 | 1(−−) | 0 | 1 | 1/3 |
5 | A− | B+ | C+ | 0 | 1(++) | 0 | 1 | 1/3 |
6 | A− | B+ | C− | 0 | 0 | 1(−−) | 1 | 1/3 |
7 | A− | B− | C+ | 1(−−) | 0 | 0 | 1 | 1/3 |
8 | A− | B− | C− | 1(−−) | 1(−−) | 1(−−) | 3 | 1 |
Now ask the simple question: If we measure the polarization at any angle, what is the probability that the polarization at any neighbor will be the same as the first? Also calculate the sum and average of the polarizations. In Table 1-2, neighbor polarization is represented by the columns AB, BC, and AC. The + and – signs in columns A, B, and C indicate either positive or negative polarizations at the given angles. Note that there are eight possible permutations, described by the column count. Thus if we find the same polarization (the same sign) for two neighbors, then we record a 1 as well as the sign in columns AB, BC, or AC. This is required to calculate the sum and the average for the respective row in the permutation table.
Now, if a polarization exists independent of measurement (local causality), as Einstein advocates, then the probability of that polarization must be ≥ 1/3. On the other hand, if Bohr is correct, and reality is defined by the act of observation, then the probability of polarization will be < 1/3. This is at the heart of Bell’s inequality. Bell does not take sides; it does not say that either is correct but provides the means of finding the truth by experimentation. As a matter of fact, in 1982, French physicist Alain Aspect created an experiment that proved once and for all that Bohr was right all along.
EPR Paradox Defeated: Bohr Has the Last Laugh
Aspect’s experiment to test Bell’s inequalities - stage 1
If both polarization filters are calibrated in the same direction, Aspect observed a correlation between the pairs of photons. They would either pass or be blocked at the same time. This correlation agreed with Einstein’s view of the photon having its polarization property predefined at the moment of emission from the source, not at the moment of measurement as quantum mechanics predicted.
If the percentage of photons passing through or being blocked is greater than or equal to the expected minimum, then Bell’s inequality is preserved and the photon polarization is defined at the moment of emission (the victory goes to Einstein and quantum mechanics is defeated).
On the other hand, if the percentage is less than the expected minimum, Bell’s inequality is violated and quantum physics is correct. The polarization is defined at the moment of measurement (Bohr wins and quantum mechanics is saved).
Aspect performed measurements of many pairs of photons at different polarization settings. The results were astounding: the measurements violated Bell’s inequality; thus it was impossible for the polarization to be predefined at the moment of emission. Quantum mechanics was correct! The photons appeared to have chosen a common polarization at the moment of measurement . Could there be some sort of unknown signal between the photons telling them to pick a common value at the moment of measurement?
Aspect’s experiment to put spooky action at a distance to the test
The optical switch is designed to send the photon in one of two directions at an extremely fast rate: 2 nanoseconds or 2 ns.
The distance between both ends of the experiment was 12 m. It takes the speed of light (traveling at 3*108 meters/second) 40 nanoseconds (ns) to go from one end of the experiment to the other.
Now, if no signal can travel faster than the speed of light, as Einstein’s relativity postulates, it should take more than 40 ns from one photon to tell the other what polarization value to choose. Because the optical switch changes at a faster rate (2 ns), the correlation between the photons should not hold. That is, the photons should not be able to choose the same polarization at the moment of measurement (no spooky action at a distance). On the other hand, if the correlation holds, things get extremely weird as some sort of signal is being transmitted to both photons faster than the speed of light.
Incredibly, the correlation held in perfect agreement with quantum mechanics, thus proving once and for all that the polarization value was chosen simultaneously by both photons at the moment of measurement faster than the speed of light. The implications were mind blowing as the distance between the photons could have been infinitely grater (e.g., for one end of the universe to the other) or even scarier, across time: from the present into the past or vice versa!
Reality Playing Tricks on Us: Is Everything Interconnected?
Aspect’s experiment proves that quantum correlations exist and that if we are to explain them, not just accept them, then we must be bound to admit that some actions occur faster than the speed of light. If that is hard to digest for some, things get even weirder. In a TV interview for the BBC, physicist John Bell said: “There is nothing we can do with this, for example, we cannot send messages or information faster than the speed of light, a fact that is also predicted by quantum mechanics. It seems as nature is playing a trick on us: extraordinary things happen behind the scenes which we cannot use.”
At the end, Bohr and Einstein never resolved their differences. They both passed away but their legacy endures. Reading through their fascinating lives, one can’t help but wonder: How would have Bohr felt by looking at the results of Alain Aspect experiment proving that he was right all along? Would he have felt happy at his triumph over Einstein? Was all this about the struggle of two egocentric geniuses trying to prove who the better man is? What do you think? I choose to believe that this was a struggle for the advancement of science. All in all, the ultimate winner over the clash of these two titans was humanity.
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![$$ H=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ {}1& -1\end{array}\right] $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_IEq6.png)
![$$ \mid 0\Big\rangle =\left[\begin{array}{c}1\\ {}0\end{array}\right] $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_IEq7.png)
![$$ \mid 1\Big\rangle =\left[\begin{array}{c}0\\ {}1\end{array}\right] $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_IEq8.png)
![$$ H\mid 0\Big\rangle =\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ {}1& -1\end{array}\right]\left[\begin{array}{c}1\\ {}0\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ {}1\end{array}\right]=\frac{1}{\sqrt{2}}\left(\left[\begin{array}{c}1\\ {}0\end{array}\right]+\left[\begin{array}{c}0\\ {}1\end{array}\right]\right)=\frac{\left|0\Big\rangle +|1\right\rangle }{\sqrt{2}} $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_IEq9.png)
![$$ H\mid 1\Big\rangle =\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ {}1& -1\end{array}\right]\left[\begin{array}{c}0\\ {}1\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ {}-1\end{array}\right]=\frac{1}{\sqrt{2}}\left(\left[\begin{array}{c}1\\ {}0\end{array}\right]-\left[\begin{array}{c}0\\ {}1\end{array}\right]\right)=\frac{\left|0\Big\rangle -|1\right\rangle }{\sqrt{2}} $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_IEq10.png)
![$$ X=\left[\begin{array}{cc}0& 1\\ {}1& 0\end{array}\right],H=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ {}1& -1\end{array}\right] $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_Equl.png)
![$$ \mid \varPsi \Big\rangle =\left[\begin{array}{c}\alpha \\ {}\beta \end{array}\right] $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_IEq11.png)
![$$ H\left[\begin{array}{c}\alpha \\ {}\beta \end{array}\right],\kern0.5em X\kern0.28em \left[\begin{array}{c}\alpha \\ {}\beta \end{array}\right],\kern0.5em U\left[\begin{array}{c}\alpha \\ {}\beta \end{array}\right]\kern0.28em where\kern0.28em U=H,X $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_Equm.png)
![$$ A=\left[\begin{array}{cc}a& b\\ {}c& d\end{array}\right]\kern0.28em then\kern0.28em {A}^T=\left[\begin{array}{cc}a& c\\ {}b& d\end{array}\right] $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_IEq12.png)
![$$ ={\left[\begin{array}{cc}a& c\\ {}b& d\end{array}\right]}^{\ast } $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_IEq13.png)
![$$ X=\left[\begin{array}{cc}0& 1\\ {}1& 0\end{array}\right] $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_IEq14.png)
![$$ \left[\begin{array}{cc}0& 1\\ {}1& 0\end{array}\right]\to $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_IEq15.png)
![$$ H=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ {}1& -1\end{array}\right] $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_IEq16.png)
![$$ \frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ {}1& -1\end{array}\right]\to $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_IEq17.png)
![$$ X=\left[\begin{array}{cc}0& 1\\ {}0& 1\end{array}\right],Y=\left[\begin{array}{cc}0& -i\\ {}i& 1\end{array}\right],Z=\left[\begin{array}{cc}1& 0\\ {}0& -1\end{array}\right] $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_IEq18.png)
![$$ \left[\begin{array}{cc}\cos \kern0.28em \theta & -\sin \theta \\ {}\sin \theta & \cos \theta \end{array}\right] $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_IEq19.png)
![$$ \left[\begin{array}{cccc}1& 0& 0& 0\\ {}0& 1& 0& 0\\ {}0& 0& 0& 1\\ {}0& 0& 1& 0\end{array}\right] $$](../images/469026_1_En_4_Chapter/469026_1_En_4_Chapter_TeX_IEq20.png)
![$$ H=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ {}1& -1\end{array}\right] $$](../images/469026_1_En_5_Chapter/469026_1_En_5_Chapter_TeX_Equa.png)
![$$ \mid 0\Big\rangle =\left[\begin{array}{c}1\\ {}0\end{array}\right] $$](../images/469026_1_En_5_Chapter/469026_1_En_5_Chapter_TeX_IEq1.png)
![$$ \mid 1\Big\rangle =\left[\begin{array}{c}0\\ {}1\end{array}\right] $$](../images/469026_1_En_5_Chapter/469026_1_En_5_Chapter_TeX_IEq2.png)
![$$ H\mid 0\Big\rangle =\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ {}1& -1\end{array}\right]\left[\begin{array}{c}1\\ {}0\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ {}1\end{array}\right]=\frac{1}{\sqrt{2}}\left(\left[\begin{array}{c}1\\ {}0\end{array}\right]+\left[\begin{array}{c}0\\ {}1\end{array}\right]\right)=\frac{\left|0\Big\rangle +|1\right\rangle }{\sqrt{2}} $$](../images/469026_1_En_5_Chapter/469026_1_En_5_Chapter_TeX_Equb.png)
![$$ H\mid 1\Big\rangle =\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ {}1& -1\end{array}\right]\left[\begin{array}{c}0\\ {}1\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ {}-1\end{array}\right]=\frac{1}{\sqrt{2}}\left(\left[\begin{array}{c}1\\ {}0\end{array}\right]-\left[\begin{array}{c}0\\ {}1\end{array}\right]\right)=\frac{\left|0\Big\rangle -|1\right\rangle }{\sqrt{2}} $$](../images/469026_1_En_5_Chapter/469026_1_En_5_Chapter_TeX_Equc.png)
![$$ A1=\frac{1}{\sqrt{2}}\left[\begin{array}{cccc}i& 0& 0& 1\\ {}0& -i& 1& 0\\ {}0& i& 1& 0\\ {}1& 0& 0& i\end{array}\right],A2=\frac{1}{2}\left[\begin{array}{cccc}i& 1& 1& i\\ {}-i& 1& -1& i\\ {}i& 1& 1& -i\\ {}-i& 1& -1& -i\end{array}\right],A3=\frac{1}{2}\left[\begin{array}{cccc}-1& -1& -1& 1\\ {}1& 1& -1& 1\\ {}1& -1& 1& 1\\ {}1& -1& -1& -1\end{array}\right] $$](../images/469026_1_En_7_Chapter/469026_1_En_7_Chapter_TeX_Equd.png)
![$$ B1=\frac{1}{2}\left[\begin{array}{cccc}i& -i& 1& 1\\ {}-i& -i& 1& -1\\ {}1& 1& -i& i\\ {}-i& i& 1& 1\end{array}\right],B2=\frac{1}{2}\left[\begin{array}{cccc}-1& i& 1& i\\ {}1& i& 1& -i\\ {}1& -i& 1& i\\ {}-1& -i& 1& -i\end{array}\right],B3=\frac{1}{\sqrt{2}}\left[\begin{array}{cccc}1& 0& 0& 1\\ {}-1& 0& 0& 1\\ {}0& 1& 1& 0\\ {}0& 1& -1& 0\end{array}\right] $$](../images/469026_1_En_7_Chapter/469026_1_En_7_Chapter_TeX_Eque.png)
![$$ A1=\frac{1}{\sqrt{2}}\left[\begin{array}{cccc}i& 0& 0& 1\\ {}0& -i& 1& 0\\ {}0& i& 1& 0\\ {}1& 0& 0& i\end{array}\right] $$](../images/469026_1_En_7_Chapter/469026_1_En_7_Chapter_TeX_IEq7.png)
![$$ A2=\frac{1}{2}\left[\begin{array}{cccc}i& 1& 1& i\\ {}-i& 1& -1& i\\ {}i& 1& 1& -i\\ {}-i& 1& -1& -i\end{array}\right] $$](../images/469026_1_En_7_Chapter/469026_1_En_7_Chapter_TeX_IEq8.png)
![$$ B1=\frac{1}{2}\left[\begin{array}{cccc}i& -i& 1& 1\\ {}-i& -i& 1& -1\\ {}1& 1& -i& i\\ {}-i& i& 1& 1\end{array}\right] $$](../images/469026_1_En_7_Chapter/469026_1_En_7_Chapter_TeX_IEq9.png)
![$$ B2=\frac{1}{2}\left[\begin{array}{cccc}-1& i& 1& i\\ {}1& i& 1& -i\\ {}1& -i& 1& i\\ {}-1& -i& 1& -i\end{array}\right] $$](../images/469026_1_En_7_Chapter/469026_1_En_7_Chapter_TeX_IEq10.png)
![$$ B3=\frac{1}{\sqrt{2}}\left[\begin{array}{cccc}1& 0& 0& 1\\ {}-1& 0& 0& 1\\ {}0& 1& 1& 0\\ {}0& 1& -1& 0\end{array}\right] $$](../images/469026_1_En_7_Chapter/469026_1_En_7_Chapter_TeX_IEq11.png)
![$$ \left[0\kern0.375em 0\kern0.375em \begin{array}{l}0\\ {}0\\ {}1\end{array}\right] $$](../images/469026_1_En_7_Chapter/469026_1_En_7_Chapter_TeX_IEq12.png)
![$$ \left[0\kern0.5em 0\kern0.5em \begin{array}{l}1\\ {}0\\ {}0\end{array}\right] $$](../images/469026_1_En_7_Chapter/469026_1_En_7_Chapter_TeX_IEq13.png)
![$$ \left[0\kern0.5em 1\kern0.5em \begin{array}{l}1\\ {}1\\ {}0\end{array}\right] $$](../images/469026_1_En_7_Chapter/469026_1_En_7_Chapter_TeX_IEq14.png)
![$$ \left[0\kern0.5em 1\kern0.5em \begin{array}{l}1\\ {}1\\ {}1\end{array}\right] $$](../images/469026_1_En_7_Chapter/469026_1_En_7_Chapter_TeX_IEq15.png)
![$$ \left[1\kern0.5em 0\kern0.5em \begin{array}{l}0\\ {}1\\ {}0\end{array}\right] $$](../images/469026_1_En_7_Chapter/469026_1_En_7_Chapter_TeX_IEq16.png)
![$$ \left[1\kern0.5em 0\kern0.5em \begin{array}{l}1\\ {}1\\ {}1\end{array}\right] $$](../images/469026_1_En_7_Chapter/469026_1_En_7_Chapter_TeX_IEq17.png)
![$$ \left[1\kern0.5em 1\kern0.5em \begin{array}{l}0\\ {}0\\ {}1\end{array}\right] $$](../images/469026_1_En_7_Chapter/469026_1_En_7_Chapter_TeX_IEq18.png)
![$$ \left[1\kern0.5em 1\kern0.5em \begin{array}{l}1\\ {}0\\ {}1\end{array}\right] $$](../images/469026_1_En_7_Chapter/469026_1_En_7_Chapter_TeX_IEq19.png)
![$$ \left[\begin{array}{cccc}1& \cdots & \cdots & 0\\ {}0& -1& & \vdots \\ {}\vdots & \cdots & \ddots & \vdots \\ {}0& \cdots & & -1\end{array}\right] $$](../images/469026_1_En_8_Chapter/469026_1_En_8_Chapter_TeX_IEq16.png)
![$$ {H}^{\otimes n}\kern0.28em \left[\begin{array}{cccc}1& \cdots & \cdots & 0\\ {}0& -1& & \vdots \\ {}\vdots & \cdots & \ddots & \vdots \\ {}0& \cdots & & -1\end{array}\right]\kern0.28em {H}^{\otimes n}={H}^{\otimes n}\kern0.28em \left(\left[\begin{array}{ccc}2& \cdots & 0\\ {}\vdots & \ddots & \vdots \\ {}0& \cdots & 0\end{array}\right]-I\right){H}^{\otimes n}={H}^{\otimes n}\kern0.28em \left[\begin{array}{ccc}2& \cdots & 0\\ {}\vdots & \ddots & \vdots \\ {}0& \cdots & 0\end{array}\right]{H}^{\otimes n}-{H}^{\otimes n}I\kern0.28em {H}^{\otimes n} $$](../images/469026_1_En_8_Chapter/469026_1_En_8_Chapter_TeX_Eque.png)
![$$ =\kern0.5em \left[\begin{array}{ccc}2/N& \cdots & 2/N\\ {}\vdots & \ddots & \vdots \\ {}2/N& \cdots & 2/N\end{array}\right]-I=\left[\begin{array}{ccc}\frac{2}{N}-1& \cdots & 2/N\\ {}\vdots & \ddots & \vdots \\ {}2/N& \cdots & \frac{2}{N}-1\end{array}\right] $$](../images/469026_1_En_8_Chapter/469026_1_En_8_Chapter_TeX_Equ1.png)
![$$ \left[\begin{array}{ccc}\frac{2}{N}-1& \cdots & 2/N\\ {}\vdots & \ddots & \vdots \\ {}2/N& \cdots & \frac{2}{N}-1\end{array}\right]\left[\begin{array}{c}{\alpha}_0\\ {}\vdots \\ {}{\alpha}_x\\ {}\vdots \\ {}{\alpha}_{N-1}\end{array}\right]\to \left[\begin{array}{c}\\ {}\vdots \\ {}2/N\sum {\alpha}_y-{\alpha}_x\\ {}\vdots \\ {}\end{array}\right]=2\mu -{\alpha}_x\kern0.28em where\kern0.28em 2/N\sum {\alpha}_y=2\mu $$](../images/469026_1_En_8_Chapter/469026_1_En_8_Chapter_TeX_Equf.png)
