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Important abbreviations: QM= quantum mechanics QT= quantum theory QP= quantum physics.

After hearing a lot of comments on QM, some comments more educated than others, I decided to start a series of threads that will chronicle and summarize the lectures. I debated whether or not to make one giant thread or 24 individual threads devoted to each lecture. In the end, I opted with the latter. There will be a total of 24 threads, each one devoted to its corresponding lecture from the series. Not anymore, I decided to consolidate them all into one.

I'm approaching these posts with the intention of educating myself and anyone else who might need some clarification on QM. Thus, I do not necessarily want to promote or encourage debate--though I don't necessarily want to discourage it either.

Think of me as the messenger and NOT the expert on QM. Everything I know will come from the lectures themselves. Feel free to ask questions. I may have glossed over, skipped, or omitted things as I transcribed them into an outline.

Also, you'll have to give me a couple lectures to really figure how to best arrange everything. I will try to put key-terms and definitions in bold, headings in CAPITAL LETTERS AND UNDERLINED, and I'll figure out the rest when I get to them. I'll include a summary at the top of each page of the content discussed within said lecture. I will try to post 2-3 lectures/threads per week. I don't want to do it any more frequently than that for fear of just being overwhelmed with too much information to digest properly. Unless I use the quote function, assume that everything written is paraphrased by me.

According to the professor, while these 3 terms actually are distinct and have separated meanings, he will use them interchangeably for the sake of simplicity.

The series will be divided into 5 sections:

Section 1: lectures 1-7, development of QT 1900-1930. Section 2: lectures 8-10, learn some basic QT. Section 3: lectures 11-18, a tour of the quantum world and quantum phenomenon. Section 4: lectures 19-22, QM and information: more contemporary look at information. Section 5: lectures 23-24, philosophical issues, implications, and speculations.

Enough with the logistics. Let's explore that rabbit hole known as Quantum Mechanics.

Note: For the purposes of these posts, within the context of these threads I will drop my promise to write in E-Prime for the sake of clarity, efficiency, and accuracy. Outside of the QM posts I will still try to post in E-Prime. All of this is probably a moot point because I'm sure no one even noticed my self-experiment in writing in E-Prime.

Summary:This lecture is an overview and introduction to the topic of QM. It defines key terms and alludes to topics that will be discussed in later lectures.

KEY DEFINITIONS: -Mechanics- the study of force and motion.

-Classical Mechanics- based on Newton’s laws of motion.

-Classical Physics- thermodynamics, electromagnetism, optics, etc.

-Quantum mechanics- the physics of the microscopic universe.Called quantum because it was thought that energy came in distinct lumps of packages or energy, distinct quanta of energy. The theory did not develop until the 1920s. QM is the basic physics of this realm of atoms, electrons, and photons.

QM studies things that are far smaller than anything observable by a microscope. How small is small?

To put things in perspective: -A single celled organism, which can be seen in a microscope, has 10^{12} atoms. -Atoms: in a 1 inch cube there are 10^{24} atoms. -Photons: a 20 watt light bulb emits 10^{20} photons in one second.

What we can do with QM: Quantum theory is the most successful theory of physics ever devised. it explains how light is absorbed and emitted by matter; how solids and liquids behave at low temperatures; and accounts for how elementary particles interact with each other.

However,

Quote: “The microscopic realm is not just a miniature copy of the world of our experience: if that were true, this wouldn’t be that interesting a subject…”

THE STRANGE IMPLICATIONS OF QM: -tells us about the laws of possibilities: that probabilities can cancel each other out, eg, if a particle can move to the left or to the right with equal probability, the probabilities will cancel each other out.

-the way particles move through space: particles move in waves and are in many places at once;

-how groups of particles act together, two particles can be exactly identical to one another;

-entanglement, an interconnectedness that can exist between two particles that are far apart;

-properties of energy—energy comes in discrete amounts (not continuous); furthermore, energy is present in empty space/vacuum;

-behavior of information—information acts in strange ways.

Despite these strange incongruities with our world of experience,

Quote: “The quantum world is our world. It is the world we live in. And QM will not just explain funny facts about the microscopic world but also about the world we live in every day.”

All physicists know how to use the mathematics behind QM: but not all of them agree about how to interpret the results of these equations. Because of this, we have three interpretations of the mathematics: THE 3 INTERPRETATIONS OF QM

-Copenhagen Interpretation—Championed by Niehl’s Bohr, QT only tells us about probabilities; it tells us nothing about actuality, and no deeper view is possible than this.

-Hidden Variables Interpretation— The probabilities reflect our ignorance of the existence of secret variables. The results of experiments are predetermined by hidden factors that we don’t know. However, this implies that the hidden factors cannot be localized in space: it needs to be part of a hidden dimension of the universe.

-Many Worlds Interpretation—All possibilites are chosen and no possibility is ever really rejected. If a particle has an equal probability of going left or right, the particle goes in both directions and two different universes branch out in which each possibility is actualized. Thus there are multiple branches of the universe in which all possibilities are actualized.

Unfortunately, I won't have internet access for several days, possibly a week. So I won't be able to update it until I get back. Good news is, once I do get back, I'll have at least 2 or 3 threads ready to go.

It's been a minute. Nonetheless, here's the second installment.

Lecture 2: The View From 1900

Summary: The history of classical physics was a battle between two competing models of reality. One model said that matter was made of discrete things, the other model said that matter was made of continuous stuff. By the 19th Century, physicists had settled upon the notion that matter is thing like and light is stuff like. This view led to the amazing unification of thermodynamics with mechanics and of optics with electromagnetism. The next goal was to unify thermodynamics with optics. This proved to be the hump that classical physics could not get over, and which eventually led to the creation of QM.

Antiquity: Throughout the history of physics there have been two competing worldviews. One view says that the basic constituents of nature are discrete and indivisible units; in short, the universe is made of things. The other view holds that the basic constituents of nature are smooth continuous substances that can be divided in any way you choose. By analogy, the former view holds that nature is made of peas and the latter holds that nature is made of mashed potatoes. Both models could be used to explain certain occurrences in nature. Fast-forward to the 17th Century: The debate between the ‘thing-people’ and the ‘stuff-people’ shifted from matter to light. Newton believed light was thing like and came in small units called corpuscles. Huygens believed that light was a continuous wave, much like sound.

Fast-forward to the 19th Century: Classical physics settled on the notion that “Matter is thing like while light is stuff like.” John Dalton showed that the atomic model of matter could explain the affinity between physics and chemistry; Maxwell and Boltzmann showed that you could understand the nature of gas if you thought it was composed of atoms; it was also discovered that heat energy is the random motion of the particles composing gas. This last discovery was a breakthrough in physics. Through this distinction—matter is a thing, light is a wave”—19th century physics was able to unify the otherwise disparate field of thermodynamics and classical mechanics.

Thomas Young proved that light behaves like a wave with his famous two-slit experiment. The experiment goes as follows: you shine a very defined light source, such as a laser, at a metal plate which has two narrow slits in it. You then observe the pattern that the light makes on a screen placed not too far from the slits. What results is a band of light on the screen broken up by dark bands. If you cover up one slit, there are no dark bands in the resulting pattern. If light were a stream of corpuscles, this wouldn’t make any sense. But if you conceive of light as a wave, then the alternating pattern of bright and dark spots can be explained by the fact that the waves form an interference pattern when they go through two slits. The physicist Maxwell concluded from this that light is a traveling electro-magnetic wave. With this discovery, the disparate field of optics and electromagnetism were unified.

Lord Kelvin pointed out that the history of physics has been a story of unifying disparate fields of study. Not only did we unify thermodynamics with mechanics, but we also unified optics and electromagnetism. The next objective for classical physics was to unify thermodynamics and optics. However, two dark clouds seemed to obstruct this path. One was the failure of scientists to discover the ether that was assumed to be necessary for light to travel through. Second, was the problem of thermal radiation—how does heat emit light? This second dark cloud in particular led to the origin of quantum physics.

Lecture 3: Two Revolutionaries—Planck and Einstein

Summary: Planck and Einstein solve three puzzles of classical physics using the quantum idea that energy comes in discrete units. This undermines the old idea that matter is made of discrete atoms and light is made of continuous waves.

Several puzzles remained unsolved by classical physics.

The 1st puzzle: the problem of thermal radiation. Light bulbs emit a bit of infrared light, a bit of visible light, and almost no ultraviolet light. However, classical physics predicts that a light bulb should emit way more ultraviolet light then it actually does: a phenomenon known as ultraviolet catastrophe. The physicist Planck proposed the idea that light isn’t secreted continuously but comes in quanta, discrete packages. His hypothesis predicts no ultraviolet catastrophe. The 2nd puzzle: the photoelectric effect: if you shine a light upon a polished metal surface, you will emit electrons from it, and you can measure how many and how strong. Classical physics thought that the strength of the electron depended on the strength of the light—i.e., the stronger the light the stronger the electron. Not so. The energy of the electron does not depend on the intensity of the light. The energy of the electrons depends on the frequency of the light. Einstein proposed the idea that photons have different energy depending on the frequency of it. This excellently explained the photoelectric effect.

(A weird picture is beginning to emerge… The double slit experiment definitively proves that light is a continuous wave. The photoelectric experiments definitively prove that light is made of photons.)

The 3rd puzzle: the problem of heat capacity. Heat capacity is how much energy you need to raise an object’s internal temperature by 1 degree Celsius. High heat capacity tells us whether an object is harder to heat or cool it. And vice versa. Classical physics predicts that any pure solid—an object made of one type of atom such as platinum or diamond—should have about the same heat capacity. This does not turn out to be true at very low temperatures. Enter Einstein who applies the quantum ideas of light also to matter. At low temperatures the atoms are fixed in place and only a few have enough energy to emit quantas of energy. And this changes from substance to substance, which depends on the vibrations of atoms of each substance.

Planck and Einstein have introduced quantum ideas to successfully explain previous puzzles of classical physics. In each case, (thermal radiation, photoelectric effect, and heat capacity) energy came in discrete units which could be discovered by Planck’s equation: E=hf ( h being Planck’s constant for energy changes). This is a direct challenge to classical physics which believed that energy is continuous and could come in any amount. From Planck and Einstein we have to conclude that light can also act like a stream of particles and that matter can also act as waves. The classical distinction between stuff and things has begun to break down at this point.

Summary:QM physics begins to embrace the idea that at the microscopic level light exhibits a particle-wave duality. De Broglie extends this particle-wave duality to include matter as well. At the subatomic level, matter and light move like waves but emit energy in quanta. According to the Born Rule, the intensity of a wave tells us the probability of finding a particle in that position. Thus, for QM everything exhibits the particle-wave duality and we can only speak about probabilities and can never say anything definitively about the quantum world.

On the one hand Young’s double-slit experiment seems to conclusively demonstrate that light is a wave. On the other hand, Einstein’s photoelectric experiment seems to demonstrate that light comes in quanta of discrete energy. Quantum Physics embraces the notion of wave-particle duality, which means that we cannot speak strictly about light as being distinctly one or the other. But, the general rule of thumb goes like this: light travels in the form of waves but light interacts in the form of discrete particles. Light has both particle and wave properties.

In 1924 Louis De Broglie extended the wave-particle duality to matter as well. He conducted an experiment in which he shot electrons at a crystal. Crystals are known for having a very regular and structured atomic formation. He observed that electrons only came out in some directions. In essence, the spaces in the structure of the crystal served like slits which caused destructive interference of the electron. A particle is characterized by its kinetic energy and its momentum. A wave is characterized by its frequency and wavelength. De Broglie connects these two together using Planck’s constant.

You can illustrate the wavelength of an electron particle but you can’t illustrate the wavelength of a baseball. Well, how far in size can you go before you can no longer illustrate the wavelength of an object? We can do them with neutrons and even atoms which are 1,000 bigger than electrons. But, in 1999, Zeilinger did a two-slit experiment using entire molecules of 60 carbon atoms—a million times bigger than an electron. And even at this size, molecules move in waves and exhibit destructive and constructive interference.

But how do we reconcile or understand this particle-wave duality? A particle has a definite position in place: either here or there. A wave is all over the place, it spreads out. How do we reconcile these two facts? Invented by Max Born, the Born Rule explains that the intensity of a wave at a point tells us the probability of finding the particle at that point.

QM cannot tell us where a particle is; it can only tell us the probability of finding a particle in a certain place. Where it actually ends up is random and a gamble. For example, going back to the double slit experiment, if you fire electrons through a slit one at a time, they will show up randomly on the screen. If you fire enough electrons through, the wave pattern of dark and bright bands will eventually emerge. The same can be done with light: shooting one photon at a time. After enough photons, the same interference pattern emerges.

The microscopic world is governed by the principle of wave-particle duality. Everything—light and matter—has both wave and particle properties. In QM, individual events are random; QM predicts only probabilities.

Quote: “And because of this, everything in nature is both thing-like and stuff-like. It’s thing-like because everything in nature is discrete; energy comes in discrete quanta. Particles come in discrete amounts 1, 2, 3, 4,…. But everything is also stuff like. Everything is described by continuous waves in space. These waves are waves of probability whose intensity gives you the probability of finding the particle. So even though the particle is discrete, the probability wave is continuous.”

Interesting read. I was expecting some of the philosophical interpretations of quantum mechanics (or things that are claimed to be quantum mechanics) that are silly, but I was pleasently surprised.

If you'd like this moved to the science forum where it will perhaps recieve comments from more knowledgable people on the subject, let me know- pm or reply

I debated about whether or not I should put it in the science forum. But I opted to put it here because I figured that there's a lot loose-headed thinking about QM in this forum. If anyone needs it, it's this forum and not that one.

On the other hand, it'd be nice to get more comments and discussion going. Although, I might be jumping the gun considering I've got 20 more lectures to go. Would I be overdoing it if I duplicated the thread in the science forum?

Naw, you've got my permission to duplicate the thread in teh science forum if you want. We don't get much traffic in there regarding bona-fide scientific discussion of this sort.

Sounds good. I will probably wait a little bit before I duplicate the thread in the science forum, at least until I definitively decide on what layout/format/delivery works best for the posts. As you can tell, my style in presentation has changed slightly from post to post--although part of that is due to the content of each lecture. Not all content can be represented in bullet-points.

Anyway, I'm working on lecture's 5 and 6 on at the moment. Expect at least another post from me later tonight.

Lecture 5: Standing Waves and Stable Atoms Summary:Niels Bohr applied the quantum ideas of Planck and Einstein to help explain the structure of the atom. The stability of the atom was due to the ‘ladder rungs of energy.’ The energy of each ladder rung was explained by de Broglie’s concept of standing waves. Finally, Erwin Schrodinger reinforced this model with a mathematical equation.

Quote: “Everything has both wave properties and particle properties. Light has light waves and also light is made of photons. Electrons are particles but they also have wave properties: frequency, wavelength, constructive and destructive interference.”

The particle and wave duality is connected in two ways: one, The Planck-de Broglie, formulas which connect the particle energy and momentum to its wave frequency and wavelength. Two, the Born rule: that the wave intensity determines the particle probability.

By the end of the 19th Century, it became clear that atoms were made of parts. But what is the internal arrangement of atoms? Atoms are thousands of times too small for a microscope to see.

Ernst Rutherford conducted the following experiment. He shot particles at a thin piece of gold foil to observer what happens. He anticipated that some of the particles will pass right through and some will be deflected at certain angles. The results of experiment revealed that some of the particles actually bounced off the gold foil and bounced right back. This is astonishing because, in the words of Rutherford, “It was as if you had taken a 16inch artillery shell, fired it at some tissue paper, and had it bounce back at you.” From this experiment Rutherford created what is known as the “solar system model of the atom,” which amounts to a dense center surrounded by moving pieces. Rutherford concluded that the particles that bounced back had hit the center, the nucleus. From the experiment Rutherford could actually measure the size of the nucleus, which turned out to be 100,000 times smaller than the atom itself. Outside of the nucleus are electrons smaller in size and mass, which orbit the nucleus in the same way the planets orbit the sun.

This model became a real puzzle for classical physics. If the electron is spinning around the nucleus, it should be emitting electromagnetic energy. And if it’s emitting energy, that means it should be getting weaker and weaker. And as it gets weaker, the orbit of the electron should spiral inward into the nucleus, causing the atom to implode. All of which should happen within the span of a microsecond. So, if Rutherford’s model is right, then according to classical physics all atoms should implode instantly.

Enter Niels Bohr who comes to work in Rutherford’s lab. Bohr tries to incorporate Einstein’s and Planck’s quantum ideas. Using these ideas he created the “Bohr model of the atom.” In this model, only discrete orbits are possible for the electron. The electron must follow the path of a specific orbit, whether big or small, and cannot move in between the orbits. And when an electron jumps from an orbit to another, it either emits or absorbs a photon. The different orbits of the electron are like the rungs of a ladder. The lowest rung has the lowest amount of energy and is closest to the nucleus. And the highest rung has the highest amount of energy and is furthest away from the center. So there are only certain energy rungs that the electron can occupy. When an electron absorbs a photon, it jumps up a rung; and when it emits a photon, it jumps down a rung.

Since there’s a bottom rung, the electron cannot lose any more energy. Which is why the atoms is stable—because there is a bottom rung of energy which cannot be passed. The energy of the electron corresponds to the spaces between the rungs. So, for example, if an electron moves from energy rung number 3 inward to energy rung number 2, it emits a ‘red’ photon. If it jumps from 4 to 2, the photon comes out green. Et cetera. Thus, Rutherford’s solar system model works because there are discrete orbits for the electrons.

De Broglie explained Bohr’s orbits with the concept of standing wave patterns. A standing wave pattern, for example, is like a piano wire that is clamped at the two ends; because of this, only distinct types of wavelengths are possible. Because the wavelengths are determined and fixed, it has a very definite and predetermined frequency. This is why the piano wire stretched at a certain length creates a pure musical tone.

Thus the electron in an orbit is like the string of a piano, it can only vibrate in certain ways, it can only have certain wave patterns, and it can only have certain frequencies, and therefore it only has certain energies.

Thus,

Quote: “Only certain de Broglie wave patterns ‘fit’ around the atom. Only certain wavelengths and frequencies are possible. Only certain electron energies are possible. Standing wave patterns [correspond to] Bohr’s orbits.”

Enter Erwin Schrodinger who creates a mathematical equation which describes de Broglie’s wave patterns. This equation allows us to work out the details of atomic structure, especially with atoms with many electrons. Physics students of QM spend 90% of their time learning to solve and apply Schrodinger’s Equation. When you solve the equation, you get standing wave patterns, each of which correspond to different energy levels on the ladder, and the energy levels change when you emit or absorb a photon of light. Schrodinger’s equation explains the jumps of Bohr’s model.

This explains: the energy levels in atoms and molecules, what energies are possible, the emission and absorption of light, you can figure out photon energies, and also the probabilities of making certain jumps in energy levels.

The idea of the wave-particle duality glimpsed by Einstein and Planck was extended by de Broglie to cover matter, then interpreted by Bohr who applied it to the atom, and then converted by Schrodinger into a mathematical equation which can explain the structure of the atom.

Summary:The quantum world has some strange properties which set a definite limit on our precision about it. The uncertainty principle says that the more accurately we know a particle’s position, the less we know about its momentum—and vice versa. This notion leads to a philosophical debate between Einstein and Bohr. Is the uncertainty of the quantum world due to our present state of ignorance? Or is the quantum world fundamentally fuzzy and indeterminate?

Particles and Waves are very different. Think of a particle as a baseball. A baseball has an exact position in space and an exact velocity. It follows a definite path and trajectory which can be described by Newton’s laws of motion. But a wave is different: it spreads out.

To understand the uncertainty principle, we need to understand the phenomenon of diffraction: After a wave passes through a barrier with an opening, just like in the single slit experiment, the wave spreads out on the other side of the barrier: that spreading out is called diffraction. The spread of the diffraction is determined by the wavelength divided by the width of the opening. In a mathematical equation, it looks like this: diffraction= λ/W. A narrow width (small W) gives us a big and spread out diffraction. A big slit with a large width (big W) gives us a more narrow diffraction.

An example to illustrate the diffraction: Imagine standing in front of a wall with a friend on the other side of that wall. Just down the hall, maybe 5 or 6 feet away, is a doorway. Your friend speaks. The sound waves from his voice hit the slit in the barrier, the door in the wall, and then spread out in a relatively wide pattern which allows you to hear your friend even though the sounds has to “travel around the corner.” But light waves, which have a wavelength a million times smaller than sound waves, do not travel as far ‘around the corner.’ Hence, we can hear our friend but we cannot see them.

To understand this, let’s look at single-slit diffraction experiment. Much like Young’s double-slit experiment, you use a laser to shine a well-defined light through a narrow slit and then project it onto a screen. It creates a broad smear of light which is much brighter at its center than on its wings. The difference between this and Young’s experiment is that we can control the width of the slit. If we make the slit wider, the pattern on the screen becomes narrower which happens because when you make the slit wider, the ratio between the wavelength and the width becomes smaller. When you make the slit narrower, the pattern gets wider because the narrower the slit (the smaller the denominator in that ratio), the stronger the effect.

Thus, there’s a tradeoff: the narrower the slit, the wider the pattern; the wider the slit, the narrower the pattern.

Quote: “Wave diffraction, together with the particle-wave duality of quantum mechanics, sets a limit on how well the particle properties are even defined in the quantum world.”

The same experiment could have been done with electrons. As an electron passes through the slit, we know its lateral position, labeled X. We do not know the electron’s exact lateral position because the slit is bigger than the electron, but we know its lateral position to a certain “uncertainty” known as ΔX. Thus, ΔX= the width of the slit. But as it passes through the slit, the electron’s lateral momentum becomes more uncertain—that is, it could fall on the screen slightly to the left, or slightly to the right, and we wouldn’t know exactly until it happens. The uncertainty in the electron’s lateral momentum is ΔP. Now, because narrow diffraction patterns correspond to wide slits, and vice versa, there is a trade off in uncertainties, a tradeoff between ΔX and ΔP. There is a tradeoff between the electron’s lateral position and the electron’s lateral momentum. The smaller one of those uncertainties is, the larger the other must be.

Quote: “And because everything has wave properties, according to the principle of wave-particle duality, that means that everything exhibits diffraction and that means that this tradeoff between uncertainty in position and uncertainty in momentum is universal and applies to everything.”

The basic tradeoff between knowing a particle’s position and knowing its momentum is called, “The uncertainty principle,” and was formulated by Werner Heisenberg into the following equation:. ΔXΔP ≥h (h= Planck’s constant).

Quote: “No particle can have a definite position and a definite momentum at the same time. The more definitely we know where the particle is (ΔX being smaller) the less definitely we can know where it is going, (ΔP is larger).”

You cannot be more certain than the principle allows. There’s an inescapable tradeoff between uncertainty in position and uncertainty in momentum. That is, you can never know the position and the momentum of an object to a greater degree than Planck’s constant, h. H=6.6 x 10^-34 Joule/Sec. This constant is very, very small. But if we apply it to the macroscopic world, if we take a baseball which is very large, we can still know its position and momentum to 12 or 15 decimal places and still respect the uncertainty principle. This is why Newtonian physics works pretty well in the macroscopic world. But to an electron, the uncertainty principle is a big deal.

Consider, for example, an electron in an atom. ΔX can be no larger than the diameter of the atom, which means that ΔP is very large, so large that we can’t even tell if the electron is moving this way or that way.

There’s another uncertainty principle in addition to the position-momentum principle, one related to time and energy: ΔT=period of time and ΔE=amount of energy need. ΔTΔE= h… This tells us that things that happen very fast (small t) have very poorly defined energy (large E). As we’ll see later on, this second uncertainty principle is of “huge, maybe even cosmic importance as we’ll see in lecture 17 and 18.”

So, the more we know a particle’s position the less we know its momentum. Heisenberg thought this was less of an uncertainty principle and more of an indeterminacy principle—that is, that the electron does not have a fixed position of momentum. Einstein thought differently. There’s a big difference between saying that an ‘electron has exact values for x and p, but we don’t know them’ and saying ‘that particle does not even have exact values for x and p.’ Said differently, tt’s one thing to say the electron follows a trajectory through space but we don’t know exactly what it is and quite another thing to say that the electron has no definite trajectory.

The difference is philosophical: one is a statement about our ignorance, the hope that we can one day overcome the uncertainty principle. The other is a statement about nature, that it is inescapably and fundamentally indeterminate.

Quote: “So are the uncertainties in quantum mechanics due to our lack of knowledge or due to nature’s lack of definiteness?”

This question is the central point of the debate between Niels Bohr and Albert Einstein over how to interpret QM.

I'm really liking these mini-lectures. I read about this kind of stuff all the time but it always seems that I can't really grasp a lot of it. Learning the subject from the ground up is helping me understand some of the other concepts I didn't get before.

By the way thanks for doing this, I'm still looking forward to the rest of the installments.

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