#### QM Basics 2 : Particle/Wave Duality and Uncertainty

<< Dallas TX, DEC 22 2023 >> Particle/Wave Duality Lets talk about some of the essential concepts we need to understand in QM. There are basically 3. A) We need waves to describe particle behavior. B) Some sub atomic particles have physical properties that are very discrete or "quantized". C) Quantum System have "inherent" uncertainty. Meaning its built in to the very nature of very small particles. How do we know about all these strange QM properties of particles ?? Much of it is through an experiment known as the Dual Slot Experiment. This fairly simple experiment repeatably demonstrates almost all the mysteries of QM. A full description of this experiment is beyond the scope of this article. There are MANY excellent descriptions on the web. But suffice to say that this experiment definitively shows that many sub atomic particles (electrons,etc) behave like waves of light energy, even though they are matter and have mass. Unfortunately, this had led to imaginations running wild and some thinking that the particle turns into a wave and then later back into a discrete particle. THIS IS NOT TRUE ! Particles are discrete objects, not waves. But these discrete objects have some properties (position,energy levels,velocity,etc) that often display wave like behavior. So, there is no such thing as a "matter wave". It is VERY important when discussing QM not to get too philosophical or meta-physical. We typically use waves as a convenient mathematical tool to describe QM behavior. This is the way Schroedingerd did it with his famous equation. However, wave theory is only one tool. Heisenberg got the same results using no waves at all. But it is commonly accepted that wave theory is the best approach understanding QM. With that being said, lets get into it. Waves, Waves, Everywhere The Schroedinger equation can be used (among other things) to find the position of a particle. In its simpler form it can help find the position, in three dimensions of an electron around the nucleus of an atom. But it can't give the exact position. It can only indicate where you will PROBABLY most likely find it. This is one inherent uncertainty of QM. So at any given time, you cant tell exactly where to find the electron. But, if you take a thousand samples and graph the results you will see a distribution that looks like a curve. This curve will indicate 300 times the electron was found at point A, 400 times at point B, and 300 times at point C. If you repeat this test again, you will get pretty much the same results. So QM IS predictable in statistical kind of way. In the real world an analogy is baseball. A batter can have a .330 batting average, but you can't really predict whats going to happen when he gets to the plate. But you can predict that after one hundred trips to the plate, he will PROBABLY get at about 33 base hits. So where are the waves in all this ?? The waves are in the sample point data over time. With the Schroedinger equation, if you look at any point in 3D space (point A) at any time T, you will get a number. This number is the probability of finding the electron there at that time. The bigger the number, the higher the chance of finding it there. If we sample the same point A say a second later, we get a different value, maybe higher, maybe lower. If you do this one thousand times, you will see this value for point A rise, fall, rise, fall, over and over again. If you would graph this thousand samples for point A over time, you see it oscillates up and down just like (you guessed it ) a wave. Here is the wave like properties of a QM particle. It is in the Data Sets. This is called a Probability Density Wave, and every particle has them. Again, it is NOT a physical wave like at the beach. Not One Wave, But Many ,,. If you have ever sat at the beach and watched the waves roll in, you might have noticed they are seldom uniform. Some are bigger than others, some smaller. Some are very well formed, others very jagged. What you are seeing is not one wave, but a combination of smaller waves merged together. When two merged waves align their peaks, it makes a bigger wave crest. When the 2 waves align their troughs, it makes a deeper combined trough. When a peak and a trough align, they tend to cancel each other and make a pretty flat wave. This phenomenon is called "wave interference" and is an important component of Wave Theory. As we have seen, this interference can be constructive (bigger combined wave) or destructive (smaller combined wave). This wave interference also affects a particles Probability Density Wave. When we previously talked about that oscillating magic number at point A at time T, that number comes from adding many different little wave values together to get the final probability number. The key here is any wave, real or probabilistic, is actually a combination of multiple smaller waves. In mathematics, this is called the Bandwidth Theorem, or more commonly a Fourier Transform. We will see the importance of these probability sub waves in the next article on Quantum Superposition . As we mentioned above, one very important use of the Schroedinger equation to statistically predict where we most likely to find a captive electron around the nucleus of an atom. It is important to note that the most common representation of an atom is incorrect. This model which depicts the electrons orbiting the nucleus like planets orbit the sun is incorrect. In reality, due to QM laws, the electrons orbit in defined cloud like regions. These orbitals are NOT circular or elliptic like planetary ones. Instead they are more like a Halo around the nucleus. There are often multiple halos around the nucleus. Each halo representing a different energy state of the electron. The Schroedinger equation lets us see where in these halos the electron is MOST likely to appear at any sample time. The equation also tells us where the electron will NEVER appear. This is one of the few times when QM is certain. When the Schroedinger equations probability result for an electron at any point in space is ZERO, the chance of itbeing there is also ZERO. These halo like orbitals, and how they hold or share electrons with other atom's orbitals are fundamental to how matter is organized in the universe. This is called the science of chemistry. In fact, studying electrons and how they behave under QM has its own sub field of study. It is called Quantum Electro Dynamics (QED). QED is way beyond the scope of these articles. But QED gives a direct example of how the microscopic QM world effects our macroscopic world. Localization and the Wave Packet. We have already learned that a sub atomic particle really has no fixed position, and could (with low probability) appear pretty much anywhere at any sample time. But in reality, we often see particles move in a predictable trajectory. An example of this is an electron beam. The beam is focused, and hits where you aim it. Not all over the solar system. So what's happening here ?? When all a particles probability sub waves are added up, you often see powerful constructive interference. This bunching up is called a "Wave-Packet". When a particle forms a persistent Wave-Packet over time, we say the particle is "Localized". Localized means not scattered all over the universe. Sometimes the Wave-Packet is transient in time,or static in 3D position for a period of time. Often how ever the bunching stays together and moves in 3D over time. This persistent, but moving Wave-Packet would cause the particle to appear to have a predictable course of movement. You can see something similar in nature. Today the internet is full of videos showing giant "rogue waves" crashing into a cruise ship and then moving on. These giant rogue waves are the constructive summation of many smaller waves that bunch up. The rogue wave is a real world example of a persistent, moving in time, Wave-Packet. In this case these are mechanical waves. In QM the bunching up is of the probability sub waves of the particle. So with Wave-Packets and Localization we can start to see how QM effects our world. In a later article, we will discuss how large groups of Quantum Objects interact to create the macroscopic world in which we live. What does "Quantum" Mean ? Before we leave this article, there is one final issue we need to understand. We use the word Quantum everywhere but what does it mean ?? A commonly used description is : "the smallest discrete unit of a phenomenon" This basically means its the smallest measurable amount of something, and you can't sub divide it further. This is important because it means everything you measure, can be measured in the units of the smallest quantum. An example again is the lowly electron. Each electron has an exact voltage. If you measure the voltage in a battery, and divide it by the charge of an electron, you will know exactly how many electrons are in that battery. Its simple arithmetic, not calculus. So every electron has a discrete "quantum" charge. All total charges, are a summation of these tiny quantum charges. Finally something in QM we can all understand ! QM does not like fractions or decimals, it only works in whole numbers. These whole numbers can be positive or negative, but they have to be whole. QM doesn't do 3/5 or 3.67. Remember a quantum is indivisible. This why the orbitals of an electron are discrete or "quantized". An electron is in orbital 1, 2, or 3. But never in between. There is no orbital 2.56. The position in space where orbital 2.56 would be, comes up with the value of ZERO in the schroedinger equation. You wont ever find the electron there. SUMMARY So, to summarize. There is no such thing as a matter wave. A particle is a discrete object, it is NOT a wave, nor does it change into one. This discrete object does have properties like, velocity, position, momentum, etc. These properties display oscillating wave like characteristics. If you sample and graph the results over time you will see the waves. This wave like behavior as computed by the Schroedinger equation, allows us to to take statistical best guesses of the value of that property at any moment in time. Also, uncertainty is ALWAYS inherent in Quantum Systems. Whatever quantum property you are measuring or predicting is inexact. You can only compute a possible range of values, and the probabilities of finding each those values. So particles are never waves, but we use waves to describe their behavior. Amazing HUH ?! Just wait, it gets even weirder (sorry ... counter-intuitive) Now on to Superposition and Entanglement !