In Part 1 of this blog article, we saw the foundations of Quantum Physics and how the results of several experiments were arranged upon each other to form a chaotic yet precise theory. At the end of that article, I gave a teaser of the Statistical Interpretation of Quantum Physics. In this part, I aim to provide a technical and concise explanation of the Statistical Interpretation of Quantum Physics and its loopholes. In the end, we shall discuss several other interpretations of Quantum Physics. But first - "What do we mean by interpretations of Quantum Physics?" - An interpretation of Quantum Physics is simply an explanation for the results and observations obtained from the experiments and theories. "Why is there an interpretation for the Quantum Theory?" - Short answer - Because the Quantum Theory we have today does not make any sense at all and Physicists are trying to make sense of it with "an interpretation" or "explanation". Some interpretations however make things worse and totally shatters our intuitions. The best example of such interpretation is the one we are going to explore in this article.
A quick recap from the last article - Objects in Quantum Physics behave both as particles and as waves. One cannot simultaneously know the position and the momentum of such objects and the outcomes of experiments cannot be predicted with certainty. Rather, one can calculate the probability for all possible outcomes. This is all that's needed to delve into the Statistical Interpretation of Quantum Mechanics which also has another name - "The Copenhagen Interpretation". According to this interpretation, you cannot completely know the state of a system in the "classical" sense. But in a "quantum" sense, you can know the state of a system with the help of a mathematical term - "the wavefunction" represented by the Greek letter psi - Ψ. This "wavefunction" is all that can be known about a quantum system. Its name is quite self-explanatory for its purpose. The wavefunction is a mathematical function responsible for the wave behavior of particles. For those of you who are unaware of what exactly a mathematical function is. A mathematical function is sort of a machine with an input and output. You plug in certain numbers (or variables) as input and you get a different or same output. There are functions of all kind in mathematics. Generally, the input of this wavefunction is a number representing the position of particle in space. So, what output does a wavefunction have for an input? It gives us a complex number called as - "Probability Amplitude" whose square (or multiplication by complex conjugate) gives the "Probability Density" for a particle to be present at a given position. I know, you might haven't understood some or all of what I just said. Let us break down the answer first. The output of this wave-function is a complex number. A complex number is a combination of a real number and an imaginary number. An imaginary number is, as the name suggests a number which does not exist. To avoid getting a pile up of unknown terms, I will not go into much depth. Why the output is a complex number will be answered later. So this complex number when multiplied with another complex number but with a reversed sign (also called as its Complex Conjugate) gives us the "probability" to find a particle at that input position. Simple as that.
Where does the waviness come in all of this abstract mathematics? - It comes from that complex number output given by the wavefunction. This output is called as "probability amplitude". These complex amplitudes represent the mathematics of waves. They are periodic i.e. their value changes by a fixed amount in fixed interval of time. Just like a guitar string vibrates up and down, there exists a wave travelling on that string. The reason complex numbers are used is because they are a neat way to represent periodic functions i.e. the sine and cosine functions by a beautiful formula called as Euler's Formula. However, these probability amplitudes are complex valued. They contain both real and imaginary numbers. Imaginary numbers as we know cant be real, thus these "probability amplitudes" alone cannot yield us the probability we need. But, by multiplying them with their complex conjugates, the imaginary part is cancelled or rather converted to real part. Thus, we square the probability amplitudes to obtain the actual probability to find a particle at a position. All of this is too technical, and for a layperson it would be convenient to only remember that the wave nature of particles is defined by their "wave-functions". But, let us just examine what all this mathematical machinery tells us about the actual physical observations. I said the wave nature of particles can be explained by their wave functions. They are complex valued functions and are harmonic (periodic i.e. repeating after fixed interval) in nature. But then for a specific input, they give out "probability amplitudes" which in turn give actual probability when squared. All this means that the particles in themselves are not wavy. Instead, it is the probability itself that is waving. Odd enough? - Welcome to the strange world of Quantum Physics. It can then be concluded that matter waves are nothing but waves of probability. Just like a light wave is a wave in the electromagnetic field. However, some quantum physicists believe that these matter waves are not at all physical, like the electromagnetic waves. These waves are quite abstract. They are in a different mathematical space called as the - "Hilbert's Space". Not the real space in which we live.
Where does the waviness come in all of this abstract mathematics? - It comes from that complex number output given by the wavefunction. This output is called as "probability amplitude". These complex amplitudes represent the mathematics of waves. They are periodic i.e. their value changes by a fixed amount in fixed interval of time. Just like a guitar string vibrates up and down, there exists a wave travelling on that string. The reason complex numbers are used is because they are a neat way to represent periodic functions i.e. the sine and cosine functions by a beautiful formula called as Euler's Formula. However, these probability amplitudes are complex valued. They contain both real and imaginary numbers. Imaginary numbers as we know cant be real, thus these "probability amplitudes" alone cannot yield us the probability we need. But, by multiplying them with their complex conjugates, the imaginary part is cancelled or rather converted to real part. Thus, we square the probability amplitudes to obtain the actual probability to find a particle at a position. All of this is too technical, and for a layperson it would be convenient to only remember that the wave nature of particles is defined by their "wave-functions". But, let us just examine what all this mathematical machinery tells us about the actual physical observations. I said the wave nature of particles can be explained by their wave functions. They are complex valued functions and are harmonic (periodic i.e. repeating after fixed interval) in nature. But then for a specific input, they give out "probability amplitudes" which in turn give actual probability when squared. All this means that the particles in themselves are not wavy. Instead, it is the probability itself that is waving. Odd enough? - Welcome to the strange world of Quantum Physics. It can then be concluded that matter waves are nothing but waves of probability. Just like a light wave is a wave in the electromagnetic field. However, some quantum physicists believe that these matter waves are not at all physical, like the electromagnetic waves. These waves are quite abstract. They are in a different mathematical space called as the - "Hilbert's Space". Not the real space in which we live.
A graph of wavefunction against position
This was the Statistical Interpretation of Quantum Physics in a nutshell. The theory on its outside may seem too fragile and impractical. But its only when you explore its mathematical construction, you realize how beautifully and flawlessly the theory describes quantum observations. However, like every scientific theory, it too has some imperfections and loopholes. The most bothering ones are the "Measurement Barrier" and the "Quantum Information Eraser Paradox". Let us understand the "Measurement Barrier" first. Remember, in part 1 of this article, we learnt the "Wave-Particle Duality". According to which, objects in the world behave both as particles and as waves depending on the circumstances. The circumstance I was talking about actually refers to "measurement"performed on that object. There exists this peculiar relationship between "measurement" performed on a system and its "state". This relationship is rather a hostile one. In the Classical World measurements performed on a system does affect its state by a very negligible amount. Thus, such measurements do not make significant changes in the system. But in the delicate Quantum World, no matter how careful one is, any measurement performed on a "system" is certain to change its "state" in an unpredictable manner. This unpredictable change of state upon measurement is called as - "Collapse of Wave-function". What it means is that prior to measurement the quantum object exists in a superposition of states. For example - a quantum object having a specific wavefunction does not really have a definite position in space. It exists everywhere at once in a superposition. It only has a certain probability to exist at any position. This behavior is just like waves, which do not have a fixed position. However, after measurement one knows the exact position of that particle. It stops behaving like wave and its wavefunction collapses to a single point. Thus, the probability of that particle to exist at that point becomes one. The graph of wavefunction contracts to a localized point. However, at which point will the particle turn out to be present is impossible to predict. The outcome is totally random. This unpredictable collapse to a random position after measurement is known as the collapse of wave-function. Physicists are unable to provide any mechanism for why such collapse occurs after measurement. Hence, the act of measurement disturbs the state of a system in an unpredictable and gives rise to the - "Measurement Barrier". The second paradox - "Quantum Information Eraser Paradox" or also known as "Delayed Choice Experiment" is a consequence of this "measurement barrier". It arises in the following way - The fact that particles behave as waves when not observed and then collapse to a localized point upon observation potentially suggests that information about the particle can be lost or destroyed.
How? - Well, we would borrow an experiment which was originally used to demonstrate wave behavior of light - The Double Slit Experiment and use the electron version of it. The one which was described in part 1 of this article. We would have an electron gun, which would fire electrons at a constant rate, ahead of it would be a plate with two narrow slits in it and next to the plate is a screen (say a phosphor screen which would produce a scintillation (spark of light) when the electrons arrive at it). The electron gun fires electrons one by one at a steady rate. Initially, the electron is fired from the gun and is detected at the screen at a specific spot. However, as time passes one would observe a peculiar pattern of electron accumulation on the screen. This pattern closely resembles the interference pattern (shown in part 1) which is observed when light is used in the experiment instead of electrons. Hence, we conclude the electrons behave as waves. But let us take a deeper look into what exactly happens. If we consider the moment when a single electron is fired from the gun, it goes towards the plate with two narrow slits. Now, here comes the weird part. We do not know exactly which slit the electron goes through. There is no way to find it out without actually observing it. Let us neglect this part which is unknown to us. We do know, that after that, the electron would arrive at our phosphor screen and would be detected at a specific point. We do know this because we are actually observing the electron. Therefore, the only moment at which we know the exact position of the electron is when we observe it at the screen. The path in between is totally unknown and when something is unknown to us we assign it a "probability". Hence, we have a 50-50 % percent probability of the electron going either through the first slit or the second. Yet, we observe an interference pattern at the screen, which could only be produced by waves. Hence, it can be concluded that the electron actually travels as a wave when it is fired from the gun. Since its a wave, it can travel through both slits at the same time and interfere with itself. It then picks up a random spot on the screen, which we cannot determine but only predict with a probability and it gets detected there. Where is the paradox in all this? - Suppose for example, I want to know through which slit the electron went. I can do this by putting up an electron detector at both slits, which might be a light source that could scatter light off the electron whenever it went past it. This way, I would be able to tell whether the electron went through first slit or the second. When this experiment is performed, one notices no interference pattern at all on the screen. That is right : No wave-like behavior at all. These particles are too shy and deny to act like waves when one is observing them. Henceforth, the wavefunction which originally determined the wave behavior of electrons is of no use. The electrons travel like ordinary particles and produce a pattern as shown below.
This means that observations performed on the electron at the slit not only forces it to behave like particles but it also destroys all the information that the electron's wavefunction carried. But, another principle of Quantum Physics states that quantum information can never be destroyed. This here is the Quantum Information Eraser Paradox, and it demands a solution to the "Measurement Barrier" to eradicate this paradox. There are many other theories out there which provide an alternative explanation for the quantum observations. Theories like - "Pilot Wave Theory", "Many World's Theory", "Hidden Variable Theories" etc. are some of the examples of different interpretations of Quantum Physics. I would not go into the detail of each. The Pilot Wave Theory explains that a particle has a "guiding wave" associated with it. The particle rides on this wave and thus behaves accordingly. Hence the name - "Pilot Wave Theory".
(Above image is of a phenomenon in fluid mechanics called as Walking droplets. These are also identified as Hydrodynamic quantum analogs, because they are analogous to the main idea of Pilot Wave Theory. The droplet in the above image moves along with the ripple or wave below it. Same is the scenario in Pilot Wave Theory, where the particle like that droplet moves according to the guiding wave associated with it. This image is an excellent visual analogy of the theory. I suggest watching this amazing video by Veritasium explaining the phenomenon - Is This What Quantum Mechanics Looks Like? )
Yet, another interpretation - "The Many World's Interpretation" by Hugh Everett suggests that the wavefunction of particles is real and it does not collapse at all. Instead all the possible alternatives happen in a different parallel universe or world. For example, if I toss a coin and get heads, it might have given the alternative result i.e. "tails" in a different Universe. There is another theory called as the "Hidden Variable Theory" which states that besides the position and momentum, a particle should have another hidden variable(s), which when known can give us the complete state of the particle without any ambiguity. Thus eliminating the uncertainty associated in Quantum Physics. As intuitive and logical these theories might sound, one can provide no subtle reason to disregard one and believe in the other. Even the "Statistical Interpretation" itself is subject to many imperfections and need to be rectified. Thus, the field of Quantum Physics is yet tender and shall definitely undergo much development in the future. Nevertheless, it still finds numerous applications in Quantum Computing, Integrated Circuits, Semiconductors, Electronics and much more. The subject in itself is a vast ocean and what I described in these two articles are only drops of such ocean.
- Thank You
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