History UsefulNotes / QuantumPhysics

Enter young physicist Max Planck[[note]]The 'a's are like the 'a' in 'father', not in 'path' or 'day'.[[/note]]. He saw that their math was correct and their conclusion incorrect. He therefore realized that the problem lay with their assumptions. He took the bold step of assuming an entirely different kind of electromagnetic radiation, one which could only exist in quanta, that is to say in discrete packets of energy rather than continuous waves. For quite some time, Planck believed that the quantization of energy was purely a formal approach, much like the electric field, rather than an actual physical phenomenon. It was only later that the world saw this as the first step in the development of Quantum Physics.\\

->''"The theory of quantum electrodynamics describes Nature as absurd from the point of view of common sense. And it agrees fully with experiment. So I hope you accept Nature as She is -- absurd."''

Think about an oncoming sound wave. You can record its audio frequency spectrum (momentum) at a certain point in time (position). However, you cannot have perfect accuracy on both quantities at the exact same instant. To measure a wave's frequencies accurately, you would need to do so over a range of time since measuring frequency requires oscillation. Because of that, you cannot match frequency with the exact position in time, only with a range of possibilities. Of course, you could always tighten that range so that there's less uncertainty with your position in time, but then you would only get back extremely low-amplitude high frequency results from that measurement, since there's less oscillation to be had.\\

Think about a sound wave. Its attributes include amplitude (the height of the wave) and frequency (how often it goes up and down). How would you measure these quantities? You can take an infinitesimal slice at a precise moment of time to measure its amplitude to a very high degree of accuracy, but then you'd lose ''all'' information about its frequency — after all a single point of data has no (or rather, undefined) oscillation. To measure frequency, you'd have to take a broader slice of the wave with its peaks and troughs. In doing so, you start to lose information on when the measurement happens and the amplitude of the wave.\\

For a more intuitive perspective, think about a sound wave. Its attributes include amplitude (the height of the wave) and frequency (how often it goes up and down). How would you measure these quantities? You can take an infinitesimal slice at a precise moment of time to measure its amplitude to a very high degree of accuracy, but then you'd lose ''all'' information about its frequency — after all a single point of data has no (or rather, undefined) oscillation. To measure frequency, you'd have to take a broader slice of the wave with its peaks and troughs. In doing so, you start to lose information on when the measurement happens and the amplitude of the wave.\\

For a more intuitive perspective, think about a sound wave. Its attributes include amplitude (the height of the wave) and frequency (how often it goes up and down). How would you measure these quantities? You can take an infinitesimal slice at a precise moment of time to measure its amplitude to a very high degree of accuracy, but then you'd lose ''all'' information about its frequency — after all a single point of data has no (or rather, undefined) oscillation. To measure frequency, you'd have to take a broader slice of the wave with all of its peaks and troughs. In doing so, you start to lose information on when the measurement happens and the amplitude of the wave.\\

That was the era of the ''Wunderkinder'' -- "child prodigies" -- in which brilliant young men of twenty-five to thirty-five years of age took the stage. The names of Heisenberg (no, not [[Series/BreakingBad that]] Heisenberg), Schrödinger, Feynman, Dirac, Freeman, and many others impressed themselves on the popular consciousness, and particularly on the scientific establishment. It all started with Max Planck in a ''succesful'' attempt to explain mathematically the mechanism of black-body radiation and strengthened by UsefulNotes/AlbertEinstein when explaining the photoelectric effect in 1905, his ''annus mirabilis'' (Miracle Year), and the world never looked back.

To solve the mathematics, Planck was forced to introduce what is now known as the Planck constant, ħ, with units of joule × seconds. This was one of the tools used by Einstein in his ''annus mirabilis'' of 1905.

Einstein saw the Planck constant and realized that Planck's understanding went beyond mere mathematical formality, but that the quantization of E=ħν held a physical reality. The energy of each photon, or particle, of light was directly equal to the product of its frequency (ν, actually the Greek letter 'nu') and Planck's constant. Thus the light striking the metal consists of many many photons, each with some chance to be absorbed by the electrons of the metal. These electrons are held by the metal with some energy Φ, and the photon has to have more energy than that to liberate it. Thus the minimum frequency is related to Planck's constant. Further, the amplitude of light is related to the energy because more amplitude means more photons to carry energy, which explains why more electrons are kicked off by a brighter light.\\

Even if we had perfect measurement techniques that had no effect on what we measured it would still be impossible to say the position and momentum of a particle at the same time. The two properties are fundamentally related in such a way that they cannot both have precise values at the same time. How is this possible? From a technical perspective is simply because they are related by the Fourier transform.

* [[http://en.wikipedia.org/wiki/Wave_function The Wave Function]]: Einstein famously said that if you couldn't describe relativity or quantum theory without resorting to mathematics then you didn't understand either. When a layman asked him to explain relativity without math one summer, Einstein repeatedly found himself stymied; his own understanding of the science was so rooted in his ability to think and communicate through mathematics that he couldn't communicate it otherwise. Although it's possible to relate the concepts of modern physics without math, to truly understand and work with them, a powerful understanding of mathematics is required. For quantum physics, visualizing states in terms of comparatively more intuitive probabilities means working with the wave functions developed by Erwin Schrödinger.\\

The wave function is a probability amplitude describing the states of a system as a function of space and time. That is, if a system is known perfectly, then you can map the system at all points in space and then follow it in time. Or vice versa. First, a probability amplitude isn't the same as a probability. A wave function has both real and imaginary parts. What happens is that you take the function multiplied by its complex conjugate to find the probability of finding a particle somewhere (for its position-space wave function) or of its momentum (for its momentum-space wave function)[[note]]Remember uncertainty, you can measure either momentum or position, but not both.[[/note]]. If a wave function were (a+bi), then its complex conjugate would be (a−bi), and their product would be (a+bi)×(a−bi) = a[-[[superscript:2]]-]+b[-[[superscript:2]]-]. \\

Wave functions are undeniably useful, but very difficult to work with, because they require taking multiple derivatives across multiple dimensions and finding complex conjugates and, worse, taking integrals across multiple dimensions. For a simple system, like a hydrogen atom, this is manageable, but the functions quickly degenerate into painful headaches. For this reason, Schrödinger's equations have been largely replaced by Heisenberg's matrices. Werner Heisenberg developed a different mathematical formulation for quantum mechanics roughly contemporaneously with Schrödinger using matrix mathematics rather than functions and integrals. At first, everyone understood Schrödinger's math, as difficult as it was to work with, and no one understood Heisenberg's matrices. Over time, though, Heisenberg's formulation has become the standard for multiple reasons, not least of which that they're much easier to work with and, once you can read a matrix, give information in a much more straightforward fashion.\\

One of the problems in working with quantum physics is the lack of commutation. Commutation is the rule of arithmetic that tells us that A × B = B × A. For quantum physics, this isn't always the case. For some properties, such as momentum and position, A × B ≠ B × A. This is the basis/result of uncertainty. AB − BA ≠ 0. For Schrödinger's wave equations, this lack of commutation was dealt with using complicated operators involving derivatives and the like, which took time to work through and, particularly for complicated systems, left many opportunities for human error. For Heisenberg's matrices, non-commutation is built right in. Matrix math is a class unto itself, but suffice it to say that the matrix equivalent of multiplication is not a commutative property and is much simpler than working in multi-variable calculus. Where the wave equations require you to do multivariable calculus, the matrix requires you to add, subtract, multiply, and divide. You'll still fill up a few pages with funny looking math, but it's much easier math.

Unlike the double slit experiment, Schrödinger's Cat isn't an actual experiment. Rather, it's satirical derision. However, many people don't realize this and assume it's some sort of actual experiment, clever notion or even a {{Koan}}. Schrödinger, like others, wasn't fond of the probabilistic theories being developed by Heisenberg and Bohr, outlined in the EPR paper of 1935. This was the Copenhagen Interpretation of Quantum theory (below).\\

The basic thought experiment is that within a box is a cat. There is also an atom of a radioactive isotope that may or may not decay. Upon decay, it will trigger a Geiger counter, that will then trigger a vial of poison that will kill the cat. According to Schrödinger, the superposition of states means that until the box is open and the system observed, the cat is simultaneously alive and dead. Schrödinger never intended this as an actual result of quantum theory, but as a ''reductio ad absurdum'' to demonstrate what was to him the ridiculousness of the Copenhagen interpretation.\\

# A system is completely described by a wave function ψ, representing an observer's subjective knowledge of the system.

The Cat Question: Schrödinger's paradox states that the cat is both alive and dead. The Copenhagen interpretation is that the Geiger counter, not the scientist, acts as the observer and collapses the wave function forcing the atom to a particle state of decayed or not decayed (not both), meaning the poison vial is either broken or not (not both), and the cat is either alive or dead (not both). A particle level superposition can't be used to set up a macroscopic superposition because that requires actively observing the particle, and observing the particle collapses the wave function. Douglas Adams later noted that the cat also acts as an observer. Creator/TerryPratchett noted that there are three states: Alive, Dead, and Bloody Furious.

These days the thought experiment is used to analyze and interpret the various schools of thought, showing their strengths, weaknesses, and potential implications. No one ever actually tries to poison a cat. Scientists aren't dicks.