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Supplement to Science Club reading for September 18th Hello Science Clubbers! Those of you who have already done your reading (“Is Quantum Mechanics Falsifiable? A computational perspective on the foundations of Quantum Mechanics,”) may have noticed that quantum computing is rather tough skating. Therefore, I have compiled a quantum mechanics “cheat sheet” to give a little background on those aspects of QM that are most apropos to the article we are discussing with Josh Grochow next week. The most crucial concept is that of “superposition,” which is defined in the cheat sheet. Due to the difficulty of this topic, Josh will give a short talk before taking questions. He will give a very brief overview of QM, focusing on superposition, then describe the difference between a classical computer bit and a quantum bit, and then dig into the article. The Big Idea of the article is that the scientific method cannot be used to test quantum mechanics. The authors propose an “interrogative method” in its place. I look forward to the superposition of our mutual states on Thursday! Laura Steward Laura’s Quantum Mechanics Cheatsheet -­‐ most of this material is excerpted from Wikipedia-­‐ Quantum mechanics is the science of the very small: the body of scientific principles that explains the behavior of matter and its interactions with energy on the scale of atoms and subatomic particles. Classical physics explains matter and energy on a scale familiar to human experience, including the behavior of astronomical bodies. It remains the key to measurement for much of modern science and technology. However, toward the end of the 19th century, scientists discovered phenomena in both the large (macro) and the small (micro) worlds that classical physics could not explain. Coming to terms with these limitations led to two major revolutions in physics – one being the theory of relativity, the other being the development of quantum mechanics. The word "quantum" in this sense means the minimum amount of any physical entity involved in an interaction. Some aspects of quantum mechanics can seem counterintuitive or even paradoxical, because they describe behavior quite different from that seen at larger length scales. In the words of Richard Feynman, quantum mechanics deals with "nature as She is – absurd." For example, the uncertainty principle of quantum mechanics means that the more closely one pins down one measurement (such as the position of a particle), the less precise another measurement pertaining to the same particle (such as its momentum) must become. In 1905, Albert Einstein suggested that not just matter but also energy could be “quantized”: the energy in a beam of light occurs in individual packets, which are now called photons. Just as light has both wave-­‐like and particle-­‐like properties, matter has wave-­‐like and particle-­‐like properties. Matter behaving as a wave was first demonstrated experimentally for electrons: A beam of electrons can exhibit diffraction, just like a beam of light or a water wave. Similar wave-­‐like phenomena were later shown for atoms and even small molecules. The concept of wave/particle duality says that neither the classical concept of "particle" nor of "wave" can fully describe the behavior of quantum-­‐scale objects, either photons or matter. Wave/particle duality is an example of the principle of complementarity in quantum physics. The quantum particle acts as a wave when passing through double slits, but as a particle when it is detected. This is a typical feature of quantum complementarity: a quantum particle will act as a wave when we do an experiment to measure its wave-­‐like properties, and like a particle when we do an experiment to measure its particle-­‐like properties. Where on the detector screen any individual particle shows up will be the 1
result of an entirely random process. However, the distribution pattern of many individual particles will mimic the diffraction pattern produced by waves. Quantum entanglement The Pauli exclusion principle says that two electrons in one system cannot be in the same state. Nature leaves open the possibility, however, that two electrons can have both states "superimposed" over each of them. Quantum superposition is a fundamental principle of quantum mechanics that holds that a physical system—such as an electron—exists partly in all its particular theoretically possible states (or, configuration of its properties) simultaneously; but when measured or observed, it gives a result corresponding to only one of the possible configurations. The principle of quantum superposition states that if a physical system may be in one of many configurations—arrangements of particles or fields—then the most general state is a combination of all of these possibilities, where the amount in each configuration is specified by a complex number. The principle was described by Paul Dirac as follows: The general principle of superposition of quantum mechanics applies to the states [that are theoretically possible without mutual interference or contradiction] ... of any one dynamical system. It requires us to assume that between these states there exist peculiar relationships such that whenever the system is definitely in one state we can consider it as being partly in each of two or more other states. The original state must be regarded as the result of a kind of superposition of the two or more new states, in a way that cannot be conceived on classical ideas. Any state may be considered as the result of a superposition of two or more other states, and indeed in an infinite number of ways. Conversely any two or more states may be superposed to give a new state... The non-­‐classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B, such that there exists an observation which, when made on the system in state A, is certain to lead to one particular result, a say, and when made on the system in state B is certain to lead to some different result, b say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes a and sometimes b, according to a probability law depending on the relative weights of A and B in the superposition process. It will never be different from both a and b [i.e, either a or b]. The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result 2
itself being intermediate between the corresponding results for the original states. Bits and Q (for Quantum) Bits In a classical system, a bit would have to be in one state or the other, but quantum mechanics allows the qubit to be in a superposition of both states at the same time, a property which is fundamental to quantum computing. The bit is the basic unit of information. It is used to represent information by computers. Regardless of its physical realization, a bit has two possible states, 0 and 1. An analogy to this is a light switch—with the off position representing 0 and the on position representing 1. A qubit has a few similarities to a classical bit, but is overall very different. There are two possible outcomes for the measurement of a qubit—usually 0 and 1, like a bit. The difference is that whereas the state of a bit is either 0 or 1, the state of a qubit can also be a superposition of both 3