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This web page location: Quantum TheoryWaves and ParticlesGreek letter psi, quantum system, mathematical expression, pair of dice, wave functionDeeper web pages: Quantum theory gives exact answers to many questions, but it can only give probabilities for some values. A probability is the likelihood of an answer being a certain value. Probability is often represented by a graph, with the highest point on the graph representing the most likely value and the lowest representing the least likely value. Scientists use a mathematical expression called a wave function to describe the characteristics of a particle that are related to time and space—such as position and velocity. The wave function helps determine the probability of these aspects being certain values. The wave function of a particle is not the same as the wave suggested by waveparticle duality. A wave function is strictly a mathematical way of expressing the characteristics of a particle. Physicists usually represent these types of wave functions with the Greek letter psi, The wave function described above does not depend on time. An isolated hydrogen atom does not change over time, so leaving time out of the atom’s wave function is acceptable. For particles in systems that change over time, physicists use wave functions that depend on time. This lets them calculate how the system and the particle’s properties change over time. Physicists represent a timedependent wave function with ?(t), where t represents time. The wave function for a single atom can only reveal the probability that an atom will have a certain set of characteristics at a certain time. Physicists call the set of characteristics describing an atom the state of the atom. The wave function cannot describe the actual state of the atom, just the probability that the atom will be in a certain state. The wave functions of individual particles can be added together to create a wave function for a system, so quantum theory allows physicists to examine many particles at once. The rules of probability state that probabilities and actual values match better and better as the number of particles (or dice thrown, or coins tossed, whatever the probability refers to) increases. Therefore, if physicists consider a large group of atoms, the wave function for the group of atoms provides information that is more definite and useful than that provided by the wave function of a single atom. An example of a wave function for a single atom is one that describes an atom that has absorbed some energy. The energy has boosted the atom’s electrons to a higher energy level, and the atom is said to be in an excited state. It can return to its normal ground state (or lowest energy state) by emitting energy in the form of a photon. Scientists call the wave function of the initial exited state ?i (with “i” indicating it is the initial state) and the wave function of the final ground state ?f (with “f” representing the final state). To describe the atom’s state over time, they multiply ?i by some function, a(t), that decreases with time, because the chances of the atom being in this excited state decrease with time. They multiply ?f by some function, b(t), that increases with time, because the chances of the atom being in this state increase with time. The physicist completing the calculation chooses a(t) and b(t) based on the characteristics of the system. At any time, the rules of probability state that the probability of finding a single atom in either state is found by multiplying the coefficient of its wave function (a(t) or b(t)) by itself. For one atom, this does not give a very satisfactory answer. Even though the physicist knows the probability of finding the atom in one state or the other, whether or not reality will match probability is still far from certain. This idea is similar to rolling a pair of dice. There is a 1 in 6 chance that the roll will add up to seven, which is the most likely sum. Each roll is random, however, and not connected to the rolls before it. It would not be surprising if ten rolls of the dice failed to yield a sum of seven. However, the actual number of times that seven appears matches probability better as the number of rolls increases. For one million or one billion rolls of the dice, one of every six rolls would almost certainly add up to seven. Similarly, for a large number of atoms, the probabilities become approximate percentages of atoms in each state, and these percentages become more accurate as the number of atoms observed increases. For example, if the square of a(t) at a certain time is 0.2, then in a very large sample of atoms, 20 percent (0.2) of the atoms will be in the initial state and 80 percent (0.8) will be in the final state. One of the most puzzling results of quantum mechanics is the effect of measurement on a quantum system. Before a scientist measures the characteristics of a particle, its characteristics do not have definite values. Instead, they are described by a wave function, which gives the characteristics only as fuzzy probabilities. In effect, the particle does not exist in an exact location until a scientist measures its position. Measuring the particle fixes its characteristics at specific values, effectively “collapsing” the particle’s wave function. The particle’s position is no longer fuzzy, so the wave function that describes it has one high, sharp peak of probability. In the 1930s physicists proposed an imaginary experiment to demonstrate how measurement causes complications in quantum mechanics. They imagined a system that contained two particles with opposite values of spin, a property of particles that is analogous to angular momentum. The physicists can know that the two particles have opposite spins by setting the total spin of the system to be zero. They can measure the total spin without directly measuring the spin of either particle. Because they have not yet measured the spin of either particle, the spins do not actually have defined values. They exist only as fuzzy probabilities. The spins only take on definite values when the scientists measure them. In this hypothetical experiment the scientists do not measure the spin of each particle right away. They send the two particles, called an entangled pair, off in opposite directions until they are far apart from each other. The scientists then measure the spin of one of the particles, fixing its value. Instantaneously, the spin of the other particle becomes known and fixed. It is no longer a fuzzy probability but must be the opposite of the other particle, so that their spins will add to zero. It is as though the first particle communicated with the second. This apparent instantaneous passing of information from one particle to the other would violate the rule that nothing, not even information, can travel faster than the speed of light. The two particles do not, however, communicate with each other. Physicists can instantaneously know the spin of the second particle because they set the total spin of the system to be zero at the beginning of the experiment. In 1997 Austrian researchers performed an experiment similar to the hypothetical experiment of the 1930s, confirming the effect of measurement on a quantum system. Article key phrases: 

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