Complex conjugates are pairs of two-component numbers used in mathematics to describe physical realities. They have a real number component and an imaginary component, with opposite signs. Multiplying two complex conjugates produces a real result, which is important in quantum mechanics. Wave equations often have an imaginary component, but by multiplying them by their complex conjugate, a physically interpretable “probability density” is obtained. This probability density is used in the discrete spectral emission of radiation from atoms and is called the “probability of birth”, after Max Born, who was awarded the Nobel Prize in Physics in 1954 for his work in this area.
In mathematics, a complex conjugate is a pair of two-component numbers called complex numbers. Each of these complex numbers has a real number component added to an imaginary component. Although their value is equal, the sign of one of the imaginary components in the pair of conjugate complex numbers is opposite to the sign of the other. Despite having imaginary components, complex conjugates are used to describe physical realities. Using complex conjugates works despite the presence of imaginary components, because when the two components are multiplied together, the result is a real number.
Imaginary numbers are defined as all numbers that when squared result in a negative real number. This can be rephrased in other terms for simplification. An imaginary number is any real number multiplied by the square root of negative one (-1) — itself unintelligible. In this form, a complex conjugate is a pair of numbers that can be written, y=a+bi and y=a–bi, where “i” is the square root of -1. Formalistically, to distinguish the two y values, one is generally written with a slash above the letter, , although an asterisk is occasionally used.
Proving that multiplying two complex conjugate numbers produces a real result, consider an example, y=7+2i and ӯ=7–2i. Multiplying these two gives yӯ=49+14i–14i–4i2=49+4=53. Such a real result from complex conjugate multiplication is important, particularly when considering systems at the atomic and subatomic levels. Often mathematical expressions for small physical systems include an imaginary component. The discipline in which this is particularly important is quantum mechanics, the non-classical physics of the very small.
In quantum mechanics, the characteristics of a physical system consisting of a particle are described by a wave equation. Everything that needs to be learned about the particle in its system can be revealed by these equations. Wave equations often have an imaginary component. By multiplying the equation by its complex conjugate, a physically interpretable “probability density” is obtained. The characteristics of the particle can be determined by mathematically manipulating this probability density.
As an example, the use of probability density is important in the discrete spectral emission of radiation from atoms. This application of probability density is called the “probability of birth”, after the German physicist Max Born. The important, closely related statistical interpretation that measuring a quantum system will give certain specific results is called Born’s rule. Max Born was awarded the Nobel Prize in Physics in 1954 for his work in this area. Unfortunately, attempts to derive Born’s rule from other mathematical derivations have had mixed results.
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