What Is Quantum Mechanics?
What is the human body and the Earth, the Sun, the Universe made of? What if the history of the universe were squeezed into the period of one year? What are the coldest and the hottest objects in the universe? What is the electromagnetic spectrum? What is a planet?
So small in fact that no Quantum Mechanical effects should be noticeable at the macroscopic level, confirming that Newtonian Mechanics is perfectly acceptable for everyday applications as required by the Correspondence Principle. Conversely, small objects like electrons have wavelengths comparable to the microscopic atomic structures they encounter in solids. Thus a Quantum Mechanical description, which includes their wave-like aspects, is essential to their understanding.
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Hopefully the foregoing discussion provides a convincing enough argument to use Quantum Mechanical ideas when dealing with electrons in solids. Next we must address the question of how exactly one describes electrons in a wave-like manner The approach suggested by Schrodinger was to postulate a function which would vary in both time and space in a wave-like manner the so-called wavefunction and which would carry within it information about a particle or system.
The time-dependent Schrodinger equation allows us to deterministically predict the behaviour of the wavefunction over time, once we know its environment. The information concerning environment is in the form of the potential which would be experienced by the particle according to classical mechanics if you are unfamiliar with the classical concept of potential an explanation is available.
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Whenever we make a measurement on a Quantum system, the results are dictated by the wavefunction at the time at which the measurement is made. It turns out that for each possible quantity we might want to measure an observable there is a set of special wavefunctions known as eigenfunctions which will always return the same value an eigenvalue for the observable.
Even if the wavefunction happens not to be one of these eigenfunctions, it is always possible to think of it as a unique superposition of two or more of the eigenfunctions, e. If a measurement is made on such a state, then the following two things will happen: The wavefunction will suddenly change into one or other of the eigenfunctions making it up.
quantum mechanics | Definition, Development, & Equations | etavuleqyz.cf
This is known as the collapse of the wavefunction and the probability of the wavefunction collapsing into a particular eigenfunction depends on how much that eigenfunction contributed to the original superposition. More precisely, the probability that a given eigenfunction will be chosen is proportional to the square of the coefficient of that eigenfunction in the superposition, normalised so that the overall probability of collapse is unity i. The measurement will return the eigenvalue associated with the eigenfunction into which the wavefunction has collapsed.
Clearly therefore the measurement can only ever yield an eigenvalue even though the original state was not an eigenfunction , and it will do so with a probability determined by the composition of the original superposition. There are clearly only a limited number of discrete values which the observable can take. We say that the system is quantised which means essentially the same as discretised.
Once the wavefunction has collapsed into one particular eigenfunction it will stay in that state until it is perturbed by the outside world.
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