Introduction to eigenstates

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Because of the
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physi ...
, statements about both the position and momentum of particles can only assign a
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking ...
that the position or
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
will have some numerical value. The uncertainty principle also says that eliminating uncertainty about position maximises uncertainty about momentum, and eliminating uncertainty about momentum maximizes uncertainty about position. A
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
assigns probabilities to all possible values of position and momentum. Schrödinger's wave equation gives wavefunction solutions, the squares of which are probabilities of where the electron might be, just as Heisenberg's probability distribution does. In the everyday world, it is natural and intuitive to think of every object being in its own eigenstate. This is another way of saying that every object appears to have a definite position, a definite
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
, a definite measured value, and a definite time of occurrence. However, the uncertainty principle says that it is impossible to measure the exact value for the momentum of a particle like an
electron The electron ( or ) is a subatomic particle with a negative one elementary charge, elementary electric charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are generally thought t ...
, given that its position has been determined at a given instant. Likewise, it is impossible to determine the exact location of that particle once its momentum has been measured at a particular instant. Therefore, it became necessary to formulate clearly the difference between the state of something that is uncertain in the way just described, such as an electron in a probability cloud, and the state of something having a definite value. When an object can definitely be "pinned down" in some respect, it is said to possess an
eigenstate In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
. As stated above, when the wavefunction collapses because the position of an electron has been determined, the electron's state becomes an "eigenstate of position", meaning that its position has a known value, an
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of the eigenstate of position. The word "eigenstate" is derived from the German/Dutch word "eigen", meaning "inherent" or "characteristic". An eigenstate is the measured state of some object possessing quantifiable characteristics such as position, momentum, etc. The state being measured and described must be
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phys ...
(i.e. something such as position or momentum that can be experimentally measured either directly or indirectly), and must have a definite value, called an eigenvalue. ("Eigenvalue" also refers to a mathematical property of square matrices, a usage pioneered by the mathematician
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
in 1904. Some such matrices are called self-adjoint operators, and represent observables in quantum mechanics.)Quantum Mechanics: A graduate level course
20 Feb 2010