Eigenvectors

FIELDS OF STUDY: Quantum Physics

ABSTRACT: Eigenvectors are mathematical descriptors of a property that has both magnitude and direction. Certain mathematical operations acting on an eigenfunction yield the original function multiplied by a constant. This constant is the eigenvalue of the operation. Vectors describe position and direction using the Cartesian coordinate system. Bra-ket notation is used to simplify complex vector and wave-function equations.

PRINCIPAL TERMS

• bra-ket notation: a system of mathematical notation developed to more easily manipulate very large vector equations.
• Cartesian coordinate system: a system that uses a pair or trio of numbers to indicate the location of a point in two-dimensional or three-dimensional space relative to a set point of origin.
• eigensystem: the set of all eigenvectors of a matrix paired with their respective eigenvalues.
• eigenvalue: the mathematical constant that, when multiplied by an eigenfunction, yields the solution to the operation performed on the function.
• matrix: a mathematical notation in which a series of coordinates is written in an array and can then be manipulated according to set rules.
• vector: the mathematical statement of a property that has both magnitude and direction.

Eigen

The German word eigen means "own," in the sense of belonging. When used as a prefix in mathematics and physics, it signifies that the property it represents is unique and complete within itself. In the mathematics of quantum mechanics, an eigenfunction is a mathematical function consisting of operators acting on a vector property. The solution of an eigenfunction is equal to the original function multiplied by a constant, which is the eigenvalue of the function. For example, the derivative of the function f(x) = ceλx, where differentiation (finding the derivative) is represented by the notation f′(x),is λceλx, which can also be written as λf(x). Thus, the constant λ (which could also be a function) is an eigenvalue of the eigenfunction f(x) = ceλx, and f′(x) = λf(x).

The quantum state of a physical system, including the energy and position of the electrons in its atoms, is best described using the Schrödinger equation. This equation is a type of wave equation, similar to the equations that describe sound waves and electromagnetic waves, because elementary particles such as electrons behave like waves on a quantum scale. Solving the Schrödinger equation for a particular system yields a wave function, represented by the symbol Ψ, which is a much more complex function than the one above. Mathematical manipulation of Ψ readily produces stable energy-state results that are eigenvalues of the eigenfunction Ψ. This can be used to describe and plot atomic orbitals.

Quantum Math

The energy state and position of an electron in an atom can be described mathematically in one of two ways. One approach is based on the mathematics of classical mechanics, using vectors and matrices. The other, based on wave mechanics, relies on differential calculus. Both express the same properties and quantities and must therefore produce the same results for specific conditions. Vector calculations use the Cartesian coordinate system (x, y, and z axes), developed by French mathematician René Descartes (1596–1650), for position. They describe changes in position and energy as matrix transformations. Another coordinate system, known as spherical coordinates, is often used in wave mechanics. This system, based on the two-dimensional polar coordinate system, uses radial distance and two angles (r, θ, and φ) for position. It describes change of position and energy with differential equations and integrals using operators. An operator denotes the function or transformation that is to be carried out on the wave function. In both forms of calculation, the complex conjugates (a − bi and a + bi) of wave functions and vector expressions are included to normalize the equations of state for the system.

The calculations are simplified using bra-ket notation, devised by British physicist Paul Dirac (1902–84). In bra-ket notation, each "bra" represents one or more operations or matrix manipulations to be performed on the state vector or wave function represented by the "ket." The product of the two, known as the dot product or the inner product, is represented as follows:

<bra | ket>

The bra is most often the complex conjugate of the ket. Thus, the product of the wave function Ψ and its complex conjugate Ψ* is written as

<Ψ | Ψ>

because <Ψ | is equal to | Ψ>*, which is the complex conjugate of | Ψ>.

Bras and kets can be mixed and matched as needed throughout the calculation sequence. The bra-ket notation does not require specific coordinates, which makes it convenient for either method of calculation used in quantum mechanics.

Eigenvectors

Every vector property has magnitude and an associated direction. Think of the forces acting on a swing as an example. Gravity acts vertically, pulling the swing down toward the ground. A person pushing on the swing applies a horizontal force to it, resulting in its displacement from the neutral vertical position. Because the motion of a swing is that of a pendulum, it can be described using wave mechanics or vector mathematics. The position of the swing and its direction of motion follow a progression of vectors relative to the swing’s neutral position. These two properties can be described in terms of the distance of the swing from its point of suspension (r) and the angle of the displacement of r from the neutral position (θ). The cyclic variation of the swing’s motion is reflected in the cyclic variation of the vectors and angular coordinates that it follows.

For any such treatment, specific conditions exist for which the resultant vector is in the same direction as the original vector, though its magnitude may differ. For a given set of vectors and operators represented by the matrix A, there are one or more vectors x that, when multiplied by A, result in a vector equal to x multiplied by λ:

Ax = λx

The value λ is therefore an eigenvalue of A, and the vector x is an eigenvector. All possible eigenvalues of a matrix, when paired with their associated eigenvectors, form an eigensystem.

Bibliography

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Bowman, Gary E. Essential Quantum Mechanics. New York: Oxford UP, 2008. Print.

Dahl, Jens P. Introduction to the Quantum World of Atoms and Molecules. Hackensack: World Scientific, 2001. Print.

Dick, Rainer. Advanced Quantum Mechanics: Materials and Photons. New York: Springer, 2012. Print.

Finn, John Michael. Classical Mechanics. Sudbury: Jones, 2010. Print.

Pereyra, Pedro. Fundamentals of Quantum Physics. New York: Springer, 2012. Print.