Bra-Ket Notation

FIELDS OF STUDY: Quantum Physics

ABSTRACT: The mathematics of quantum mechanics developed from the vectors and matrices of classical mechanics and the differential calculus and integration of wave mechanics. Bra-ket notation provides an independent, simple system of calculation.

PRINCIPAL TERMS

• dot product: the product of the lengths of two vectors and the cosine of the angle between them; also called the inner product or the scalar product.
• eigenfunction: any mathematical function for which the solution yields the original function multiplied by a constant (an eigenvalue).
• quantum mechanics: the branch of physics that deals with matter interactions on a subatomic scale, based on the concepts that energy is quantized, not continuous, and that elementary particles exhibit wavelike behavior.
• quantum state: the condition of a physical system as defined by its associated quantum attributes.
• spin: the intrinsic angular momentum of a subatomic particle.
• vector: a mathematical representation of a property whose value has both magnitude and direction.
• wave function: a function that describes the quantum state of a system and represents the probability of finding the system in a given state at a given time.

Vectors, Matrices, and Dirac Notation

Many physical quantities, including velocity and force, consist of two distinct components: magnitude and direction. Such quantities are called vectors. The counterpart of the vector is the scalar, which has magnitude but no direction. For example, the velocity of an object describes not just its speed—a scalar quantity that measures how fast the object is moving—but also the direction of travel. A car driving north at sixty miles per hour has a different velocity from a car driving west at sixty miles per hour, even though their speeds are the same.

In plane, or Euclidean, geometry, it is easy to see how vectors function. A classic example is a boat crossing a flowing river. If the boat travels directly across the river under the force of its motor, it will reach the opposite shore somewhere downstream of its initial target due to the force of the current. The point at which the boat will touch the shore is determined by the overall vector of the boat’s motion. This vector is the sum of the two force vectors (the motor and the current) acting on it. In such a case, the two vectors function relative to a common origin point, and the resultant vector is obtained by adding them together.

Vectors can also be multiplied. The result of vector multiplication is called the dot product, so named because of the notation used. If two vectors a and b are multiplied together, the operation is written as a ∙ b. The dot product is a scalar quantity. It is equal to the quantity abcosθ, where θ is the angle between the two vectors a and b.

A series of vectors and operations can be represented as a matrix rather than as equations. This allows them to be more easily manipulated. Such manipulations are best performed using bra-ket notation, also called Dirac notation, after its developer, English physicist Paul Dirac (1902–84). In the simplest form of bra-ket notation, the operator of the equations is the "bra," and the vectors being operated on are the "ket."

Bra-ket notation is a specialized form of mathematical notation that separates the common operator from the vectors being operated on. In quantum mechanics, it is used to describe the quantum state of a physical system. The ket, written in the format | ket>, represents the quantum state in an abstract vector space. The bra, written in the format <bra |, represents a function that translates the abstract state into specific conditions. A bra-ket pair is written as <bra | ket>, representing the dot product of the bra and the ket.

Subatomic Particles and Wave Mechanics

Simple vector relationships are readily solved without bra-ket notation. However, the vectors describing the behavior of subatomic particles are much more complex. The basic premise of quantum mechanics is that energy is absorbed and emitted in discrete units, called "quanta" (the plural of "quantum"). Quantum mechanics also asserts that subatomic particles behave as both particles and waves, a concept known as wave-particle duality. The behavior of these particles is determined by their quantum states, described by specific quantum numbers. One such quantum number is spin, which represents the angular momentum of the particle. Different types of particles have different spins.

Subatomic particles are modeled using wave mechanics. In effect, they are treated as "matter waves." These waves are described by the Schrödinger equation, devised by Austrian physicist Erwin Schrödinger (1887–1961). Solutions to the Schrödinger equation are called wave functions, represented by the Greek symbol psi, Ψ. A wave function can be derived from the Schrödinger equation that will describe the behavior of any given particle, including its spin. This function is termed an eigenfunction if the Hamiltonian operator (H) acting on it is equal to the energy operator (E) of the system. This is normally written as

HΨ = EΨ

which is the time-independent form of the Schrödinger equation. The energy operator corresponds to the total energy of the system. The Hamiltonian operator is a function representing the sum of the system’s kinetic energy operator and its potential energy operator. (If the equation is time-dependent, the Hamiltonian operator shows how the energy of the system changes over time.) Thus, E represents the eigenvalues of the function H, such that for n = 1, 2, 3 . . . ,

HΨn = EnΨn

Bra-ket notation is used to simplify the calculations carried out on the wave function. Dirac developed this system in order to reconcile the different approaches to quantum theory taken by Schrödinger and German physicist Werner Heisenberg (1901–76). Schrödinger’s approach relied on wave mechanics, Heisenberg’s on vector and matrix calculations. Bra-ket notation simplifies the expression and solving of these complex mathematical relationships, allowing them to be addressed apart from any preexisting coordinate system.

Continuing with Bra-Ket Notation

Bra-ket notation is particularly valuable in quantum physics, as it allows very large vector statements to be manipulated in compact form. However, it is also useful in other fields, particularly those incorporating concepts from quantum mechanics. Arguably, any theoretical studies that involve both vector or matrix calculation and differential calculus can benefit from the use of bra-ket notation.

Bibliography

Beddard, Godfrey. Applying Maths in the Chemical & Biomolecular Sciences: An Example-Based Approach. New York: Oxford UP, 2009. Print.

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

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

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

Finn, John Michael. Classical Mechanics. 2008. Sudbury: Jones, 2010. Print.

Pereyra, Pedro. Fundamentals of Quantum Physics: Textbook for Students of Science and Engineering. Berlin: Springer, 2012. Print.