In the early days of quantum theory, P. A. M. (Paul Adrian Maurice) Dirac created a powerful and concise formalism for it which is now referred to as Dirac notation or bra-ket (bracket ) notation. Two major mathematical traditions emerged in quantum mechanics: Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics. These distinctly different computational approaches to quantum theory are formally equivalent, each with its particular strengths in certain applications. Heisenberg’s variation, as its name suggests, is based matrix and vector algebra, while Schrödinger’s approach requires integral and differential calculus. Dirac’s notation can be used in a first step in which the quantum mechanical calculation is described or set up. After this is done, one chooses either matrix or wave mechanics to complete the calculation, depending on which method is computationally the most expedient.
Kets, Bras, and Bra-Ket Pairs
In Dirac’s notation what is known is put in a ket, So, for example, expresses the fact that a particle has momentum p. It could also be more explicit: , the particle has momentum equal to 2; , the particle has position 1.23. represents a system in the state ψ and is therefore called the state vector. The ket can also be interpreted as the initial state in some transition or event.
The bra represents the final state or the language in which you wish to express the content of the ket For example, is the probability amplitude that a particle in state ψ will be found at position x = .25. In conventional notation we write this as ψ(x=.25), the value of the function ψ at x = .25. The absolute square of the probability amplitude,
, is the probability density that a particle in state ψ will be found at x = .25. Thus, we see that a bra-ket pair can represent an event, the result of an experiment. In quantum mechanics an experiment consists of two sequential observations - one that establishes the initial state (ket) and one that establishes the final state (bra).
If we write we are expressing ψ in coordinate space without being explicit about the actual value of x. is a number, but the more general expression is a mathematical function, a mathematical function of x, or we could say a mathematical algorithm for generating all possible values of , the probability amplitude that a system in state has position x. For the ground state of the well-known particle-in-a-box of unit dimension
However, if we wish to express ψ in momentum space we would write,
How one finds this latter expression will be discussed later. The major point here is that there is more than one language in which to express
The most common language for chemists is coordinate space (x, y, and z, or r, θ, and φ, etc.), but we shall see that momentum space offers an equally important view of the state function. It is important to recognize that are formally equivalent and contain the same physical information about the state of the system. One of the tenets of quantum mechanics is that if you know you know everything there is to know about the system, and if, in particular, you know you can calculate all of the properties of the system and transform , if you wish, into any other appropriate language such as momentum space. A bra-ket pair can also be thought of as a vector projection - the projection of the content of the ket onto the content of the bra, or the “shadow” the ket casts on the bra. For example,
is the projection of the state ψ onto the state . It is the amplitude (probability amplitude) that a system in state ψ will be subsequently found in state . It is also what we have come to call an overlap integral. The state vector can be a complex function (that is have the form, a + ib, or exp(-ipx), for example, where Given the relation of amplitudes to probabilities mentioned above, it is necessary that , the projection of ψ onto itself is real. This requires that is the complex conjugate of then , a real number.
Ket-Bra Products - Projection Operators
Having examined kets , bras , and bra-ket pairs , it is now appropriate to study projection operators which are ket-bra products Take the specific example of operating on the state vector This operation reveals the contribution of or the length of the shadow that casts on
We are all familiar with the simple two-dimensional vector space in which an arbitrary vector can be expressed as a linear combination of the unit vectors (basis vectors, basis states, etc) in the mutually orthogonal x- and y-directions. We label these basis vectors For the two-dimensional case the projection operator which tells how
contribute to an arbitrary vector In other words, . This means, of course, that is the identity operator: . This is also called the completeness condition and is true if span the space under consideration.
For discrete basis states the completeness condition is : For continuous basis states, such as position, the completeness condition is:
If is normalized (has unit length) then We can use Dirac’s notation to express this in coordinate space as follows.
In other words integration of over all values of x will yield 1 if is normalized. Note how the continuous completeness relation has been inserted in the bra-ket pair on the left. Any vertical bar | can be replaced by the discrete or continuous form of the completeness relation. The same procedure is followed in the evaluation of the overlap integral, , referred to earlier.
Now that a basis set has been chosen, the overlap integral can be evaluated in coordinate space by traditional mathematical methods.