Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev

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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 Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev) 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,Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev  So, for example, Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev expresses the fact that a particle has momentum p. It could also be more explicit: Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev, the particle has momentum equal to 2;  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev , the particle has position 1.23. Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev 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 braDirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev represents the final state or the language in which you wish to express the content of the ket  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev For example,  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRevis 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,

Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev, 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 writeDirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev we are expressing ψ in coordinate space without being explicit about the actual value of x.Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev is a number, but the more general expression Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev is a mathematical function, a mathematical function of x, or we could say a mathematical algorithm for generating all possible values of Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev , the probability amplitude that a system in state Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev has position x.  For the ground state of the well-known particle-in-a-box of unit dimension

Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev However, if we wish to express ψ in momentum space we would write,  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev

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 expressDirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev

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 thatDirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev 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    Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev     you know everything there is to know about the system, and if, in particular, you know  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev you can calculate all of the properties of the system and transform Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev, 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,

Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev is the projection of the state ψ onto the state Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev. It is the amplitude (probability amplitude) that a system in state ψ will be subsequently found in state Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev.  It is also what we have come to call an overlap integral. The state vector Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev can be a complex function (that is have the form, a + ib, or exp(-ipx), for example, where  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev  Given the relation of amplitudes to probabilities mentioned above, it is necessary that  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev , the projection of ψ onto itself is real. This requires that  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRevis the complex conjugate of  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev thenDirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev , a real number.

 


Ket-Bra Products - Projection Operators
Having examined kets  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev  , bras  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev , and bra-ket pairs  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev , it is now appropriate to study projection operators which are ket-bra products  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev Take the specific example of  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev operating on the state vector  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev  This operation reveals the contribution of  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev  or the length of the shadow that  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev casts on  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev

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  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev For the two-dimensional case the projection operator which tells how

Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev contribute to an arbitrary vector  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev  In other words,  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev .  This means, of course, that  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRevis the identity operator:  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev . This is also called the completeness condition and is true if   Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev span the space under consideration.
For discrete basis states the completeness condition is :  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev For continuous basis states, such as position, the completeness condition is:  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev 

If  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev is normalized (has unit length) then  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev We can use Dirac’s notation to express this in coordinate space as follows.

Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev

In other words integration of  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev over all values of x will yield 1 if  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev  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,  Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev  , referred to earlier.

Dirac Notation for State Vectors - General Formalism of Wave Mechanics, Quantum Mechanics, CSIR-NET Physics Notes | EduRev

Now that a basis set has been chosen, the overlap integral can be evaluated in coordinate space by traditional mathematical methods.

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