4.7: Dirac Notation
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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 \( \langle \, | \, \rangle\)) notation. Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics.
Kets
In Dirac’s notation what is known is put in a ket, \(| \, \rangle\). So, for example, \(| p \rangle\) expresses the fact that a particle has momentum \(p\). It could also be more explicit: \(| p=2 \rangle\), the particle has momentum equal to 2; \(| x=1.23 \rangle\), the particle has position 1.23. \(| \Psi \rangle\) represents a system in the state \( \Psi \) and is therefore called the state vector. The ket can also be interpreted as the initial state in some transition or event.
Bras
The bra \(\langle \, | \) represents the final state or the language in which you wish to express the content of the ket \(| \, \rangle\). For example,\( \langle 0.25 | \Psi \rangle\) is the probability amplitude that a particle in state \( \Psi\) will be found at position \(x = 0.25\). In conventional notation we write this as \( \Psi(x=0.25) \), the value of the function \( \Psi \) at \(x\)=0.25. The absolute square of the probability amplitude, \( \left| \langle x=0.25| \Psi \rangle \right|^2\), is the probability density that a particle in state \( \Psi \) will be found at \(x\) = 0.25. 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).
Bra-Ket Pairs (dot products)
It is so called because the inner product (or dot product) of two states is denoted by a bracket, \(\langle Φ|Ψ \rangle\), consisting of a left part, \(\langle Φ|\), (the bra), and a right part, \(|Ψ\rangle\), (the ket). The notation was introduced in 1939 by Paul Dirac,[1] and is also known as Dirac notation. Bra-ket notation is widespread in quantum mechanics: almost every phenomenon that is explained using quantum mechanics—including a large proportion of modern physics—is usually explained with the help of bra-ket notation.
If we write \( \langle x| \Psi \rangle \), we are expressing \( \Psi \) in coordinate space without being explicit about the actual value of \(x\). \(\langle 0.25 | \Psi \rangle \) is a number, but the more general expression \(\rangle x | \Psi \rangle \) is a mathematical function, a mathematical function of \(x\), or we could say a mathematical algorithm for generating all possible values of \( \langle x| \Psi \rangle \), the probability amplitude that a system in state \( | \Psi \rangle \) has position \(x\).
Example
For the ground state of the well-known particle-in-a-box of unit dimension.
\[\langle x | \Psi \rangle = \Psi(x) \ 2^{1/2} \sin (\pi x) \]
However, if we wish to express \( \Psi \) in momentum space we would write
\[ \langle p | \Psi \rangle = \Psi(p) = 2^{1/2} \dfrac{e^{-ip} +1}{\pi^2 - p^2} \]
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 \( | \Psi \rangle \). The most common language for chemists is coordinate space (\(x\), \(y\), and \(z\), or \(r\), \(\theta\), and \(\phi\), etc.), but we shall see that momentum space offers an equally important view of the state function. It is important to recognize that \( \langle x| \Psi \rangle \) and \( \langle p| \Psi \rangle \) 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 \( | \Psi \rangle \), you know everything there is to know about the system, and if, in particular, you know \( \langle x| \Psi \rangle \), you can calculate all of the properties of the system and transform \( \langle x| \Psi \rangle \), 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 (i.e., a dot product) - 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, \( \langle \Phi | \Psi \rangle \) is the projection of state \(\Psi\) onto state \(\Phi\). It is the amplitude (probability amplitude) that a system in state \( | \Psi \rangle \) will be subsequently found in state \(| \Phi \rangle \). It is also what we have come to call an overlap integral.
The \( | \Psi \rangle \) state vector can be a complex function (that is have the form, \(a + ib\), or \(exp(-ipx)\), for example, where \(i = \sqrt{-1}\)). Given the relation of amplitudes to probabilities mentioned above, it is necessary that \( \langle \Psi | \Psi \rangle \), the projection of \( | \Psi \rangle \) onto itself is real. This requires that
\[ \langle \Psi | = | \Psi \rangle^*\]
where \( | \Psi \rangle^* \) is the complex conjugate of \( | \Psi \rangle \). So if \( | \Psi \rangle = a + ib \) then \( \langle \Psi | = a - ib \), which yields \( \langle \Psi | \Psi \rangle = a^2 + b^2\) , a real number.
References
- Chester, M. Primer of Quantum Mechanics ; Krieger Publishing Co.:Malabar, FL, 1992.
- Das, A.; Melissinos, A. C. Quantum Mechanics: A Modern Introduction; Gordon and Breach Science Publishers: New York, 1986.Feynman, R. P.; Leighton, R. B.; Sands, M. The Feynman Lectures on Physics, Vol.3 ;Addison-Wesley: Reading, 1965.
- Martin, J. L. Basic Quantum Mechanics ; Claredon Press, Oxford, 1981.