12.3 Dirac notation

12.3 Dirac notation

In the Dirac bra-ket formalism, position and momentum eigenstates will be denoted $|q⟩$ and $|k⟩$ respectively, with

$$Q|q⟩=q|q⟩,P|k⟩=k|k⟩$$

Arbitrary states $|\psi ⟩$ can be thought of as determined by coeficients

$$⟨q|\psi ⟩=\psi (q),⟨k|\psi ⟩=\tilde{\psi }(k)\text{\hspace{1em}(12.3)}$$

with respect to either the $|q⟩$ or |k⟩ basis. The use of the bra-ket formalism requires some care however since states like $|q⟩$ or $|k⟩$ are elements of $S^{\prime }(\mathbf{R})$ that do not correspond to any element of $\mathcal{S}(\mathbf{R})$ Given elements |$\psi$⟩ in $\mathcal{S}(\mathbf{R})$ , they can be paired with elements of $S^{\prime }(\mathbf{R})$ like $⟨q|$ and $⟨k|$ as in equation 12.3 to get numbers. When working with states like $|\grave{q^{\prime }}⟩$ and $|k^{\prime }⟩$ , one has to invoke and properly interpret distributional relations such as

$$⟨q|q^{\prime }⟩=\delta (q-q^{\prime }),⟨k|k^{\prime }⟩=\delta (k-k^{\prime })$$

Equation 12.1 is written in Dirac notation as

$$|\psi ⟩={\int }_{-\infty }^{\infty }|q⟩⟨q|\psi ⟩dq$$

and 12.2 as

$$|\psi ⟩={\int }_{-\infty }^{\infty }|k⟩⟨k|\psi ⟩dk$$

The resolution of the identity operator of equation 4.6 here is written

$$1={\int }_{-\infty }^{\infty }|q⟩⟨q|dq={\int }_{-\infty }^{\infty }|k⟩⟨k|dk$$

The transformation between the |q⟩ and |k⟩ bases is given by the Fourier transform, which in this notation is

$$⟨k|\psi ⟩={\int }_{-\infty }^{\infty }⟨k|q⟩⟨q|\psi ⟩dq$$

where

$$⟨k|q⟩=\frac{1}{\sqrt{2\pi }}{e}^{-ikq}$$

and the inverse Fourier transform

$$⟨q|\psi ⟩={\int }_{-\infty }^{\infty }⟨q|k⟩⟨k|\psi ⟩dk$$

where

$$⟨q|k⟩=\frac{1}{\sqrt{2\pi }}{e}^{ikq}$$

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