Changes

<math>f_\psi\left(x\right) = \left\langle x,\psi\right\rangle, x\in\mathcal{H}</math>

<math>f_\psi\left(x\right) = \left\langle x,\psi\right\rangle, x\in\mathcal{H}</math>

In the Dirac Notation, the functional defined by <math>\psi</math> is instead denoted by:

In the Dirac Notation, the functional defined by <math>\psi</math> is instead represented by:

<math>\left\langle\psi\right| = f_\psi\left(x\right)</math>

<math>\left\langle\psi\right| \dot= f_\psi\left(x\right)</math>

And in this case, <math>\left\langle\psi\right|</math> is called a "bra".  When the bra is written next to a ket <math>\left|\varphi\right\rangle</math>, we understand the pair to form a "bracket" giving us an inner product:

And in this case, <math>\left\langle\psi\right|</math> is called a "bra".  When the bra is written next to a ket <math>\left|\varphi\right\rangle</math>, we understand the pair to form a "bracket" giving us an inner product:

<math>Q=\sum_{n}\lambda_n\left|n\right\rangle\left\langle n\right|</math>

<math>Q=\sum_{n}\lambda_n\left|n\right\rangle\left\langle n\right|</math>

The kets may also represent a [[continuous]] set of states.  In this, Dirac also found it necessary to develop what is known as the [[Dirac Delta Function]].  Then, for a continuous set of complete kets indexed by the continuous variables <math>x^\prime</math> and <math>x^{\prime\prime}</math>:

The kets may also represent a [[continuous]] set of states.  In such circumstances (which would include, for example, a [[free particle]]), Dirac also found it necessary to develop what is known as the [[Dirac Delta Function]] as an analogue to the [[Kronecker Delta Function]].  For a continuous set of complete kets indexed by the continuous variables <math>x^\prime</math> and <math>x^{\prime\prime}</math>:

<math>\left\langle x^\prime|x^{\prime\prime}\right\rangle = \delta\left(x^\prime-x^{\prime\prime}\right)</math>

<math>\left\langle x^\prime|x^{\prime\prime}\right\rangle = \delta\left(x^\prime-x^{\prime\prime}\right)</math>