Editing Bra–ket notation (section)

==Pitfalls and ambiguous uses==

There are some conventions and uses of notation that may be confusing or ambiguous for the non-initiated or early student.

===Separation of inner product and vectors===
A cause for confusion is that the notation does not separate the inner-product operation from the notation for a (bra) vector. If a (dual space) bra-vector is constructed as a linear combination of other bra-vectors (for instance when expressing it in some basis) the notation creates some ambiguity and hides mathematical details. We can compare bra–ket notation to using bold for vectors, such as <math>\boldsymbol \psi</math>, and <math>(\cdot,\cdot)</math> for the inner product. Consider the following dual space bra-vector in the basis <math>\{|e_n\rangle\}</math>, where <math>\{\psi_n\}</math> are the complex number coefficients of <math>\langle \psi | </math>:
<math display="block">\langle\psi| = \sum_n \langle e_n| \psi_n</math>

It has to be determined by convention if the complex numbers <math>\{\psi_n\}</math> are inside or outside of the inner product, and each convention gives different results.

<math display="block">\langle\psi| \equiv (\boldsymbol\psi, \cdot ) = \sum_n (\boldsymbol e_n, \cdot ) \, \psi_n</math>
<math display="block">\langle\psi| \equiv (\boldsymbol\psi, \cdot ) = \sum_n (\boldsymbol e_n \psi_n, \cdot ) = \sum_n (\boldsymbol e_n, \cdot ) \, \psi_n^*</math>

===Reuse of symbols===
It is common to use the same symbol for ''labels'' and ''constants''. For example, <math>\hat \alpha |\alpha\rangle = \alpha |\alpha \rangle</math>, where the symbol <math>\alpha</math> is used simultaneously as the ''name of the operator'' <math>\hat \alpha</math>, its ''eigenvector'' <math>|\alpha\rangle</math> and the associated ''eigenvalue'' <math>\alpha</math>. Sometimes the ''hat'' is also dropped for operators, and one can see notation such as <math>A |a\rangle = a |a \rangle</math>.<ref>{{harvnb|Sakurai|Napolitano|2021}} Sec 1.2, 1.3</ref>

===Hermitian conjugate of kets===
It is common to see the usage <math>|\psi\rangle^\dagger = \langle\psi|</math>, where the dagger (<math>\dagger</math>) corresponds to the Hermitian conjugate. This is however not correct in a technical sense, since the ket, <math>|\psi\rangle</math>, represents a [[vector (mathematics and physics)|vector]] in a complex Hilbert-space <math>\mathcal{H}</math>, and the bra, <math>\langle\psi|</math>, is a [[linear functional]] on vectors in <math>\mathcal{H}</math>. In other words, <math>|\psi\rangle</math> is just a vector, while <math>\langle\psi|</math> is the combination of a vector and an inner product.

===Operations inside bras and kets===
This is done for a fast notation of scaling vectors. For instance, if the vector <math>|\alpha \rangle</math> is scaled by <math>1/\sqrt{2}</math>, it may be denoted <math>|\alpha/\sqrt{2} \rangle</math>. This can be ambiguous since <math>\alpha</math> is simply a label for a state, and not a mathematical object on which operations can be performed. This usage is more common when denoting vectors as tensor products, where part of the labels are moved '''outside''' the designed slot, e.g. <math>|\alpha \rangle = |\alpha/\sqrt{2} \rangle_1 \otimes |\alpha/\sqrt{2} \rangle_2</math>.