kb/data/en.wikipedia.org/wiki/Bra–ket_notation-7.md

title	chunk	source	category	tags	date_saved	instance
Bra–ket notation	8/8	https://en.wikipedia.org/wiki/Bra–ket_notation	reference	science, encyclopedia	2026-05-05T14:40:03.882193+00:00	kb-cron

One ignores the parentheses and removes the double bars. Moreover, mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one, and they usually use not an asterisk but an overline (which the physicists reserve for averages and the Dirac spinor adjoint) to denote complex conjugate numbers; i.e., for scalar products mathematicians usually write

    ⟨
    ϕ
    ,
    ψ
    ⟩
    =
    ∫
    ϕ
    (
    x
    )
    
      
        
          ψ
          (
          x
          )
        
        ¯
      
    
    
    d
    x
    
    ,
  

{\displaystyle \langle \phi ,\psi \rangle =\int \phi (x){\overline {\psi (x)}}\,dx\,,}

whereas physicists would write for the same quantity

    ⟨
    ψ
    
      |
    
    ϕ
    ⟩
    =
    ∫
    d
    x
    
    
      ψ
      
        ∗
      
    
    (
    x
    )
    ϕ
    (
    x
    )
     
    .
  

{\displaystyle \langle \psi |\phi \rangle =\int dx\,\psi ^{*}(x)\phi (x)~.}

== See also ==

Angular momentum diagrams (quantum mechanics) Inner product space n-slit interferometric equation Quantum state

== Notes ==

== References == Dirac, P. A. M. (1939). "A new notation for quantum mechanics". Mathematical Proceedings of the Cambridge Philosophical Society. 35 (3): 416–418. Bibcode:1939PCPS...35..416D. doi:10.1017/S0305004100021162. S2CID 121466183. Also see his standard text, The Principles of Quantum Mechanics, IV edition, Clarendon Press (1958), ISBN 978-0198520115 Grassmann, H. (1862). Extension Theory. History of Mathematics Sources. 2000 translation by Lloyd C. Kannenberg. American Mathematical Society, London Mathematical Society. Cajori, Florian (1929). A History Of Mathematical Notations Volume II. Open Court Publishing. p. 134. ISBN 978-0-486-67766-8. {{cite book}}: ISBN / Date incompatibility (help) Shankar, R. (1994). Principles of Quantum Mechanics (2nd ed.). ISBN 0-306-44790-8. Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1965). The Feynman Lectures on Physics. Vol. III. Reading, MA: Addison-Wesley. ISBN 0-201-02118-8. Sakurai, J J; Napolitano, J (2021). Modern Quantum Mechanics (3rd ed.). Cambridge University Press. ISBN 978-1-108-42241-3.

== External links == Richard Fitzpatrick, "Quantum Mechanics: A graduate level course", The University of Texas at Austin. Includes:

1. Ket space
2. Bra space
3. Operators
4. The outer product
5. Eigenvalues and eigenvectors Robert Littlejohn, Lecture notes on "The Mathematical Formalism of Quantum mechanics", including bra–ket notation. University of California, Berkeley. Gieres, F. (2000). "Mathematical surprises and Dirac's formalism in quantum mechanics". Rep. Prog. Phys. 63 (12): 1893–1931. arXiv:quant-ph/9907069. Bibcode:2000RPPh...63.1893G. doi:10.1088/0034-4885/63/12/201. S2CID 10854218.