View source for Heisenberg representation

One of the principal possible equivalent representations (together with the [[Schrödinger representation|Schrödinger representation]] and the representation of interaction, cf. [[Interaction, representation of|Interaction, representation of]]) of the dependence of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h0468501.png" /> and the wave functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h0468502.png" /> on the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h0468503.png" /> in quantum mechanics and in quantum field theory. In the Heisenberg representation the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h0468504.png" /> depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h0468505.png" />, while the wave functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h0468506.png" /> do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h0468507.png" />, and are connected with the corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h0468508.png" />-independent operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h0468509.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h04685010.png" />-dependent wave functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h04685011.png" /> in the Schrödinger representation by a unitary transformation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h04685012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where the Hermitian operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h04685013.png" /> is the complete [[Hamiltonian|Hamiltonian]] of the system, which is independent of time. That it is possible to introduce the Heisenberg representation, the Schrödinger representation and the representation of interaction, and that they are equivalent, is due to the fact that it is not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h04685014.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h04685015.png" /> by themselves but only the average value of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h04685016.png" /> in the state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h04685017.png" /> that must be invariant with respect to unitary transformations of the type (1) and, consequently, the average value should not depend on the selection of the representation. Differentiation of (1) with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h04685018.png" /> yields an equation for the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h04685019.png" /> in the Heisenberg representation that contains complete information on the variation of the state of the quantum system with the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h04685020.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h04685021.png" /></td> </tr></table>

where the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h04685022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046850/h04685023.png" /> do not usually commute.

Named after W. Heisenberg, who introduced it in 1925 in a matrix formulation of quantum mechanics.



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Mehra,   H. Rechenberg,   "The historical development of quantum theory" , '''1–4''' , Springer  (1982)</TD></TR></table>