View source for Schrödinger representation

One of the basic possible (together with the [[Heisenberg representation|Heisenberg representation]] and the interaction representation (cf. [[Interaction, representation of|Interaction, representation of]])) equivalent representations of the dependence on time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s0834201.png" /> of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s0834202.png" /> and wave functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s0834203.png" /> in quantum mechanics and quantum field theory. In the Schrödinger representation the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s0834204.png" /> corresponding to physical dynamical quantities do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s0834205.png" />; thus, the solution of the [[Schrödinger equation|Schrödinger equation]]

<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/s/s083/s083420/s0834206.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

can be formally expressed by the [[Hamilton operator|Hamilton operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s0834207.png" />, which is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s0834208.png" />, in the form

<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/s/s083/s083420/s0834209.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s08342010.png" />, being the initial value, does not depend on time, and the wave function in the Schrödinger representation depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s08342011.png" /> and contains all information with respect to changes in the state of the system when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s08342012.png" /> changes. The mean value of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s08342013.png" /> in the Schrödinger representation

<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/s/s083/s083420/s08342014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

<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/s/s083/s083420/s08342015.png" /></td> </tr></table>

depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s08342016.png" /> as a result of the dependence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s08342017.png" /> of the wave functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s08342018.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s08342019.png" /> can be also considered as the mean value of the time-dependent operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s08342020.png" /> over the wave functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s08342021.png" />, which do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083420/s08342022.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/s/s083/s083420/s08342023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

i.e. as the mean value of an operator in the Heisenberg representation. The invariance property of the mean value (which should be observable and have physical meaning) under unitary transformations of type (4) means that the Schrödinger representation, the Heisenberg representation and the interaction representation are equivalent.

The Schrödinger representation was called after E. Schrödinger, who introduced it in 1926 when formulating an equation in quantum mechanics that was later called the Schrödinger equation.



====Comments====
Instead of Schrödinger representation one uses sometimes Schrödinger picture.

Equation (2) is correct for time-independent Hamiltonian operators only (cf. [[Schrödinger equation|Schrödinger equation]]).