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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]]
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<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>
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{{TEX|done}}
  
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
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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  $  t $
 +
of operators  $  A $
 +
and wave functions  $  \psi $
 +
in quantum mechanics and quantum field theory. In the Schrödinger representation the operators  $  A _ {S} $
 +
corresponding to physical dynamical quantities do not depend on  $  t $;
 +
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/s0834209.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{1 }
 +
i\hbar
 +
\frac{\partial  \psi ( t) }{\partial  t }
 +
  = H \psi ( t)
 +
$$
  
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
+
can be formally expressed by the [[Hamilton operator|Hamilton operator]]  $  H $,  
 +
which is independent of  $  t $,  
 +
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/s08342014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{2 }
 +
\psi ( t)  \equiv  \psi _ {s} ( t)  = e ^ {- i t H / \hbar } \psi ( 0),
 +
$$
  
<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>
+
where  $  \psi ( 0) $,
 +
being the initial value, does not depend on time, and the wave function in the Schrödinger representation depends on  $  t $
 +
and contains all information with respect to changes in the state of the system when  $  t $
 +
changes. The mean value of the operator  $  A _ {S} $
 +
in the Schrödinger representation
  
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" />:
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$$ \tag{3 }
 +
\overline{A}\;  \equiv  \overline{A}\; _ {S}  = \
 +
( \psi _ {S} ( t), A _ {S} \psi _ {S} ( t) ) =
 +
$$
  
<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>
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$$
 +
= \
 +
( \psi ( 0), e ^ {+ i t H / \hbar } A _ {S} e ^ {- i t H / \hbar } \psi ( 0) )
 +
$$
 +
 
 +
depends on  $  t $
 +
as a result of the dependence on  $  t $
 +
of the wave functions  $  \psi _ {S} ( t) $.  
 +
$  \overline{A}\; $
 +
can be also considered as the mean value of the time-dependent operator  $  A _ {H} $
 +
over the wave functions  $  \psi _ {H} $,
 +
which do not depend on  $  t $:
 +
 
 +
$$ \tag{4 }
 +
A _ {H} ( t) =  e ^ {+ i t H / \hbar } A _ {S} e ^ {- i t H / \hbar } ; \ \
 +
\psi _ {H}  =  \psi ( 0)  =  e ^ {it H / \hbar } \psi _ {S} ( t),
 +
$$
  
 
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.
 
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.
 
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====
 
====Comments====

Latest revision as of 08:12, 6 June 2020


One of the basic possible (together with the Heisenberg representation and the interaction representation (cf. Interaction, representation of)) equivalent representations of the dependence on time $ t $ of operators $ A $ and wave functions $ \psi $ in quantum mechanics and quantum field theory. In the Schrödinger representation the operators $ A _ {S} $ corresponding to physical dynamical quantities do not depend on $ t $; thus, the solution of the Schrödinger equation

$$ \tag{1 } i\hbar \frac{\partial \psi ( t) }{\partial t } = H \psi ( t) $$

can be formally expressed by the Hamilton operator $ H $, which is independent of $ t $, in the form

$$ \tag{2 } \psi ( t) \equiv \psi _ {s} ( t) = e ^ {- i t H / \hbar } \psi ( 0), $$

where $ \psi ( 0) $, being the initial value, does not depend on time, and the wave function in the Schrödinger representation depends on $ t $ and contains all information with respect to changes in the state of the system when $ t $ changes. The mean value of the operator $ A _ {S} $ in the Schrödinger representation

$$ \tag{3 } \overline{A}\; \equiv \overline{A}\; _ {S} = \ ( \psi _ {S} ( t), A _ {S} \psi _ {S} ( t) ) = $$

$$ = \ ( \psi ( 0), e ^ {+ i t H / \hbar } A _ {S} e ^ {- i t H / \hbar } \psi ( 0) ) $$

depends on $ t $ as a result of the dependence on $ t $ of the wave functions $ \psi _ {S} ( t) $. $ \overline{A}\; $ can be also considered as the mean value of the time-dependent operator $ A _ {H} $ over the wave functions $ \psi _ {H} $, which do not depend on $ t $:

$$ \tag{4 } A _ {H} ( t) = e ^ {+ i t H / \hbar } A _ {S} e ^ {- i t H / \hbar } ; \ \ \psi _ {H} = \psi ( 0) = e ^ {it H / \hbar } \psi _ {S} ( t), $$

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).

How to Cite This Entry:
Schrödinger representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schr%C3%B6dinger_representation&oldid=17978
This article was adapted from an original article by V.D. Kukin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article