Difference between revisions of "Schrödinger representation"
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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]] | ||
| − | + | $$ \tag{1 } | |
| + | i\hbar | ||
| + | \frac{\partial \psi ( t) }{\partial t } | ||
| + | = H \psi ( t) | ||
| + | $$ | ||
| − | + | can be formally expressed by the [[Hamilton operator|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. | 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).
Schrödinger representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schr%C3%B6dinger_representation&oldid=17978