Configuration-Space Based Representations in Quantum Theory| Configuration-Space Based Representations in Quantum Theory |
Schrödinger Picture / Representation
> s.a. representations in quantum theory; quantum field theory.
* Idea: The representation
on L2(\(\cal C\), dμ), where
\(\cal C\) is the configuration space, in which states are time-dependent,
observables time-independent.
* Use: It is not convenient
for the relativistic theory, since it treats time differently from the space
coordinates.
@ References:
Hiley & Dennis a1809 [Dirac-Bohm picture];
Stoica a1906
[multi-layered field representation in 3D space].
Heisenberg Picture / Representation > s.a. quantum field theory.
* Idea: The representation on
L2(\(\cal C\), dμ) in which
states are time-independent, and observables time-dependent operators.
* Relationships: Compared to
the Schrödinger representation,
ψH = exp{iHt/\(\hbar\)} ψS(t), and AH(t) = exp{iHt/\(\hbar\)} AS exp{−iHt/\(\hbar\)};
the time evolution of the operators is given by i\(\hbar\) ∂AH/∂t
= [AH, H]; The Hamiltonian operator is the same.
* Advantages: Constant phase
shifts in the operators due, e.g., to a constant potential are cancelled;
The equations of motion for the operators are formally identical to
the classical ones; It is convenient for relativistic theory.
* Disadvantages: It
is difficult to solve practical problems with it.
@ Compared to Schrödinger representation:
Faria et al PLA(02);
Nikolić PLA(04)qp/03;
de la Madrid qp/05-conf [for unbounded operators, and rigged Hilbert space];
Solomon a0706 [in quantum field theory];
Aharonov a1303-fs [and non-local quantum phenomena];
Partovi a1305
[and Eulerian vs Lagrangian descriptions of fluid dynamics];
de Gosson a1404 [inequivalence];
Franson & Brewster a1811 [limitations].
Interaction Picture / Representation
* Idea: Both states and
operators are time-dependent, with evolutions governed by different parts
of the Hamiltonian:
H'I(t) ψI(t) = i\(\hbar\) ∂ψI(t)/∂t , and i\(\hbar\) ∂AI(t)/∂t = [AI(t), H0] ,
where H = H0 + H', a free and an interaction part, and the relationship with Schrödinger representation quantities is
ψI(t) = exp{iH0t/\(\hbar\)} ψS(t) , and H'I(t) = exp{iH0t/\(\hbar\)} H'S exp{−iH0t/\(\hbar\)} .
* Advantage: Takes into
account only the non-trivial evolution of states; The free H
evolves the operators.
* Disadvantage: It does
not exist in general for a relativistic theory.
* Remark: This representation
is often implicitly used in ordinary quantum mechanics, when ignoring
"the rest of the world".
@ In quantum field theory: Biswas a0807 [transformation to free fields not unitary].
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