Interaction picture

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In quantum mechanics, the Interaction picture (or Dirac picture) is an intermediate between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of observables.

Warning: Operator equations which hold in the interaction picture don't necessarily hold in the Schrödinger or the Heisenberg picture. This is because the operators A_I, A_S and A_H are not the same but are related by unitary transformations. Unfortunately, most textbooks and articles omit the subscript which can often lead to confusion and mistakes when an unwary student applies an equation for one picture to another picture.

[edit] Switching pictures

To switch into the interaction picture, we divide the Schrödinger picture Hamiltonian into two parts, $H S = H 0, S + H 1, S$ . Then the state vector is defined as

$| \psi_{I}(t) \rang = e^{i H_{0, S} t / \hbar} | \psi_{S}(t) \rang$

Operators transform between the pictures as

$A_{I} = e^{i H_{0} t / \hbar} A_{S} e^{-i H_{0} t / \hbar}$ .

The Schrödinger equation then becomes in this picture:

$i \hbar \frac{d}{dt} | \psi_{I} (t) \rang = H_{1, I} | \psi_{I} (t) \rang$ .

This equation is referred to as the Schwinger-Tomonaga equation.

If the operator $A S$ is time independent (classically), then the corresponding time evolution for $A I (t)$ is ruled by:

$i\hbar\frac{d}{dt}A_I(t)=\left[A_I(t),H_0\right]\;.$

In the interaction picture the operators evolve in time like the operators in the Heisenberg picture with an Hamiltonian $H' = H 0$ .

The purpose of this picture is to shunt all the time dependence due to H₀ onto the operators, leaving only H_{1, I} affecting the time-dependence of the state vectors.

The interaction picture is convenient when considering the effect of a small interaction term, H_{1, S}, being added to the Hamiltonian of a solved system, H_{0, S}. By switching into the interaction picture, you can use time-dependent perturbation theory to find the effect of H_{1, I}.

[edit] References

Townsend, John S. (2000). A Modern Approach to Quantum Mechanics, 2nd ed.. Sausalito, CA: University Science Books. ISBN 1-891389-13-0.

Category: Quantum mechanics

Interaction picture

From Wikipedia, the free encyclopedia

[edit] Switching pictures

[edit] References

[edit] See also

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