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. Fig. ... In quantum mechanics, a state function is a linear combination (a superposition) of eigenstates (the exception is if a state function is itself an eigenstate). ... In quantum mechanics in the Heisenberg picture the state vector, |ψ> does not change with time, and an observable A satisfies In some sense, the Heisenberg picture is more natural and fundamental than the Schrödinger picture, especially for relativistic theories. ... Quite literally, quantum state describes the state of a quantum system. ... In mathematical formulations of quantum mechanics, an operator is a linear transformation from a Hilbert space to itself. ... In physics, particularly in quantum physics a system observable is a property of the system state that can be determined by some sequence of physical operations. ...

Warning: Operator equations which hold in the interaction picture don't necessarily hold in the Schrodinger 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 for another picture.

Switching pictures

To switch into the interaction picture, we divide the Schrödinger picture Hamiltonian into two parts, H = H₀ + H₁. Then the state vector is defined as: The Hamiltonian, denoted H, has two distinct but closely related meanings. ...

Operators transform between the pictures as

.

The Schrödinger equation then becomes in this picture: In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ...

.

This equation is referred to as the Schwinger- Tomonaga equation. Julian Seymour Schwinger (February 12, 1918 -- July 16, 1994) was an American theoretical physicist. ... Sin-Itiro Tomonaga Sin-Itiro Tomonaga or Shinichirō Tomonaga (朝永 振一郎 Tomonaga Shinichirō, March 31, 1906–July 8, 1979) was a Japanese physicist, influential in the development of quantum electrodynamics, work for which he was jointly awarded the Nobel Prize in Physics in 1965 along with Richard Feynman and Julian Schwinger. ...

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, H₁, being added to the Hamiltonian of a solved system, H₀. By switching into the interaction picture, you can use time-dependent perturbation theory to find the effect of H_{1, I}. In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. ...

References

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

See also

• Bra-ket notation
• Schrödinger equation
• Haag's theorem