The Interaction Picture

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Quantum Mechanics A
Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian \mathcal{H}, it describes how a state |\Psi\rangle evolves in time.
Basic Concepts and Theory of Motion
UV Catastrophe (Black-Body Radiation)
Photoelectric Effect
Stability of Matter
Double Slit Experiment
Stern-Gerlach Experiment
The Principle of Complementarity
The Correspondence Principle
The Philosophy of Quantum Theory
Brief Derivation of Schrödinger Equation
Relation Between the Wave Function and Probability Density
Stationary States
Heisenberg Uncertainty Principle
Some Consequences of the Uncertainty Principle
Linear Vector Spaces and Operators
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The Schrödinger Equation in Dirac Notation
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One-Dimensional Bound States
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Motion in a Periodic Potential
Summary of One-Dimensional Systems
Harmonic Oscillator Spectrum and Eigenstates
Analytical Method for Solving the Simple Harmonic Oscillator
Coherent States
Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
The Interaction Picture
The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
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Free Particle in Spherical Coordinates
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Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
Differential Cross Section and the Green's Function Formulation of Scattering
Central Potential Scattering and Phase Shifts
Coulomb Potential Scattering

The interaction, or Dirac, picture is a hybrid between the Schrödinger and Heisenberg pictures. In this picture, both the operators and the state vectors are time dependent; the time dependence is split between the vectors and the operators. This is achieved by splitting the Hamiltonian \hat{H} into two parts - an exactly solvable, or "bare", part \hat{H}_0 and a "peturbation", \hat{V}(t):

\hat{H}=\hat{H}_0+\hat{V}(t)

Let us now take a solution |\Psi(t)\rangle of the Schrödinger equation for the full Hamiltonian and "factor out" the time dependence due to the "bare" part of the Hamiltonian:

|\Psi(t)\rangle=e^{-i\hat{H}_0t/\hbar}|\Psi_I(t)\rangle

In this way, we have defined the state vector |\Psi_I(t)\rangle in the interaction picture. If we substitute this into the Schrödinger equation, we find that this vector satisfies

i\hbar\frac{d}{dt}|\Psi_I(t)\rangle=\hat{V}_I(t)|\Psi_I(t)\rangle,

where \hat{V}_I(t)=e^{i\hat{H}_0t/\hbar}\hat{V}(t)e^{-i\hat{H}_0t/\hbar} is the "perturbation" in the interaction picture. In other words, the time evolution of the state vector in the interaction picture is governed entirely by the "perturbation" part of the Hamiltonian.

We may see that the same relation between the "perturbation" in the interaction picture and the same in the Schrödinger picture is also satisfied by all operators. If we consider the expectation value of an operator, we may rewrite it in terms of the interaction picture state vectors as follows:

\langle\hat{A}\rangle(t)=\langle\Psi(t)|\hat{A}|\Psi(t)\rangle=\langle\Psi_I(t)|e^{i\hat{H}_0t/\hbar}\hat{A}e^{-i\hat{H}_0t/\hbar}|\Psi_I(t)\rangle

Similarly to how we defined the Heisenberg picture operators, we may define the operator \hat{A}_I(t) in the interaction picture as

\hat{A}_I(t)=e^{i\hat{H}_0t/\hbar}\hat{A}e^{-i\hat{H}_0t/\hbar}.

We therefore see that the time dependence of operators in the interaction picture is dictated entirely by the "bare" part of the Hamiltonian.

We may derive equations of motion for operators in the interaction picture, just as we did for the Heisenberg picture; these equations are

\frac{d}{dt}\hat{A}_{I}(t)=-\frac{i}{\hbar}\left [\hat{A},\hat{H}_0\right ]+\left (\frac{\partial\hat{A}}{\partial t}\right )_{I}.

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