The Interaction Picture

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.

Physical Basis of Quantum Mechanics

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

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

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:

$|\Psi (t)\rangle =e^{-i{\hat {H}}_{0}t/\hbar }|\Psi _{I}(t)\rangle$

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 ${\hat {V}}_{I}(t)=e^{i{\hat {H}}_{0}t/\hbar }{\hat {V}}(t)e^{-i{\hat {H}}_{0}t/\hbar }$ 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:

$\langle {\hat {A}}\rangle (t)=\langle \Psi (t)|{\hat {A}}|\Psi (t)\rangle =\langle \Psi _{I}(t)|e^{i{\hat {H}}_{0}t/\hbar }{\hat {A}}e^{-i{\hat {H}}_{0}t/\hbar }|\Psi _{I}(t)\rangle$

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

${\hat {A}}_{I}(t)=e^{i{\hat {H}}_{0}t/\hbar }{\hat {A}}e^{-i{\hat {H}}_{0}t/\hbar }.$

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

Equation of motion :

${\text{i}}\hbar {\frac {\partial }{\partial t}}\left|{\alpha ,t}\right\rangle _{I}=-H_{o}e^{{\frac {i}{\hbar }}H_{o}t}\left|{\alpha ,t}\right\rangle _{S}+e^{{\frac {i}{\hbar }}H_{o}t}(H_{0}+V)\left|{\alpha ,t}\right\rangle _{S}$

$e^{{\frac {i}{\hbar }}H_{o}t}Ve^{{\frac {-i}{\hbar }}H_{o}t}{\text{ . }}e^{{\frac {i}{\hbar }}H_{o}t}\left|{\alpha ,t}\right\rangle _{S}$

If we call firstpart "$V_{I}\!$" and second part "$\left|{\alpha ,t}\right\rangle _{I}$" ,

it turns out :

${\text{=}}V_{I}\left|{\alpha ,t}\right\rangle _{I}$

so;

${\text{i}}\hbar {\frac {\partial }{\partial t}}\left|{\alpha ,t}\right\rangle _{I}=V_{I}\left|{\alpha ,t}\right\rangle _{I}$

${\frac {d}{dt}}A_{I}(t)={\frac {1}{i\hbar }}\left[{A_{I},H_{o}}\right]+{\frac {\partial A_{I}}{\partial t}}$

and this equation of motion evolves with $H_{o}\!$.