Interaction Picture: Bridging Schrödinger and Heisenberg Frameworks

Table of Contents

1. Introduction
2. Motivation for the Interaction Picture
3. Schrödinger vs Heisenberg vs Interaction Picture
4. Definition of the Interaction Picture
5. Time Evolution in the Interaction Picture
6. State Vector Evolution
7. Operator Evolution
8. The Role of the Interaction Hamiltonian
9. Dyson Series and Time-Ordered Exponentials
10. Application to Time-Dependent Perturbation Theory
11. Example: Two-Level System Driven by a Field
12. Relation to the Dirac Picture
13. Use in Quantum Field Theory
14. Advantages and Limitations
15. Interaction Picture in Quantum Optics and Computing
16. Conclusion

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1. Introduction

The interaction picture, also called the Dirac picture, is an intermediate representation in quantum mechanics. It combines elements of both the Schrödinger and Heisenberg pictures and is particularly useful in time-dependent perturbation theory and quantum field theory.

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2. Motivation for the Interaction Picture

When dealing with a time-dependent Hamiltonian that has both solvable and perturbative components, the interaction picture simplifies calculations:

• The free part of the Hamiltonian governs operator evolution
• The interaction part governs state evolution

This separation is ideal for perturbative expansions.

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3. Schrödinger vs Heisenberg vs Interaction Picture

Picture	State Evolution	Operator Evolution
Schrödinger	\( |\psi(t)\rangle \) evolves	Operators fixed
Heisenberg	State fixed	\( \hat{A}(t) \) evolves
Interaction	\( |\psi_I(t)\rangle \) evolves	\( \hat{A}_I(t) \) evolves

The interaction picture allows splitting time evolution across both states and operators.

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4. Definition of the Interaction Picture

Let the full Hamiltonian be:

• \( \hat{H}_0 \): free Hamiltonian (solvable)
• \( \hat{H}_{\text{int}}(t) \): interaction Hamiltonian (perturbative)

The state in the interaction picture is:

\[
|\psi_I(t)\rangle = e^{i\hat{H}_0 t/\hbar} |\psi_S(t)\rangle
\]

The operator in the interaction picture is:

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5. Time Evolution in the Interaction Picture

Time evolution is governed by:

Where:

This makes the interaction picture ideal for time-dependent perturbation theory.

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6. State Vector Evolution

The evolution of the state is driven by the interaction Hamiltonian:

\[
|\psi_I(t)\rangle = \hat{U}_I(t, t_0) |\psi_I(t_0)\rangle
\]

Where \( \hat{U}_I \) satisfies:

\[
i\hbar \frac{d}{dt} \hat{U}_I(t, t_0) = \hat{H}_I(t) \hat{U}_I(t, t_0), \quad \hat{U}_I(t_0, t_0) = \hat{I}
\]

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7. Operator Evolution

Operators evolve using only the free Hamiltonian:

\[
\hat{A}_I(t) = e^{i\hat{H}_0 t/\hbar} \hat{A}_S e^{-i\hat{H}_0 t/\hbar}
\]

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8. The Role of the Interaction Hamiltonian

The interaction Hamiltonian \( \hat{H}_{\text{int}} \):

• Drives transitions between unperturbed eigenstates
• Encodes external fields, coupling, and perturbations
• Is assumed to be small compared to \( \hat{H}_0 \)

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9. Dyson Series and Time-Ordered Exponentials

The solution to the state evolution is given by the Dyson series:

Where \( \mathcal{T} \) is the time-ordering operator, ensuring proper ordering of non-commuting terms.

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10. Application to Time-Dependent Perturbation Theory

Used to compute transition probabilities:

\[
P_{i \to f}(t) = |\langle f | \hat{U}_I(t, t_0) | i \rangle|^2
\]

Expanding \( \hat{U}_I \) in a series yields successive orders of perturbation, widely used in quantum optics and spectroscopy.

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11. Example: Two-Level System Driven by a Field

Consider:

\[
\hat{H}(t) = \hat{H}_0 + V \cos(\omega t) \hat{\sigma}_x
\]

Interaction picture separates:

• Free evolution from \( \hat{H}_0 \)
• Coupling from \( V \cos(\omega t) \hat{\sigma}_x \)

Used to study Rabi oscillations and quantum gates.

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12. Relation to the Dirac Picture

The interaction picture is often referred to as the Dirac picture, especially in field theory. It’s the standard choice in scattering theory, S-matrix formulation, and perturbative QFT.

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13. Use in Quantum Field Theory

• Fields evolve like in Heisenberg picture
• States evolve with interaction Hamiltonian
• Wick’s theorem and Feynman diagrams derive from this framework

Essential for computing scattering amplitudes.

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14. Advantages and Limitations

Advantages:

• Ideal for perturbation theory
• Clean separation of solvable and interaction parts
• Adaptable to numerical methods

Limitations:

• Requires \( \hat{H}_0 \) to be exactly solvable
• Breaks down for strong interactions

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15. Interaction Picture in Quantum Optics and Computing

• Describes laser-atom interactions
• Used in Jaynes–Cummings model
• Basis for interaction-frame Hamiltonians in quantum control
• Underlies pulse shaping in qubit manipulation

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16. Conclusion

The interaction picture bridges the dynamics of the Schrödinger and Heisenberg pictures, providing a flexible and powerful framework for handling time-dependent problems in quantum mechanics. It’s essential for perturbative approaches and underpins key calculations in quantum optics, field theory, and quantum information science.

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