Table of Contents

1. Introduction
2. What Is Quantum Dynamics?
3. Schrödinger Equation and Time Evolution
4. Unitary Evolution and the Role of the Hamiltonian
5. Schrödinger, Heisenberg, and Interaction Pictures
6. Evolution of Wavefunctions
7. Evolution of Operators
8. Energy Eigenstates and Stationary States
9. Superposition and Phase Evolution
10. Probability Conservation and Unitarity
11. Quantum Tunneling Dynamics
12. Quantum Harmonic Oscillator Dynamics
13. Quantum Spin Dynamics
14. Entanglement and Nonlocal Dynamics
15. Decoherence and Open System Dynamics
16. Quantum Control and Coherent Manipulation
17. Quantum Dynamics in Field Theory
18. Conclusion

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1. Introduction

Quantum dynamics is the study of how quantum systems evolve over time. It governs everything from electron transitions and atomic motion to quantum computing operations and particle interactions in quantum field theory. The dynamics are driven by the system’s Hamiltonian and manifest as changes in wavefunctions or operators, depending on the chosen picture.

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2. What Is Quantum Dynamics?

Quantum dynamics answers the question: How does a quantum state evolve in time?

It provides a framework for:

• Predicting probabilities of measurement outcomes
• Modeling quantum interference, entanglement, and coherence
• Describing atomic and molecular motion
• Simulating quantum circuits and algorithms

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3. Schrödinger Equation and Time Evolution

The fundamental equation governing dynamics is the time-dependent Schrödinger equation (TDSE):

\[
i\hbar \frac{\partial}{\partial t} |\psi(t)\rangle = \hat{H}(t) |\psi(t)\rangle
\]

This first-order linear differential equation determines the deterministic, unitary evolution of quantum states.

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4. Unitary Evolution and the Role of the Hamiltonian

The Hamiltonian \( \hat{H} \) encapsulates the energy and interactions of the system and acts as the generator of time translations. Time evolution is described by a unitary operator:

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

For time-independent \( \hat{H} \):

\[
\hat{U}(t) = e^{-i\hat{H}t/\hbar}
\]

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5. Schrödinger, Heisenberg, and Interaction Pictures

Picture	States	Operators
Schrödinger	Time-dependent	Fixed
Heisenberg	Fixed	Time-dependent
Interaction	Both evolve (split roles)	Both evolve (split roles)

All pictures are equivalent in terms of physical predictions.

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6. Evolution of Wavefunctions

In position representation:

\[
\psi(x, t) = \langle x | \psi(t) \rangle
\]

Evolves under:

\[
i\hbar \frac{\partial \psi(x,t)}{\partial t} = \left( -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) \right)\psi(x,t)
\]

This partial differential equation governs particle motion and tunneling.

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7. Evolution of Operators

In the Heisenberg picture:

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

With dynamics governed by:

\[
\frac{d\hat{A}}{dt} = \frac{i}{\hbar}[\hat{H}, \hat{A}]
\]

This provides insights into symmetries and conservation laws.

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8. Energy Eigenstates and Stationary States

If \( \hat{H} |\phi_n\rangle = E_n |\phi_n\rangle \), then:

\[
|\psi(t)\rangle = e^{-iE_n t/\hbar} |\phi_n\rangle
\]

These states evolve only by a global phase, leading to stationary probability distributions.

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9. Superposition and Phase Evolution

For a superposition:

\[
|\psi(t)\rangle = c_1 e^{-iE_1 t/\hbar} |\phi_1\rangle + c_2 e^{-iE_2 t/\hbar} |\phi_2\rangle
\]

Relative phase \( (E_1 – E_2)t \) causes interference and oscillations, crucial in quantum computation and control.

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10. Probability Conservation and Unitarity

The norm of the quantum state is preserved:

\[
\langle \psi(t) | \psi(t) \rangle = \text{constant}
\]

Implying that total probability remains 1. This is ensured by the unitarity of \( \hat{U}(t) \).

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11. Quantum Tunneling Dynamics

Quantum particles can tunnel through potential barriers:

• Described by time-dependent wave packets
• Non-zero probability of finding the particle in classically forbidden regions
• Key in nuclear decay, semiconductors, and scanning tunneling microscopes

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12. Quantum Harmonic Oscillator Dynamics

In energy eigenstates:

\[
E_n = \hbar \omega (n + \frac{1}{2})
\]

Wavefunctions oscillate with time via phase:

\[
\psi_n(x, t) = \psi_n(x) e^{-iE_n t/\hbar}
\]

Superpositions exhibit quantum beats and coherent dynamics.

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13. Quantum Spin Dynamics

For a spin-1/2 system in a magnetic field:

\[
\hat{H} = -\gamma \vec{B} \cdot \vec{\sigma}
\]

Results in Rabi oscillations and spin precession — foundational in NMR and quantum control.

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14. Entanglement and Nonlocal Dynamics

Entangled systems exhibit nonlocal correlations that evolve coherently:

• Bell states remain entangled over time
• Evolution can entangle or disentangle systems
• Important in teleportation, quantum cryptography, and quantum algorithms

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15. Decoherence and Open System Dynamics

In real systems:

• Interaction with environment causes decoherence
• Evolution described by master equations and density matrices
• Leads to classical-like behavior without measurement

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16. Quantum Control and Coherent Manipulation

Quantum dynamics enables:

• Precise control over qubit evolution
• Pulse sequences and gates in quantum computers
• Coherent population transfer in atomic systems (STIRAP, Rabi pulses)

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17. Quantum Dynamics in Field Theory

Quantum fields evolve using:

\[
\frac{d\hat{\phi}(\vec{x}, t)}{dt} = \frac{i}{\hbar}[\hat{H}, \hat{\phi}(\vec{x}, t)]
\]

Field theory dynamics involve creation and annihilation operators, scattering amplitudes, and Feynman diagrams.

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18. Conclusion

Quantum dynamics governs the evolution of everything from atomic particles to quantum computers. Whether through wavefunctions or operator equations, understanding these dynamics is essential for interpreting, predicting, and manipulating the behavior of quantum systems. It forms the beating heart of quantum mechanics, computation, and modern physics.

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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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