Time Evolution: Schrödinger Picture in Quantum Mechanics

Table of Contents

1. Introduction
2. Quantum Time Evolution Overview
3. Schrödinger vs Heisenberg Pictures
4. The Time-Dependent Schrödinger Equation (TDSE)
5. Hamiltonian and the Generator of Time Evolution
6. Time Evolution Operator
7. Properties of the Evolution Operator
8. Solving the TDSE for Time-Independent Hamiltonians
9. Example: Free Particle Evolution
10. Example: Quantum Harmonic Oscillator
11. Superposition and Coherence over Time
12. Unitary Evolution and Probability Conservation
13. Time Evolution in Hilbert Space
14. Adiabatic Approximation
15. Role in Quantum Computing and Information
16. Conclusion

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1. Introduction

In the Schrödinger picture of quantum mechanics, the state vector evolves in time, while observables (operators) remain fixed. This framework, introduced by Erwin Schrödinger, is the most commonly used representation and forms the foundation of most quantum mechanical calculations and simulations.

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2. Quantum Time Evolution Overview

Quantum evolution is deterministic between measurements and governed by the Hamiltonian of the system. The change of state over time is described by a linear differential equation — the time-dependent Schrödinger equation.

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3. Schrödinger vs Heisenberg Pictures

Aspect	Schrödinger Picture	Heisenberg Picture
States	Evolve in time: \(	\psi(t)\rangle \)
Operators	Constant in time	Evolve in time
Emphasis	State dynamics	Observable dynamics

In this article, we focus on the Schrödinger picture.

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4. The Time-Dependent Schrödinger Equation (TDSE)

The central equation is:

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

Where:

• \( |\psi(t)\rangle \) is the quantum state at time \( t \)
• \( \hat{H} \) is the Hamiltonian operator (total energy)
• \( \hbar \) is the reduced Planck constant

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5. Hamiltonian and the Generator of Time Evolution

The Hamiltonian acts as the generator of time translations. It determines how quantum states change with time.

If \( \hat{H} \) is time-independent, the solution to TDSE can be formally written using the time evolution operator.

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6. Time Evolution Operator

The time evolution operator \( \hat{U}(t) \) satisfies:

\[
|\psi(t)\rangle = \hat{U}(t) |\psi(0)\rangle
\]

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

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

This operator is unitary, meaning it preserves the norm of the quantum state.

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7. Properties of the Evolution Operator

• Unitarity: \( \hat{U}^\dagger(t) \hat{U}(t) = \hat{I} \)
• Initial condition: \( \hat{U}(0) = \hat{I} \)
• Composition: \( \hat{U}(t_2) \hat{U}(t_1) = \hat{U}(t_2 + t_1) \)
• If \( [\hat{H}(t), \hat{H}(t’)] = 0 \), then evolution is exponential

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8. Solving the TDSE for Time-Independent Hamiltonians

Let \( \hat{H} |\phi_n\rangle = E_n |\phi_n\rangle \) be the eigenstates of \( \hat{H} \).

Then the general solution:

\[
|\psi(t)\rangle = \sum_n c_n e^{-i E_n t/\hbar} |\phi_n\rangle
\]

Where:

• \( c_n = \langle \phi_n | \psi(0) \rangle \)
• \( |\phi_n\rangle \) form a complete orthonormal basis

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9. Example: Free Particle Evolution

For a free particle with:

\[
\hat{H} = \frac{\hat{p}^2}{2m}
\]

In position space:

\[
\psi(x, t) = \int_{-\infty}^\infty \phi(p) e^{ipx/\hbar} e^{-i p^2 t / (2m\hbar)} \frac{dp}{\sqrt{2\pi\hbar}}
\]

Demonstrates wave packet spreading over time.

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10. Example: Quantum Harmonic Oscillator

With:

\[
\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m\omega^2 \hat{x}^2
\]

The eigenstates \( |\phi_n\rangle \) evolve as:

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

Where \( E_n = \hbar \omega \left(n + \frac{1}{2}\right) \)

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11. Superposition and Coherence over Time

Time evolution preserves quantum coherence:

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

Relative phase changes over time lead to interference effects.

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12. Unitary Evolution and Probability Conservation

Since \( \hat{U}(t) \) is unitary:

\[
\langle \psi(t) | \psi(t) \rangle = \langle \psi(0) | \psi(0) \rangle
\]

This ensures total probability remains 1 — a core requirement of quantum theory.

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13. Time Evolution in Hilbert Space

Quantum states evolve in Hilbert space trajectories governed by unitary operators:

• The evolution traces a path on the unit sphere in Hilbert space
• Governed by linear combinations of basis vectors with evolving phases

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14. Adiabatic Approximation

If \( \hat{H}(t) \) changes slowly:

• System remains in the instantaneous eigenstate
• Basis evolves slowly
• Used in quantum control and quantum computing (adiabatic quantum computing)

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15. Role in Quantum Computing and Information

In quantum circuits:

• Time evolution is simulated by unitary gates
• Gates like \( U(t) = e^{-iHt} \) represent quantum operations
• Accurate time control essential for coherence, entanglement, and quantum speedups

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16. Conclusion

The Schrödinger picture provides a dynamic view of quantum mechanics where the state vector evolves over time under the influence of the Hamiltonian. This picture captures the evolution of quantum probability amplitudes, wave packet dynamics, and interference, and plays a critical role in quantum simulations, chemistry, and quantum technologies. Mastery of time evolution is essential for any serious understanding of quantum behavior.

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