Free Particle Solutions: Quantum Motion Without Potential

Table of Contents

1. Introduction
2. The Classical vs Quantum View of Free Motion
3. The Concept of a Free Particle in Quantum Mechanics
4. Schrödinger Equation for a Free Particle
5. General Plane Wave Solutions and Their Interpretation
6. Normalization and the Role of the Delta Function
7. Momentum Eigenstates and Their Significance
8. Constructing Physical States: Superposition and Wave Packets
9. Time Evolution of Wave Packets: Dispersion and Spreading
10. Gaussian Wave Packets: A Detailed Study
11. Probability Density, Probability Current, and Continuity
12. Energy and Momentum Expectation Values
13. Heisenberg Uncertainty Principle in Free Motion
14. Free Particle in Higher Dimensions
15. Real-World Applications and Importance
16. Conclusion

1. Introduction

The free particle model is one of the most fundamental yet insightful topics in quantum mechanics. It serves as the baseline for understanding wave behavior, uncertainty, and quantum dynamics without the complication of external potentials. Despite the simplicity of a “potential-free” scenario, the quantum treatment reveals rich and non-intuitive behavior, highlighting the wave-particle duality of matter.

2. The Classical vs Quantum View of Free Motion

In classical mechanics, a free particle is one that moves with constant velocity in a straight line due to the absence of external forces. Newton’s first law governs its motion.

In quantum mechanics, the free particle does not follow a deterministic trajectory. Instead, its behavior is described by a wavefunction that evolves in space and time, governed by the Schrödinger equation. This wavefunction encodes the probabilistic nature of its location and momentum.

3. The Concept of a Free Particle in Quantum Mechanics

A free particle is defined as one whose potential energy \( V(x) \) is zero everywhere. The particle is not influenced by any fields or barriers.

This scenario allows us to focus solely on the kinetic energy of the particle:

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

Where \( \hat{p} = -i\hbar \frac{d}{dx} \) is the momentum operator.

4. Schrödinger Equation for a Free Particle

In one spatial dimension, the time-dependent Schrödinger equation is:

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

Here:

• \( \psi(x, t) \): wavefunction of the particle
• \( m \): mass of the particle
• \( \hbar \): reduced Planck’s constant

This second-order partial differential equation governs how the wavefunction evolves over time.

5. General Plane Wave Solutions and Their Interpretation

A standard solution is the plane wave:

\[
\psi_k(x, t) = A e^{i(kx – \omega t)}
\]

Where:

• \( k \): wave number, related to momentum \( p = \hbar k \)
• \( \omega = \frac{\hbar k^2}{2m} \): angular frequency
• \( E = \hbar \omega \): energy of the particle

Plane waves are idealized, infinite-extent solutions representing a particle with definite momentum but uncertain position. These solutions are not square-integrable, which means they are not physically realizable alone but are still mathematically crucial.

6. Normalization and the Role of the Delta Function

Since plane waves extend to infinity, they cannot be normalized in the usual sense. Instead, we use Dirac delta normalization:

\[
\langle \psi_{k’} | \psi_k \rangle = \delta(k – k’)
\]

This normalization allows us to construct physical, normalizable states using wave packets, which are superpositions of plane waves.

7. Momentum Eigenstates and Their Significance

The momentum operator in position space is:

\[
\hat{p} = -i\hbar \frac{d}{dx}
\]

Plane waves are eigenfunctions of this operator:

\[
\hat{p} \psi_k(x) = \hbar k \psi_k(x)
\]

This means that a particle in a plane wave state has a definite momentum \( \hbar k \) but completely uncertain position.

8. Constructing Physical States: Superposition and Wave Packets

To represent a localized particle, we construct a wave packet by integrating over many momentum states:

\[
\psi(x, t) = \int_{-\infty}^{\infty} \phi(k) e^{i(kx – \omega t)} dk
\]

Where:

• \( \phi(k) \): momentum space distribution, often chosen as a Gaussian

This results in a localized wavefunction with both position and momentum uncertainties.

9. Time Evolution of Wave Packets: Dispersion and Spreading

Unlike classical particles, quantum wave packets spread over time due to dispersion. This occurs because each component wave has a different velocity, leading to destructive interference in some regions and constructive in others.

The shape of the packet broadens with time, reflecting increasing uncertainty in position.

10. Gaussian Wave Packets: A Detailed Study

Consider a Gaussian wave packet at \( t = 0 \):

\[
\psi(x, 0) = \left( \frac{1}{2\pi \sigma_0^2} \right)^{1/4} \exp\left( -\frac{x^2}{4\sigma_0^2} \right)
\]

Its time evolution is:

\[
\psi(x, t) = \left( \frac{1}{2\pi \sigma_t^2} \right)^{1/4} \exp\left( -\frac{x^2}{4\sigma_t^2} + i \theta(x,t) \right)
\]

Where:

\[
\sigma_t = \sigma_0 \sqrt{1 + \left( \frac{\hbar t}{2m\sigma_0^2} \right)^2 }
\]

Key points:

• The width \( \sigma_t \) increases over time
• \( \theta(x, t) \) is a phase factor
• The shape remains Gaussian but spreads out

11. Probability Density, Probability Current, and Continuity

The probability density is:

\[
\rho(x, t) = |\psi(x, t)|^2
\]

The probability current is:

\[
j(x, t) = \frac{\hbar}{2mi} \left( \psi^* \frac{\partial \psi}{\partial x} – \psi \frac{\partial \psi^*}{\partial x} \right)
\]

These satisfy the continuity equation:

\[
\frac{\partial \rho}{\partial t} + \frac{\partial j}{\partial x} = 0
\]

Ensuring conservation of total probability.

12. Energy and Momentum Expectation Values

For a wave packet \( \psi(x, t) \), the expectation values are:

• Momentum:
  \[
  \langle \hat{p} \rangle = \int \psi^*(x, t) (-i\hbar \frac{d}{dx}) \psi(x, t) dx
  \]
• Energy:
  \[
  \langle \hat{H} \rangle = \int \psi^*(x, t) \left( -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \right) \psi(x, t) dx
  \]

These values remain constant over time for a free particle.

13. Heisenberg Uncertainty Principle in Free Motion

For a Gaussian wave packet:

\[
\Delta x \Delta p = \frac{\hbar}{2}
\]

As time evolves:

• \( \Delta x \) increases
• \( \Delta p \) remains constant

This reflects the quantum spreading of the particle’s position distribution.

14. Free Particle in Higher Dimensions

In 2D and 3D, solutions generalize to:

\[
\psi(\vec{r}, t) = \int \phi(\vec{k}) e^{i(\vec{k} \cdot \vec{r} – \omega t)} d^n k
\]

Free particle behavior is important for describing propagating beams, scattering, and field quantization.

15. Real-World Applications and Importance

Free particle models are crucial in:

• Electron microscopy
• Quantum optics (e.g., laser beam propagation)
• Neutron and X-ray diffraction
• Scattering theory
• Semiconductor modeling

They serve as the starting point for perturbation methods and Green’s function techniques.

16. Conclusion

Though the free particle lacks potential energy, its quantum description is rich and foundational. It introduces central ideas like plane waves, momentum eigenstates, wave packet dynamics, and quantum uncertainty. Mastery of this simple system builds the groundwork for understanding interactions, measurements, and quantum fields.