Free-particle wavefunction and wave-packets

Definition

A free-particle wavefunction is a solution of the Schrödinger equation for a particle moving in a region where the potential energy is zero, $V(x)=0$. A wave-packet is a localized wavefunction formed by combining a continuous range of plane waves with different wave numbers and frequencies, representing a particle with a finite position spread and momentum spread.

Main Content

1. Free-Particle Wavefunction and Plane Wave Solutions

• For a free particle, the time-dependent Schrödinger equation becomes
  $$i\hbar \frac{\partial \psi}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2}$$ since the potential energy $V(x)=0$. Solving this equation gives wavefunctions of the form
  $$\psi(x,t)=Ae^{i(kx-\omega t)}$$ where $A$ is the amplitude, $k$ is the wave number, and $\omega$ is the angular frequency.

• These solutions are called plane waves because their phase is constant on planes in higher dimensions and because they represent a wave extending infinitely in space. For a free particle, the wave number and energy are related by $$p=\hbar k,\qquad E=\hbar\omega=\frac{\hbar^2 k^2}{2m}$$ showing that the momentum and energy are directly linked to the wave properties.

• A plane wave is an exact mathematical solution, but it is not physically realistic for an actual particle because its probability density $|\psi|^2=|A|^2$ is constant everywhere. This means the particle would be equally likely to be found anywhere in space, so the particle is completely delocalized. Such a state is useful for theory, especially in scattering and momentum analysis, but not for describing a particle at a specific location.

2. Wave-Packets and Localization of Particles

• A wave-packet is obtained by adding together many plane waves with slightly different wave numbers and frequencies: $$\psi(x,t)=\int A(k)e^{i(kx-\omega t)}\,dk$$ The coefficient $A(k)$ determines how much each plane wave contributes. By choosing a suitable distribution, such as a Gaussian distribution, the resulting wavefunction becomes localized around a certain position.

• Wave-packets represent particles more realistically because they have a concentrated probability density in a limited region of space. The width of the packet indicates the uncertainty in position. A narrow packet means the particle is more localized, while a broad packet means it is less localized.

• According to the Heisenberg uncertainty principle, position and momentum cannot both be known exactly: $$\Delta x\,\Delta p \ge \frac{\hbar}{2}$$ A sharply localized wave-packet requires many wave numbers, meaning a large spread in momentum. Thus, localization in space and certainty in momentum are mutually limited. This is a fundamental feature of quantum particles and shows why wave-packets are physically meaningful.

3. Time Evolution, Group Velocity, and Dispersion

• A wave-packet does not remain perfectly unchanged with time. Because its component plane waves have different wavelengths and therefore different phase velocities, the packet spreads out as it moves. This spreading is called dispersion. For a free particle, the energy depends on $k^2$, so different $k$-components travel differently in phase.

• The speed at which the overall packet moves is the group velocity, given by $$v_g=\frac{d\omega}{dk}$$ For a free particle, using $\omega=\frac{\hbar k^2}{2m}$, we get $$v_g=\frac{\hbar k}{m}=\frac{p}{m}$$ which matches the classical particle velocity. This is an important result because it links quantum wave behavior to classical motion.

• The phase velocity is $$v_p=\frac{\omega}{k}=\frac{\hbar k}{2m}$$ which is only half of the group velocity for a free particle. The phase velocity is not the physical speed of the particle; instead, the group velocity corresponds to the motion of the packet as a whole. In practice, the packet’s center moves with the group velocity while the packet gradually spreads due to dispersion.

Working / Process

1. Start with the time-dependent Schrödinger equation for a free particle by setting the potential energy $V=0$.
2. Solve the equation to obtain plane-wave solutions and then combine many such solutions using superposition to construct a wave-packet.
3. Use the de Broglie relations, uncertainty principle, and group velocity concept to interpret the physical meaning of the wavefunction and its time evolution.

Advantages / Applications

• Helps describe real particles more accurately than a single plane wave because it provides localization in space.
• Explains fundamental quantum ideas such as momentum spread, uncertainty principle, and the difference between phase velocity and group velocity.
• Used in scattering theory, electron beams, quantum transport, and the study of particle motion in nearly free regions.

Summary

• A free-particle wavefunction is a solution of the Schrödinger equation when $V=0$.
• Plane waves are exact solutions, but they are not localized and therefore represent idealized states.
• Wave-packets are formed by superposition of plane waves and provide a realistic description of particles with finite position and momentum spreads.