Wave Function in Quantum Mechanics

Last Updated on September 16, 2024 by Max

The wave function, denoted by \(\psi\), is a fundamental concept in quantum mechanics that fully describes the quantum state of a particle. It encodes all relevant information about the particle’s position, momentum, and energy. The wave function is essential for understanding key quantum phenomena, including superposition, where particles exist in multiple states simultaneously, entanglement, where particles become interconnected in ways that their properties are correlated, and the probabilistic nature of quantum measurements.

This article explains the wave function of a quantum system, what it represents, its main properties, and how it helps us understand and use quantum mechanics in modern technology.

Mathematical Representation of the Wave Function

A wave function is a complex-valued function of space and time:

\[\psi(\mathbf{r}, t) = \langle \mathbf{r} | \psi(t) \rangle,\]

where \(\mathbf{r}\) represents the position vector in three-dimensional space, and \(t\) represents time. The wave function is typically defined over a Hilbert space, a mathematical structure that provides the framework for quantum states.

Physical Interpretation: Probability Amplitude

The modulus squared of the wave function gives the probability density of finding a particle at a certain position and time:

\[|\psi(\mathbf{r}, t)|^2 = \psi^*(\mathbf{r}, t) \psi(\mathbf{r}, t).\]

This probability density function, introduced by Max Born, is fundamental to quantum mechanics. To ensure that the total probability is equal to one, the wave function must be normalized as,

\[\int_{-\infty}^{\infty} |\psi(\mathbf{r}, t)|^2 \, d^3r = 1.\]

Properties of a Wave Function

A valid wave function must satisfy several key properties as mentioned below.

1. Normalization:
   The wave function must be normalized to ensure that the total probability of finding a particle in all space equals one,
   \[\int_{-\infty}^{\infty} |\psi(\mathbf{r}, t)|^2 \, d^3r = 1.\]
2. Continuity:
   The wave function and its first spatial derivative must be continuous. This ensures no abrupt changes in probability distribution or its flow.
3. Single-Valued:
   The wave function must be single-valued at any point in space, implying only one value of \( \psi(\mathbf{r}, t) \) for a given position \( \mathbf{r} \) and time \( t \).
4. Square Integrability:
   The wave function must be square integrable over all space as,
   \[\int_{-\infty}^{\infty} |\psi(\mathbf{r}, t)|^2 \, d^3r < \infty.\]
   This ensures a finite total probability of finding the particle somewhere in space.
5. Differentiability:
   The wave function should be at least once differentiable, which is necessary for calculating physical quantities like momentum.
6. Complex Nature:
   The wave function is generally complex, with both real and imaginary parts:
   \[\psi(\mathbf{r}, t) = \Re(\psi) + i \Im(\psi).\]
7. Orthonormality:
   For a set of wave functions representing different quantum states of a system, they must be orthonormal as given by,
   \[\int_{-\infty}^{\infty} \psi_n^*(\mathbf{r}, t) \psi_m(\mathbf{r}, t) \, d^3r = \delta_{nm}.\]
8. Symmetry Properties:
   The wave function must obey specific symmetry properties depending on the nature of the particles. For fermions (particles with half-integer spin), the wave function must be antisymmetric under particle exchange, while for bosons (particles with integer spin), it must be symmetric.

Schrödinger Equation: Evolution of the Wave Function

The time evolution of a quantum system is governed by the Schrödinger equation, a fundamental differential equation relating the wave function to the Hamiltonian operator \( \hat{H} \) of the system. For a non-relativistic particle moving in a potential \( V(\mathbf{r}, t) \), the time-dependent Schrödinger equation is given by,

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

where \( \hbar \) is the reduced Planck’s constant, \( m \) is the mass of the particle, and \( \nabla^2 \) is the Laplacian operator. This equation determines how the wave function evolves over time, given an initial condition.

Wave Function for Different Quantum Systems

1. Free Particle:
   For a free particle (no potential, \( V = 0 \)), the Schrödinger equation simplifies, and the solution is typically a plane wave, which can be expressed as,
   \[\psi(\mathbf{r}, t) = A e^{i(\mathbf{k} \cdot \mathbf{r} – \omega t)},\]
   where \( A \) is a normalization constant, \( \mathbf{k} \) is the wave vector, and \( \omega = \frac{\hbar k^2}{2m} \) is the angular frequency.
2. Particle in a Potential Well:
   For a particle in a one-dimensional infinite potential well of width \( a \), the wave function must satisfy boundary conditions \( \psi(0) = \psi(a) = 0 \). The allowed solutions are sinusoidal,
   \[\psi_n(x) = \sqrt{\frac{2}{a}} \sin \left( \frac{n \pi x}{a} \right), \quad n = 1, 2, 3, \ldots\]
   Each solution corresponds to a discrete energy level \( E_n = \frac{n^2 \pi^2 \hbar^2}{2m a^2} \).
3. Harmonic Oscillator:
   The quantum harmonic oscillator, a model for particles in a quadratic potential, has wave functions expressed in terms of Hermite polynomials \( H_n(x) \) as,
   \[\psi_n(x) = \left( \frac{m \omega}{\pi \hbar} \right)^{1/4} \frac{1}{\sqrt{2^n n!}} H_n\left( \sqrt{\frac{m \omega}{\hbar}} x \right) e^{-\frac{m \omega x^2}{2 \hbar}},\] with corresponding energy levels \( E_n = \hbar \omega \left( n + \frac{1}{2} \right) \).

Superposition Principle and Wave Function Collapse

The wave function can exist in a superposition of states. If a system can exist in states \( \psi_1 \) and \( \psi_2 \), any linear combination \( c_1 \psi_1 + c_2 \psi_2 \) is also a valid wave function. This superposition principle leads to interference effects, a hallmark of quantum mechanics.

Upon measurement, however, the wave function ‘collapses‘ to one of its eigenstates. This collapse is instantaneous and non-deterministic, reflecting the probabilistic nature of quantum outcomes.

Wave Function in Multiple-Particle Systems

For systems involving multiple particles, the wave function depends on the coordinates of all particles. For two particles, the wave function is given by,

\[\psi(\mathbf{r}_1, \mathbf{r}_2, t).\]

If the particles are indistinguishable, such as electrons, the wave function must satisfy specific symmetry requirements. It must be antisymmetric for fermions (such as electrons and protons) and symmetric for bosons (such as photons).

Conclusion

The wave function is a fundamental concept in quantum mechanics, giving a complete description of a quantum system. Its behavior is described by the Schrödinger equation, and its probabilistic interpretation represents a significant shift from classical physics. Understanding the wave function is crucial for comprehending quantum phenomena and their uses in modern technology.