Symmetry and Identical Particles

Oct 13, 2025 by miodeya.com 53 views

Introduction

In quantum mechanics, the study of identical particles is crucial for understanding various phenomena in physics. When dealing with identical particles, the concept of symmetry plays a vital role in determining the possible states of the system. In this article, we will explore the case of two non-interacting identical particles placed in a two-dimensional infinite square well. We will derive the wavefunction for the system and discuss the implications of symmetry on the possible states.

The Single-Particle Spatial Wavefunction

The single-particle spatial wavefunction for a particle in a two-dimensional infinite square well is given by:

\psi_{n_x,n_y}(x,y) = \sqrt{\frac{4}{L^2}} \sin\left(\frac{n_x\pi x}{L}\right) \sin\left(\frac{n_y\pi y}{L}\right)

where $n_x$ and $n_y$ are positive integers, $L$ is the side length of the square well, and $x$ and $y$ are the coordinates of the particle.

Symmetry and Identical Particles

When dealing with identical particles, the wavefunction must be symmetric under the exchange of particles. This means that if we swap the positions of the two particles, the wavefunction must remain unchanged. Mathematically, this can be expressed as:

\Psi(x_1,y_1,x_2,y_2) = \Psi(x_2,y_2,x_1,y_1)

where $\Psi(x_1,y_1,x_2,y_2)$ is the wavefunction of the system with particles at positions $(x_1,y_1)$ and $(x_2,y_2)$ .

Derivation of the Two-Particle Wavefunction

To derive the wavefunction for the two-particle system, we can use the single-particle wavefunction as a basis. We can write the two-particle wavefunction as a linear combination of the single-particle wavefunctions:

\Psi(x_1,y_1,x_2,y_2) = \sum_{n_x,n_y} c_{n_x,n_y} \psi_{n_x,n_y}(x_1,y_1) \psi_{n_x,n_y}(x_2,y_2)

where $c_{n_x,n_y}$ are the coefficients of the linear combination.

Symmetry and the Two-Particle Wavefunction

To ensure that the wavefunction is symmetric under the exchange of particles, we can impose the following condition:

\Psi(x_1,y_1,x_2,y_2) = \Psi(x_2,y_2,x_1,y_1)

This condition can be used to determine the coefficients $c_{n_x,n_y}$ .

Solving for the Coefficients

To solve for the coefficients $c_{n_x,n_y}$ , we can use the condition of symmetry. We can write the wavefunction as:

\Psi(x_1,y_1,x_2,y_2) = \sum_{n_x,n_y} c_{n_x,n_y} \sin\left(\frac{n_x\pi x_1}{L}\right) \sin\left(\frac{n_y\pi y_1}{L}\right) \sin\left(\frac{n_x\pi x_2}{L}\right) \sin\left(\frac{n_y\pi y_2}{L}\right)

Using the condition of symmetry, we can show that the coefficients $c_{n_x,n_y}$ must satisfy the following condition:

c_{n_x,n_y} = c_{n_y,n_x}

This means that the coefficients are symmetric under the exchange of $n_x$ and $n_y$ .

Determining the Possible States

To determine the possible states of the system, we can use the condition of symmetry to impose constraints on the coefficients $c_{n_x,n_y}$ . We can show that the coefficients must satisfy the following condition:

c_{n_x,n_y} = \frac{1}{2} (c_{n_x} + c_{n_y})

where $c_{n_x}$ and $c_{n_y}$ are the coefficients of the single-particle wavefunctions.

Conclusion

In this article, we have derived the wavefunction for two non-interacting identical particles placed in a two-dimensional infinite square well. We have shown that the wavefunction must be symmetric under the exchange of particles, and we have used this condition to impose constraints on the coefficients of the wavefunction. We have also determined the possible states of the system, and we have shown that the coefficients must satisfy certain conditions to ensure symmetry.

References

[1] Messiah, A. (1961). Quantum Mechanics. North-Holland Publishing Company.
[2] Landau, L. D., & Lifshitz, E. M. (1977). Quantum Mechanics: Non-relativistic Theory. Pergamon Press.
[3] Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley Publishing Company.

Q: What is the significance of identical particles in quantum mechanics?

A: Identical particles are crucial in quantum mechanics because they exhibit symmetry under the exchange of particles. This symmetry is a fundamental concept in quantum mechanics and has far-reaching implications for the behavior of particles.

Q: What is the single-particle spatial wavefunction for a particle in a 2D infinite square well?

A: The single-particle spatial wavefunction for a particle in a 2D infinite square well is given by:

\psi_{n_x,n_y}(x,y) = \sqrt{\frac{4}{L^2}} \sin\left(\frac{n_x\pi x}{L}\right) \sin\left(\frac{n_y\pi y}{L}\right)

where $n_x$ and $n_y$ are positive integers, $L$ is the side length of the square well, and $x$ and $y$ are the coordinates of the particle.

Q: What is the condition for the wavefunction to be symmetric under the exchange of particles?

A: The wavefunction must satisfy the following condition:

\Psi(x_1,y_1,x_2,y_2) = \Psi(x_2,y_2,x_1,y_1)

This means that if we swap the positions of the two particles, the wavefunction must remain unchanged.

Q: How do we derive the two-particle wavefunction from the single-particle wavefunction?

A: We can use the single-particle wavefunction as a basis to derive the two-particle wavefunction. We can write the two-particle wavefunction as a linear combination of the single-particle wavefunctions:

\Psi(x_1,y_1,x_2,y_2) = \sum_{n_x,n_y} c_{n_x,n_y} \psi_{n_x,n_y}(x_1,y_1) \psi_{n_x,n_y}(x_2,y_2)

where $c_{n_x,n_y}$ are the coefficients of the linear combination.

Q: What is the condition for the coefficients to ensure symmetry under the exchange of particles?

A: The coefficients must satisfy the following condition:

c_{n_x,n_y} = c_{n_y,n_x}

This means that the coefficients are symmetric under the exchange of $n_x$ and $n_y$ .

Q: How do we determine the possible states of the system?

A: We can use the condition of symmetry to impose constraints on the coefficients $c_{n_x,n_y}$ . We can show that the coefficients must satisfy the following condition:

c_{n_x,n_y} = \frac{1}{2} (c_{n_x} + c_{n_y})

where $c_{n_x}$ and $c_{n_y}$ are the coefficients of the single-particle wavefunctions.

Q: What are the implications of symmetry on the behavior of particles?

A: Symmetry has far-reaching implications for the behavior of particles. It determines the possible states of the system and affects the behavior of particles in various ways.

Q: What are some real-world applications of identical particles in quantum mechanics?

A: Identical particles have numerous real-world applications in quantum mechanics, including:

Superconductivity
Superfluidity
Bose-Einstein condensation
Quantum computing

Q: What are some common misconceptions about identical particles in quantum mechanics?

A: Some common misconceptions about identical particles in quantum mechanics include:

Identical particles are always indistinguishable
Identical particles always exhibit symmetry
Identical particles always behave in the same way

These misconceptions can lead to incorrect conclusions and a lack of understanding of the behavior of particles in quantum mechanics.

Q: What are some resources for further learning about identical particles in quantum mechanics?

A: Some resources for further learning about identical particles in quantum mechanics include:

Quantum Mechanics by Lev Landau and Evgeny Lifshitz
Quantum Mechanics by J. J. Sakurai
Symmetries in Quantum Mechanics by A. Messiah
Online courses and tutorials on quantum mechanics and identical particles.