Symmetric, antisymmetric and parity

[Sacroiliac]

Let me see if I can make it clearer.

Problem 5.5 In David Griffiths “Introduction to Quantum Mechanics” says:

Imagine two non interacting particles, each of mass m, in the infinite square well. If one is in the state psi_n and the other in state psi_m orthogonal to psi_n, calculate < (x₁ - x₂) ² >, assuming that (a) they are distinguishable particles, (b) they are identical bosons, (c) they are identical fermions.

(a) a² [1/6 – (1/2pi²)(1/n² + 1/m²)]

(b) The answer to (a) - (128*a²*m²n²) / (pi⁴(m² - n²)⁴)

But this last term is present only when m,n have opposite parity.

(c) The answer to (a) plus the term added in (b) with the same stipulation as in (b)

What does this mean? It seams to be saying that all three particles would have the same separation unless their states have opposite parity. Is this correct? Bosons and Fermions would have the same separation unless their states have odd parities? I never heard of this before, how does this work?

 

[Aaron Barnard]

Yes, this is correct. This is due to the wave nature of the particles and the Pauli exclusion principle for fermions. If the two particles have the same state, then their wave functions overlap and the two particles cannot be separated. If the two particles have states of opposite parity, then their wave functions do not overlap, and so they can be separated. For bosons, the wave functions can overlap, but the total wave function must remain symmetric, leading to a larger separation between the particles.

 

[R0man]

In quantum mechanics, particles can be described by their wavefunctions, which contain information about their position, momentum, and other properties. In the case of identical particles, such as the two non-interacting particles in the infinite square well, the wavefunction must also take into account the symmetry of the system.

Symmetry refers to the behavior of the wavefunction under exchange of the two particles. If the wavefunction remains unchanged, it is called symmetric. If the wavefunction changes sign, it is called antisymmetric.

Parity is a specific type of symmetry that refers to the behavior of the wavefunction under reflection. If the wavefunction remains unchanged, it is called even parity. If the wavefunction changes sign, it is called odd parity.

In the problem described, the calculation of < (x1 - x2) 2 > depends on the symmetry of the system. For distinguishable particles, the calculation is straightforward and does not depend on their states. However, for identical particles, the calculation also takes into account the symmetry of their wavefunctions.

In the case of identical bosons, the wavefunction must be symmetric, which leads to the additional term in the calculation. This term only appears when the states of the particles have opposite parity, meaning that they are in odd parity states. Similarly, for identical fermions, the wavefunction must be antisymmetric, which also leads to an additional term in the calculation. This term also only appears when the states of the particles have opposite parity, meaning that they are in even parity states.

In summary, the calculation of < (x1 - x2) 2 > in this problem takes into account the symmetry of the system, which is determined by the types of particles and their states. This can result in different values for the separation of the particles, depending on the symmetry of their wavefunctions. This concept is important in understanding the behavior of identical particles in quantum mechanics.