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In quantum mechanics, eigenfunctions play a crucial role. An eigenfunction is a special type of function that, when acted upon by an operator (often the Hamiltonian in physics), yields the same function scaled by a constant factor called the eigenvalue.
When we say an eigenfunction is nondegenerate, this means that no other eigenfunction shares the same eigenvalue.
This is significant because it implies a unique relationship between the eigenfunction and its eigenvalue.
If you have a nondegenerate eigenfunction, there is no ambiguity in its response to the operator—it will behave in a unique way that cannot be replicated by some other function with the same eigenvalue.
In our context, if a symmetric Hamiltonian has a nondegenerate eigenfunction, that function must be either symmetric or antisymmetric.
This uniqueness helps us conclude that the wave function has a definite symmetry property.

The Hamiltonian is a critical operator in quantum mechanics, representing the total energy of the system. A symmetric Hamiltonian is one that remains unchanged if you swap the particles involved.
This symmetry means that for any pair of particles, the Hamiltonian H(1,2) is the same as H(2,1).
Symmetric Hamiltonians are common in systems of identical particles, where it doesn't matter which particle is labeled as '1' and which as '2'.
The symmetry of the Hamiltonian ensures that the solutions (the wave functions) must also respect this symmetry.
When we say the wave function u(1,2) is an eigenfunction of the symmetric Hamiltonian H, with eigenvalue E, it implies that the application of the Hamiltonian does not alter the symmetry of the wave function.

• If the Hamiltonian is symmetric, then the wave function must either remain the same (symmetric) or change sign (antisymmetric) under particle exchange.
• This characteristic comes from the nature of how the symmetric operation of the Hamiltonian applies consistently across permutations.

Permutation symmetry refers to the invariance of a system under the rearrangement of its components. For a wave function with multiple variables (like u(1,2,...,n) where each number corresponds to a different particle), permutation symmetry means that swapping any two particles should not change the fundamental form of the wave function.
This concept is crucial in systems of identical particles. There are two main types of permutation symmetry for wave functions: symmetric and antisymmetric.

• A symmetric wave function does not change when you swap two particles: u(1,2) = u(2,1).
• An antisymmetric wave function changes sign when you swap two particles: u(1,2) = -u(2,1).

For example, in systems such as bosons or fermions, symmetric and antisymmetric functions respectively describe their behavior due to the statistics they follow.
Permutation symmetry ensures coherence in the description of particles and simplifies understanding their collective behavior.
In our context, the nondegenerate eigenfunction of a symmetric Hamiltonian must inherently respect this symmetry, meaning it must act symmetrically or antisymmetrically when any two particles are permuted. This ensures the wave function’s properties remain consistent with the underlying physics of the system.