Show that, if a wave function \(u(1,2, \ldots, n)\) is an energy eigenfunction of a symmetric hamiltonian that corresponds to a nondegenerate eigenvalue, it is either symmetric or antisymmetric.

In quantum mechanics, a **wave function** is a mathematical description of the quantum state of a particle or system of particles. The wave function is typically represented by the symbol \( u(x) \), where \( x \) represents the spatial coordinates of one or more particles. It encapsulates all the information about a system's physical properties.
The square of the absolute value of the wave function, \[ |u(x)|^2 \], gives the probability density of finding a particle at a given point in space. Because of this, the wave function must be normalized, meaning its total probability must be one: \[ \int |u(x)|^2 \, dx = 1 \].
Wave functions can have different symmetries. For example, they can be either **symmetric** or **antisymmetric**:

• A **symmetric wave function** stays the same when any two particles are exchanged.
• An **antisymmetric wave function** changes its sign when any two particles are exchanged.

These properties are particularly significant in systems where particles are indistinguishable, such as electrons in an atom.

In quantum mechanics, an eigenvalue problem is often expressed as \[ H u = E u \], where \( H \) is the Hamiltonian operator, \( u \) is the eigenfunction (or wave function), and \( E \) is the eigenvalue.
A **nondegenerate eigenvalue** means there is a single, unique wave function corresponding to that eigenvalue. In contrast, a **degenerate eigenvalue** would have multiple, linearly independent wave functions.

• When the eigenvalue \( E \) is nondegenerate, it indicates that the wave function associated with it is unique.

For instance, if \( H u = E u \) and \( H v = E v \) for the same eigenvalue \( E \), the nondegenerate condition implies \( u = \beta v \) (where \( \beta \) is a constant).
Understanding whether an eigenvalue is degenerate or nondegenerate is crucial because it affects the symmetry properties of the wave functions. It's especially significant in the context of symmetric Hamiltonians.

A **permutation operator** \( P \) is used to exchange the coordinates of particles in a system. For example, if we have a wave function \( u(x_1, x_2) \) and apply a permutation operator \( P \) to exchange \( x_1 \) and \( x_2 \), we get \( Pu(x_1, x_2) = u(x_2, x_1) \).

• In the context of a symmetric Hamiltonian, the Hamiltonian \( H \) remains invariant under such permutations: \[ HPu = P H u = E P u \].

When the wave function \( u \) is an eigenfunction with a nondegenerate eigenvalue, the symmetry of the Hamiltonian ensures that \( P u \) is also an eigenfunction with the same eigenvalue.
Given that \( u \) and \( P u \) are eigenfunctions of the same nondegenerate eigenvalue, they must be linearly dependent. Therefore, \( P u = \beta u \), where \( \beta \) is a constant. Since applying the permutation operator twice results in the identity operation, we have \[ P P u = u \], leading to \[ \beta^2 = 1 \]. This implies \( \beta = \pm 1 \).

• If \( \beta = 1 \), the wave function is symmetric.
• If \( \beta = -1 \), the wave function is antisymmetric.

Hence, any eigenfunction of a symmetric Hamiltonian with a nondegenerate eigenvalue must be either symmetric or antisymmetric.