Proving NonDegeneracy of Energy Eigenfunctions

Prove that the energy eigenfunction of bound states in one-dimensional potential is non-degenerate by considering two energy eigenfunctions $\psi_1(x)$ and $\psi_2(x)$ with the same eigenvalue $E$.

In this problem, you are tasked with proving that the energy eigenfunctions associated with bound states in a one-dimensional potential are non-degenerate. This topic engages with one of the essential concepts of quantum mechanics: the uniqueness of quantum states under particular constraints and the broader implications of these principles in understanding physical systems. Non-degeneracy in quantum mechanics implies that for a given energy level, there is only one independent quantum state. In practical terms, this ensures that observable quantities computed from these states remain consistent and predictable within a given context.

To tackle this problem, you should recall the properties and implications of Hermitian operators in quantum mechanics. Since the Hamiltonian in one-dimensional quantum systems is Hermitian, it guarantees real eigenvalues and orthogonal eigenfunctions under different eigenvalues. However, proving non-degeneracy involves showing that if two eigenfunctions share the same eigenvalue, they must be linearly dependent, meaning they represent the same physical state. Begin by considering the properties of the wave functions, particularly their boundary conditions and normalization. These considerations, coupled with applying the appropriate theorems such as the Sturm-Liouville theory or Wronskian considerations, can effectively demonstrate that in bound states, energy eigenfunctions cannot be degenerate.

This type of problem not only reinforces your understanding of energy quantization but also delves into the mathematical rigor needed to firmly establish quantum principles. Such exercises hone your problem-solving skills in handling conditions and assumptions that underpin the standard quantum mechanical framework. They emphasize reasoning within the structure of linear differential equations that define quantum states, which is crucial for more advanced topics you'll encounter later in quantum mechanics studies.