Wave mechanics represents a quantum state by a wavefunction and observables by operators acting on it. For a nonrelativistic spinless particle in a potential energy , the Schrödinger equation readsThe spatial shape and boundary conditions determine possible energy eigenfunctions; the Born rule gives their experimental interpretation as probability amplitudes.
For the infinite square well, vanishing at both walls selects standing sine waves and hence discrete energy levels. This explains quantization through a boundary-value problem rather than by imposing a separate allowed-orbit rule. Erwin Schrödinger's first 1926 paper developed quantization as an eigenvalue problem for the hydrogen atom. Matrix and wave mechanics equivalence explains how the same physical predictions arise from matrix mechanics in a suitable orthonormal basis.[ref]
Quantisierung als Eigenwertproblem (Erste Mitteilung) by Erwin Schrödinger. 1926, pp. 361–363: quantization as an eigenvalue problem and the vibrating-string analogy
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Wave mechanics is a branch of physics that deals with the study of waves and their behavior in various mediums. It forms an integral part of the broader field of quantum mechanics, particularly in the context of describing the behavior of particles at the quantum level. In wave mechanics, physical systems are described using wave functions, which encapsulate the properties of particles, such as their position and momentum.