When Erwin Schrödinger developed his famous equation in 1925-1926, he formulated it as a wave equation for matter waves (inspired by de Broglie). He was thinking in terms of differential equations and their solutions (wavefunctions, ψ) in configuration space. His primary goal was to find an equation that yielded the correct quantized energy levels for systems like the hydrogen atom.

Key points:

1. Wave Mechanics Focus: Schrödinger’s approach was initially called “wave mechanics.” He viewed ψ as describing some sort of wave amplitude, although its precise physical interpretation was initially unclear (Max Born later provided the probabilistic interpretation of |ψ|²).
2. Differential Equation: He derived a partial differential equation whose eigenvalue solutions corresponded to the allowed energy states.
3. No Abstract Vector Space (Initially): He wasn’t thinking in terms of abstract vectors in an infinite-dimensional complex vector space with inner products, operators, etc., which is the hallmark of the Hilbert space formulation.
4. Hilbert Space Formalism Came Later: The rigorous mathematical framework of quantum mechanics using Hilbert spaces was largely developed slightly later, most notably by John von Neumann in the late 1920s and early 1930s (culminating in his 1932 book “Mathematical Foundations of Quantum Mechanics”). Von Neumann showed how both Schrödinger’s wave mechanics and Heisenberg’s matrix mechanics could be seen as different representations within the same abstract Hilbert space structure.
5. Implicit Connection: While Schrödinger didn’t explicitly use the formalism of Hilbert space, the space of square-integrable functions (L² space) which contains the solutions (wavefunctions) to his equation is mathematically an example of a Hilbert space. So, the mathematical structure was implicitly present, but the explicit recognition and utilization of Hilbert space as the fundamental mathematical arena for quantum states came later.

Several key factors drove physicists like Paul Dirac and mathematicians like John von Neumann to adopt and develop the Hilbert space formalism for quantum mechanics (QM), moving beyond Schrödinger’s initial wave mechanics formulation:

1. Unification of Wave and Matrix Mechanics:
   • In the mid-1920s, physics had two seemingly different, yet successful, formulations of QM: Heisenberg’s Matrix Mechanics (using infinite matrices for observables) and Schrödinger’s Wave Mechanics (using differential equations and wavefunctions).
   • Physicists needed a single, underlying mathematical structure that could encompass both. Von Neumann, in particular, rigorously demonstrated that both were mathematically equivalent representations within the abstract framework of Hilbert spaces. Wavefunctions became vectors in an infinite-dimensional Hilbert space (L²), and operators in Wave Mechanics (like x or -iħ d/dx) corresponded to the infinite matrices of Matrix Mechanics when expressed in appropriate bases.
2. Need for Mathematical Rigor:
   • Schrödinger’s differential equation approach, while intuitive for physicists trained in classical waves, had potential mathematical subtleties (e.g., defining the domain of operators like momentum, handling boundary conditions, ensuring solutions were well-behaved).
   • Matrix mechanics dealt with infinite matrices, which also required careful mathematical treatment.
   • Hilbert spaces provided a well-defined, rigorous mathematical foundation. Concepts like completeness, inner products, norms, and the precise definition of self-adjoint operators (crucial for observables having real eigenvalues) put QM on solid mathematical ground.
3. Incorporating Probability (Born Rule):
   • Max Born’s interpretation stated that |ψ(x)|² represents a probability density. To get a total probability of 1, the wavefunction ψ must be square-integrable (∫|ψ(x)|² dx = 1).
   • The space of square-integrable functions (L² space) is the prototypical infinite-dimensional Hilbert space. The inner product <φ|ψ> = ∫φ*(x)ψ(x) dx naturally arises in this context and is essential for calculating probabilities of transitions between states (|<φ|ψ>|²) and expectation values. Hilbert space formalizes this requirement beautifully.
4. Superposition Principle (Linearity):
   • QM is fundamentally linear: if |ψ₁> and |ψ₂> are possible states, then any linear combination a|ψ₁> + b|ψ₂> is also a possible state.
   • Vector spaces are the natural mathematical structure for describing linearity and superposition. Hilbert spaces are specific types of vector spaces (with an inner product and completeness) ideally suited for QM’s needs.
5. Abstract Representation of States and Observables:
   • Physicists realized that a quantum state is a more abstract concept than just a wave in physical space. It could represent intrinsic properties like spin, which don’t have a simple spatial wavefunction equivalent.
   • Hilbert space allows states to be represented as abstract vectors |ψ>. Observables (like position, momentum, energy, spin) are represented as linear operators (specifically, Hermitian or self-adjoint operators) acting on these vectors. The possible results of measuring an observable are the eigenvalues of the corresponding operator. This provides a general and powerful framework applicable to any quantum system.
6. Dirac’s Bra-Ket Notation:
   • Dirac introduced the elegant and powerful bra-ket notation (<φ|, |ψ>, <φ|A|ψ>). This notation is perfectly tailored for calculations in Hilbert space.
   • |ψ> (ket) represents a state vector.
   • <φ| (bra) represents a vector in the dual space (related to the original Hilbert space via the inner product).
   • <φ|ψ> naturally represents the inner product.
   • This notation emphasized the abstract vector space nature of quantum states and simplified manipulations, making the Hilbert space approach more intuitive and usable for physicists.

In essence, Hilbert space provided the necessary unification, rigor, generality, and interpretational clarity that was needed for quantum mechanics to mature into a complete and consistent physical theory. It offered the perfect mathematical language to express the core principles of superposition, probability, and the relationship between states and observables.