The Mathematical Representation of Quantum Mechanics

The mathematical formulations of quantum mechanics involve mathematical formalisms that enable a precise and thorough description of the theory. This mathematical framework primarily relies on a subfield of functional analysis, specifically Hilbert spaces, which are a type of linear space. These formulations differ from mathematical frameworks for physics theories developed before the early 1900s due to the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces (mainly L2 space) and operators acting on these spaces. In essence, physical observables like energy and momentum transitioned from being values of functions on phase space to eigenvalues, or more accurately, spectral values of linear operators in Hilbert space.

Today, these quantum mechanics formulations remain in use. Central to these descriptions are the concepts of quantum states and quantum observables, which significantly differ from those employed in earlier models of physical reality. Although the mathematics allows for the calculation of numerous experimentally measurable quantities, there is a definitive theoretical limit to the values that can be measured simultaneously. Heisenberg first clarified this limitation through a thought experiment, and the non-commutativity of operators representing quantum observables mathematically represents it within the new formalism.

Before quantum mechanics emerged as a distinct theory, the mathematics used in physics mainly involved formal mathematical analysis, starting with calculus and progressing in complexity up to differential geometry and partial differential equations. Probability theory found use in statistical mechanics. Geometric intuition played a crucial role in the first two, leading to the formulation of relativity theories entirely in terms of differential geometric concepts. The phenomenology of quantum physics surfaced between approximately 1895 and 1915. In the 10 to 15 years preceding the development of quantum mechanics (around 1925), physicists continued to conceptualize quantum theory within the bounds of classical physics and particularly within the same mathematical structures. The Sommerfeld-Wilson-Ishiwara quantization rule, formulated entirely on the classical phase space, exemplifies this most profoundly.

In the late 19th century, Planck developed the concept of energy quanta while working on the blackbody spectrum, which later helped address the ultraviolet catastrophe issue in classical physics. Einstein then expanded on this idea, proposing that these energy quanta were actual particles, which came to be known as photons. While Bohr and Sommerfeld made attempts to adapt classical mechanics to explain these phenomena, the underlying mathematical framework for quantum theory remained uncertain. In 1923, de Broglie extended the idea of wave-particle duality, suggesting that it applied to all physical systems, including electrons and other particles.

The period from 1925 to 1930 marked a turning point in the development of quantum mechanics, as key mathematical foundations were established by scientists such as Schrödinger, Heisenberg, Born, Jordan, von Neumann, Weyl, and Dirac. They were able to unify several different approaches using a new set of concepts. Furthermore, the physical interpretation of quantum mechanics became clearer with the discovery of Heisenberg's uncertainty relations and Bohr's introduction of the principle of complementarity. These advancements laid the groundwork for our modern understanding of quantum mechanics and its applications in various scientific fields.

The "new quantum theory" emerged with Werner Heisenberg's matrix mechanics, which successfully replicated the observed quantization of atomic spectra. Later, Schrödinger developed wave mechanics, which was considered more accessible due to its use of familiar differential equations. Within a year, the equivalence of these two theories was established.

Schrödinger initially misunderstood the probabilistic nature of quantum mechanics, and it was Max Born who introduced the interpretation of the wave function's absolute square as the probability distribution of a point-like object's position. This idea became the foundation of the Copenhagen interpretation of quantum mechanics. Paul Dirac, another key figure in quantum mechanics, demonstrated the equivalence of Schrödinger's wave mechanics and Heisenberg's matrix mechanics, as well as their connection to classical Hamilton–Jacobi equations. Dirac's work led to the development of the modern abstract quantum mechanics framework based on Hilbert space.

The complete mathematical formulation, known as the Dirac–von Neumann axioms, is credited to John von Neumann's 1932 book Mathematical Foundations of Quantum Mechanics, with prior contributions from Hermann Weyl. The mathematical framework for quantum mechanics continues to evolve, but most discussions and extensions are based on the shared assumptions of von Neumann's foundational work.

In the rigorous formulation of quantum mechanics, a system's state is represented by a vector in a complex Hilbert space. Physical properties are represented by observables, which are Hermitian linear operators acting on the Hilbert space. Quantum states can be eigenstates or quantum superpositions. When an observable is measured, the outcome follows the Born rule, which assigns probabilities to its eigenvalues. Measurement causes the quantum state to collapse.

The time evolution of a quantum state is described by the Schrödinger equation, with the Hamiltonian representing the total energy of the system. The time-evolution operator is unitary, which guarantees deterministic time evolution. Some wave functions produce time-independent probability distributions, such as eigenstates of the Hamiltonian.

In quantum mechanics, a system's state is represented by a vector |ψ⟩ in a complex Hilbert space 𝓗. This vector obeys ⟨ψ,ψ⟩ = 1 and is well-defined up to a complex number of modulus 1. Physical properties are represented by observables, denoted as A, which are Hermitian linear operators acting on the Hilbert space.

A quantum state can be an eigenstate, in which case it satisfies A|ψ⟩ = a|ψ⟩, where a is the eigenvalue corresponding to the observable in that eigenstate. More generally, a quantum state can be a superposition, written as |ψ⟩ = ∑_i c_i |ψ_i⟩, where |ψ_i⟩ are eigenstates and c_i are complex coefficients.

When an observable is measured, the probability of obtaining the eigenvalue λ is given by the Born rule: P(λ) = |⟨λ|ψ⟩|^2 or, more generally, P(λ) = ⟨ψ,P_λψ⟩, where P_λ is the projector onto the eigenspace associated with λ. After the measurement, the quantum state collapses to |λ⟩ or P_λψ/√⟨ψ,P_λψ⟩.

The time evolution of a quantum state is described by the Schrödinger equation:

iħ(d/dt)|ψ(t)⟩ = H|ψ(t)⟩

where H denotes the Hamiltonian, and ħ is the reduced Planck constant. The solution of this equation is given by:

|ψ(t)⟩ = U(t)|ψ(0)⟩ = e^(-iHt/ħ)|ψ(0)⟩

where U(t) = e^(-iHt/ħ) is the unitary time-evolution operator, ensuring deterministic time evolution. Some wave functions produce time-independent probability distributions, such as eigenstates of the Hamiltonian (H|ψ⟩ = E|ψ⟩).

Analytic solutions of the Schrödinger equation exist for only a few simple model Hamiltonians, but there are methods for finding approximate solutions, such as perturbation theory and semiclassical equations of motion.