Hilbert Space: The Mathematical Foundation of Quantum Reality

During a seminar where Richard Feynman explored quantum physics and the concept of Hilbert space, the German mathematician and philosopher David Hilbert posed a simple yet profound question: “What is Hilbert space?” This scenario implies the importance of Hilbert space. But yes, why Hilbert space and what does it mean? In this piece of writing I try to unveil the deep meaning of it and why this space made the fundamental foundation of quantum physics and chemistry. 

Let’s begin with the idea of a mathematical space. In simple terms, a mathematical space is a framework defined by a specific set of rules or properties. Any entity that satisfies these predefined rules is said to exist within that mathematical space. Whenever we work in a particular mathematical space it is common that we are bound to that space and not allowed to go beyond that space. For example- vector space is also a mathematical space defined by some rules. Those rules include vector addition, multiplication  by different numbers called scalers. The operation in vector space must satisfy some certain requirements called vector axioms. For instance some axioms are mentioned in the following figure.

A crucial property of vector space is that a subset of vector space  is a basis of its elements that are linearly independent and span the vector space. But as i mentioned before we are not permitted to go beyond this space by any means. If we do that our calculation will be inconsistent. So in short we can say the basis of vector space should be finite. Let’s give you an example- we can just take the set of all powers of x, starting with zeroth power. Using this basis any polynomial can be expressed as some linear combination of elements of this basis. 

Now, let us examine if we extended these linear combinations to infinite. I am giving you a youtube video link that can show how the f(x) function graph turns into ex if we are approaching infinity.  However, here’s the catch: when we work with a finite basis set, f(x) remains within our vector space, since polynomials are vectors. But  ex is not a polynomial, meaning it no longer fits within that finite vector space — it exists outside of it. That’s quite absurd — it’s like gathering a pile of stones only to end up with an apple pie at the end. What you get is completely different from what you started with. So, what is the problem? Actually this problem arises with the infinite. Because infinity is not a number it’s just a concept. That's why with a finite number of polynomials we indeed have a polynomial and hence a vector. To better understand, see the following figure.  When we start to add more extra dots then it will go outside of our vector space. 

Hey, stop here. We are starting our discussion for Hilbert space and quantum physics but where is it? Let’s come to our discussion. In quantum physics/chemistry wave function ψ is a fundamental concept. This wavefunction holds all the necessary information of a particle (suppose electron). Since any quantum state can be expressed as a linear combination of basis states, the wave function itself represents a superposition of all the possible states the particle can occupy.  Mathematically, 

∣i⟩ are basis states (like atomic orbitals or energy eigenfunctions). ci are complex coefficients (probability amplitudes). In quantum mechanics, the dimension of the basis set describing a system can be infinite or even continuous. Consequently, if the wave function ψ were confined to an ordinary finite-dimensional vector space, the formalism would become mathematically inconsistent, as the vector space definition restricts the representation to a finite basis. The requirement of an infinite or even continuous set of basis states therefore goes beyond the scope of a simple vector space, leading to the need for a more general mathematical framework — the Hilbert space. Got it.

A Hilbert space is a refined form of vector space with additional structure. It is a complete vector space endowed with an inner product, enabling the measurement of lengths and angles between vectors. Unlike ordinary vector spaces, it not only allows vector addition and scalar multiplication but also provides a framework for defining distances and angles through the inner product. The concept is named after the German mathematician David Hilbert, who developed the abstract foundation of infinite-dimensional spaces that play a fundamental role in modern quantum theory.

In summary, in quantum physics and chemistry, the Hilbert space serves as a fundamental framework that enables a precise and consistent description of quantum systems, allowing us to explore and understand the quantum world accurately. The next section of the article will delve into the mathematical rigor of Hilbert space, providing a deeper understanding of its structure and properties.

References:
1. https://www.youtube.com/watch?v=_kJUUxjJ_FY

2. https://en.wikipedia.org/wiki/Vector_space

3. https://en.wikipedia.org/wiki/Hilbert_space