Hilbert Spaces: Why do we need them in Quantum Mechanics?

Disclaimer: This text is based on the content of "Lectures on Quantum Mechanics" by Jean-Louis Basdevant, specifically Chapter 6, Section 1. It is an excellent (I mean excellent for sure) resource for graduate-level readers, providing a deep and intuitive understanding of quantum mechanics. I highly recommend this book to anyone looking to explore the subject further.

In the world of quantum mechanics and mathematical analysis, the concept of Hilbert space plays a fundamental role. It is not merely a theoretical construct but a powerful framework that allows us to address complex problems geometrically and intuitively. This article provides a deep dive into the principles of Hilbert space and its profound implications, particularly in the context of quantum mechanics.

Understanding the Foundations: Plancherel’s Theorem

Let us begin by considering two wave functions and their Fourier transforms. By applying Plancherel’s theorem, we establish an important equivalence:

This relationship, which equates integrals in both real and momentum space, reveals a deeper underlying structure. Mathematicians quickly recognize that these integrals represent scalar products, and this insight forms the basis of a powerful abstraction: treating functions as vectors in a vector space. This extraordinary idea, pioneered by the likes of Banach, Hilbert, and Fréchet, introduced the use of geometric language to solve analytical problems.

The Geometry of Hilbert Space

The transition to viewing functions as vectors in an abstract space is analogous to our understanding of ordinary geometry. In geometry, a vector can be represented in different coordinate systems, but its properties, such as length and angles (scalar products), remain invariant. Similarly, the state of a quantum system can have various representations, each with distinct advantages.

Before diving deeper into the intricacies of Hilbert space, let’s revisit some fundamental concepts of Hermitian spaces, which are finite-dimensional complex vector spaces. These notions serve as a stepping stone to understanding the infinite-dimensional case.

Hermitian Scalar Product in Finite Dimensions

In a two-dimensional Hermitian space, a vector u and the Hermitian scalar product of vectors u and v is defined as:can be represented as:

This scalar product is positive definite, allowing us to define the norm of a vector:

For matrices, the Hermitian conjugate M? of a matrix M is obtained by transposing and taking the complex conjugate of its elements. A matrix is called Hermitian if M=M?. Such matrices have real eigenvalues, and their eigenvectors form an orthonormal basis.

Extending to Infinite Dimensions: Square Integrable Functions

In quantum mechanics, we often deal with square-integrable functions, which are part of the function space L2(R). A function f(x) is considered square-integrable if:

This set of functions forms a complex vector space (any linear combination of square-integrable functions is square-integrable), where the Hermitian scalar product of two functions f and g is defined as:

This product satisfies Hermitian symmetry:

This allows us to define the norm of ||f||:

Algebraically, it is exactly the same as in finite-dimensional spaces considered above. It is convergence that is different, that is topological properties.

The Legacy of Hermite: The Quantum Harmonic Oscillator

In 1860, Charles Hermite made a significant discovery, without being aware of that, while studying an eigenvalue problem (someone said that everything in maths is an eigenvalue problem):

The eigenfunctions of this problem are the well-known Hermite functions, which form an orthonormal basis in L2(R). These functions are not only normalized but also complete, meaning any square-integrable function f(x) can be expressed as:

where the coefficients Cn are given by:

This expansion highlights a remarkable property: the Hermite functions form a Hilbert basis, allowing us to represent any function in L2(R) as a series of orthogonal components.

The Essence of Hilbert Space

The space L2(R) exhibits three crucial properties that define a Hilbert space:

• It is a complex vector space.
• It has a Hermitian, positive-definite scalar product.
• It contains orthonormal bases, such as the Hermite functions.

Despite being infinite-dimensional, the algebraic properties of Hilbert space are akin to those of finite-dimensional Euclidean spaces. Concepts like the Pythagorean theorem and the triangle inequality still apply, which is a testament to its robust geometric structure.

The Geometric Language of Quantum Mechanics

That is the great discovery! Square-integrable functions form a Hilbert space! This space is infinite-dimensional. For mathematicians it is more interesting to study than a two-dimensional space, but psychologically, for us it is essentially similar. We are not concerned with topological properties (even though we mention some of them, and they play an important role in more elaborate quantum mechanical problems). The algebraic rules are the same as in finite-dimensional spaces.

Conclusion

The introduction of Hilbert space has been revolutionary, providing a unifying framework for both mathematics and quantum mechanics. By treating functions as vectors and employing geometric language, we can tackle problems with greater clarity and efficiency. The insights of pioneers like Hermite, Hilbert, and others have given us powerful tools to navigate the complexities of infinite-dimensional spaces, laying the foundation for modern quantum theory.

In the realm of quantum mechanics, understanding Hilbert space is not merely an academic exercise; it is the key to unlocking the mysteries of the quantum world.