Hilbert Spaces & Quantum Mechanics: Algebra, Geometry & Analysis

[mmwave]

I browsed a book by Byron & Fuller "Math. Physics" and read the following:

Algebra, Geometry & Analysis are joined when functions are treated as vectors in a vector space. This makes Hilbert spaces extremely useful in QM.(paraphrased but that's the jist of it)

Comments on this?

If it's true than I am interested in studying Hilbert spaces to better understand QM. When I took QM my teacher was asked "What good is this stuff on Hilbert Spaces?". His reply was that every space & every function used in QM is a Hilbert Space so don't worry about it - it's of no consequence at all.

 

[selfAdjoint]

The states of a quantum are represented numerically in a Hilbert space, which is a mathematical object. The simplest Hilbert space, for the simplest quantum systems, is the complex numbers. Only a little more compicated are sets of vectors with complex components. In field theory the vectors have infinitely many components, like the coefficients of a Fourier series.

Things the experimenters do are represented by mathematical operators that act on the vectors and turn them into other vectors. Sometimes an operator only multiplies a vector by a number. In that case the vector is called an eigenvector of the operator and the number is called an eigenvalue. There are some operators, called Hermitian operators, for which the eigenvalues are always real (as opposed to complex) numbers. All the genuine operators in quantum mechanics are assumed to be Hermitian. The eigenvalues represent things that can be measured in experiments. They are called observables.

 

[R0man]

The statement that functions in QM can be treated as vectors in a Hilbert space is indeed true and is a fundamental concept in quantum mechanics. This approach allows for a more elegant and powerful mathematical framework to describe quantum systems and their properties.

Hilbert spaces provide a rigorous mathematical structure for dealing with the complex wave functions that describe quantum states. By treating these functions as vectors, we can use the tools of linear algebra, such as inner products and orthogonal projections, to analyze and manipulate them.

Furthermore, the geometry of Hilbert spaces plays a crucial role in understanding the properties of quantum systems. The concept of orthogonality, which is a fundamental property of Hilbert spaces, is closely related to the concept of observables and their measurements in quantum mechanics.

In addition, the analysis aspect of Hilbert spaces allows us to study the convergence and completeness of quantum states, which is essential for understanding the probabilistic nature of quantum systems.

Overall, the study of Hilbert spaces is crucial for a deep understanding of quantum mechanics. It allows us to mathematically describe and analyze quantum systems in a precise and elegant manner. So, if you are interested in studying quantum mechanics, delving into the theory of Hilbert spaces is definitely worth your time and effort.