# Mathematical formulation of quantum mechanics

I am reading a book on quantum mechanics, but it is difficult to understand.

Quantum mechanics is roughly formulated as follows:

• Physicsl state is a normalized ray $$\{e^{i\theta}\psi|\theta \in \mathbb{R}\}$$, where $$\psi$$ is a vector in a Hilbert space $$\mathcal{H}$$.
• Observables are self-adjoint linear operator of $$\mathcal{H}$$.
• Measurable value is a eigenvalue of a observable.
• When we measure an observable $$A$$ in a state $$\psi$$, the probability of measuring a value $$a$$ is the square of the (norm of the) projection of $$\psi$$ onto the eigenspace of $$a$$.

Is it possible to explain the reason for formulating this way?

For example, "We use function space (i.e. Hilbert space) rather than point space (i.e. Euclidean space or manifold) for state space because states are indeterminate" (I don't know this is true).

• Is "because it works" a sufficient answer? – BioPhysicist Mar 19 '19 at 3:40

The whole confusing I suppose was the understanding of the usage of Hilbert space and Hermitian operator. Without bringing in formal Copenhagen interpretation, the whole ideal above was to introduce you familiar with the Hermitian operator, which emphasized, here, on the real eigenvalues.

Think that when you do an measurement, you instrument would probably return you an value, i.e. $$5mV$$, a measurement which has real numbers. That's what the author want you to understand, that the formalism would give you the benefit that the result/measurement of your operator, which was the eigenvalue of the Hermitian operator, was real.

The fist sentence about $$e^{i\theta}$$ was the phase factor. I don't think you need to worry about it right now as most time in your book it won't matter much.

The last one involves the multiplication of complex conjugate, which was because of the usage from complex space. Your representation of probability here was no longer a real value, but a complex number. In order for it to make sense, you need to do a multiplication with respect to its complex conjugate and make it into a real number.

Basically, it because in quantum mechanics, you almost always operated in complex space, and you need Hermitian operator and multiplication of complex conjugate to make the results into real number so that they would make sense.

As for "why hilbert space", right now I think we can just stop at it's "convenient", that it gave you the satisfactorily in the degree of freedom of vector space, which was infinite. Notice Hilbert space(a construction of space, or "generalized euclidean space"), even in geometrical representation, was usually locally euclidean, and it can certainly contain manifold. Most of the time in quantum, people assume the "smooth" property, which has connection with manifold. It's hard to describe it without bringing in some formal definition. But mainly, the author want you to know that he/she's operating in a possibly infinite dimensional space, local Euclidean, and smooth.

Quantum mechanics as a theory did not start with a formal mathematical formulation, as the one you are reading.

It started with the Bohr model which attempted to describe the spectra of atoms and was within a classical physics framework, a planetary model with some constraints imposed by hand on angular momentum, so that discrete energy levels would appear in the atomic spectra. It was successful because it predicted the Balmer etc series which phenomenologically fitted the data.

The Bohr model successfully predicted the energies for the hydrogen atom, but had significant failures that were corrected by solving the Schrodinger equation for the hydrogen atom.

In these solutions is where complex numbers enter, and the basic postulates of quantum mechanics were set down.

These are the postulates that identify measurements and observations with the solutions of the quantum mechanical equations. Postulates for physics theories are extra axioms imposed on the mathematical solutions so that prediction for measurements can be made.

Hilbert spaces etc are a mathematical formalism created so that one need not refer to the specific solutions of each quantum mechanical equation ( dirac, klein gordon, quantized maxwell) entering a problem, and be able to calculate the crossection of an interaction using pertubation theory and Feynman diagrams..

Thus you have to have a background on why quantum mechanics was necessary, and how the postulates developed , and not get landed immediately into Hilbert spaces, which are a very useful formalism but depend on the underlying quantum mechanical postulates and whose validity rests on : it describes the data and predicts new set ups.