I'm currently learning some basic functional analysis. Yesterday I arrived at the spectral theorem of self-adjoint operators. I've heard that this theorem has lots of applications in Quantum Mechanics.
But let me first state the formulation of the theorem that I'm using:
Let $H$ be a Hilbert space. There's a 1-to-1-correspondence between self-adjoint operators $A$ on $H$
and spectral measures $P^{A}$ given by
$$A~=~\int_{\mathbb{R}} \lambda ~dP^{A}.$$
($\lambda$ denotes a constant, $\mathbb{R}$ denotes the real numbers.)
A corollary is:
Let $g:\mathbb{R}\to\mathbb{R}$ be a function. (Again: $\mathbb{R}$ denotes the set of real numbers.) Then:
$$g(A)~:=~\int_{\mathbb{R}} g(\lambda)~ dP^{g(A)}$$
$$P^{g(A)}(\Delta) ~=~ P^{A}(g^{-1}(\Delta))$$
where $\Delta$ denotes a set in the $\sigma$-algebra of $\mathbb{R}$.
Okay. Now this is the theorem. First I don't really the application of the corollary in Quantum mechanics. I've heard that suppose you're given an operator $A$ this means that it's easy for you to define operators like $\exp(A)$, especially on infinite dimensional Hilbert spaces. This indeed could be useful in quantum mechanics. Especially when thinking about the "time-evolution operator" of a system.
However then I say: Why do you make things so complicated?
Suppose you want to calculate $\exp(A)$. Why don't you define
$$\exp(A)~:=~1+A+1/2 A^2 + \ldots $$
and require convergence with respect to the operator norm.
An example: Consider the vectorspace spanned by the monomials $1,x,x^2,\ldots$ and let $A=d/dx$.
Then you can perfectly define
$$\exp(d/dx)~:=~1+ d/dx + 1/2 d^2/dx^2 + \ldots $$
and require convergence with respect to the operator norm.
In addition to that I've heard that the spectral theorem gives a full description of all
self-adjoint operators. Now why is that the case? I mean okay..there's a one to one correspondence between self-adjoint operators and spectral measures…but why does this give me any information about "the inner structure of the operator"? (And why is there this $\lambda$ in the integral? Looks somehow like an eigenvalue of $A$? But I'm just guessing)
I'd me more than happy, if you could provide me with some intuition and ideas
of how the theorem can be used.
Best Answer
It is true that a lot of quantum mechanics can be taught and understood without much knowledge of the mathematical foundations, and usually it is. Since QM is a mandatory class at many faculties that future experimental physicists have to attend, too, this also makes sense. But for future theoretical and mathematical physicists, it may pay off to learn a little bit about the math, too.
A little anecdote: John von Neumann once said to Werner Heisenberg that mathematicians should be grateful for QM, because it led to the invention of a lot of beautiful mathematics, but that mathematicians repaid this by clarifying, e.g., the difference between a selfadjoint and a symmetric operator. Heisenberg asked: "What is the difference?"
That's correct. The benefit of the spectral theorem is that you can define f(A) for any selfadjoint (or more generally, normal) operator for any bounded Borel function. This comes in handy in many proofs in operator theory.
That's correct, too. Spectral measures are much much simpler objects than selfadoint operators, that's why. Futhermore, you can use the spectral theorem to prove that every selfadjoint operator is unitarily equivalent to a multiplication operator (multiply f(x) by x). From an abstract viewpoint, this is a very satisfactory characterization. It does not help much for concrete calcuations in QM, though.
BTW: On a more advanced level, you'll need to understand the spectral theorem to understand what a mass gap is in Yang-Mills Theory (millenium problem).
Hint: In QFT in Minkowski-Spacetime, one usually assumes that there is a continuous representation of the Poincaré group, especially of the commutative subgroup of translations, on the Hilbert space that contains all physical states. The operators that form the representation have a common spectral measure, this is an application of the SNAG-theorem. The support of this spectral measure is bounded away from zero, that's the definition of the mass gap.