In physics we often don’t have the luxury to work with a discrete number of dimensions or values. In these cases, our vector spaces become infinite dimensional. But how can we deal with a countably infinite number of basis vectors? Here we will show how we can take the notion of countably infinite vector spaces and demonstrate how we can take a discrete vector space to a continuous one that we can describe with functions.

Hilbert Space

<aside> 💡 Hilbert Space: Infinite dimensional inner-product space that is complete (i.e. with enough structure that bases exists)

→ This structure comes in the form of Cauchy completeness

</aside>

• More on the structure

A Hilbert Space is thus equipped with

• An inner product: $\braket{\cdot|\cdot}$

  • The inner product allows for the definition of a distance, $d(f,g)=\sqrt{\braket{f-g|f-g}}$ → Distances have the following properties
    • $d(f,g)=0$ if and onlt if the $f,g$ are the same and positive otherwise (positive definite)
    • Symmetric in arguments: distance from f to g is the same as g to f
      • We can also pull out scalar multiples
    • Satisfies the triangle inequality: the sum of the norms of $f$ and $g$ bound the vector sum: $||f+g||\le||f||+||g||$
  • Distances give use a way to interpret limits and convergence and define completeness

• Complete: Giving us the ability to write our vectors with a set of bases $\ket{e_n}$

  $$ \ket v=\sum_{n=1}^\infin v_n\ket e_n $$

  • $L^2(\R)$: The Space of Square Integrable Functions ($\psi:\R\to\Complex$) is an example of an infinite dimensional complete Inner-product space.

    • A slight caveat is that the notation $L^2(\R)$ is really $\mathcal{L}^2(\R)/\mathcal N$ the set of equivalence relations that is modulo measureable functions that vanish. Another way to say this is that it’s the quotient of $\mathcal{L}^2(\R)$ (square integrable functions) by the kernel of our inner-product norm
      • If we consider just square integrable functions, there are some functions with a norm of $0$, but we require the inner product to only be $0$ for the zero vector (it violates positive definiteness)
      • If we define an equivalence relation $f\sim g$ between functions that agree “almost everywhere” (specifically, they differ on a set of Lebesgue measure zero) and the space of equivalence classes (functions all in the same equivalent class correspond to the same state) under this equivalence relation is $L^2(\R)$.
      • Physically this makes sense: ultimately in Quantum Mechanics the value of $|\psi(x)|^2$ is a probability density which we can’t directly measure. You can only measure the probability $P(I)=\int_I|\psi(x)|^2dx$. Two wavefunctions that only differ on a set of Levesgue measure zero will give the same probabilty and thus the same results. Similar to how changing the phase of our states ($e^{i\theta}$) does not ultimately change our physically measurable quantities
    • The uncountable $\ket x$ delta-function basis that we’ll introduce it too “fine” and picks out all these irrelevant unphysical degrees of freedom
      • Ultimately we restrict the space to vectors of finite length and scalar products

  • $\ell^2(\N)=\bigg\{(x_n){n\in\N}\bigg|\sum{n\in\N}|x_n|^2<\infty\bigg\}$ the space of all square integrable complex sequences is another example of an infinite dimensional complete inner product space

    • Inner-product:

      $$ \braket{g|f}=\sum_{n=1}^\infty g^*_nf_n $$

  • It can be shown that all infinite-dimensional, separable, complex Hilbert spaces are isomorphic to the set of square integrable complex sequences $\ell^2(\N)$

Notice that we now have countably infinte sum of basis vectors to deal with.

Functions as an Infinite Dimensional Hilbert Space:

You can go through all the axioms for a vector space defined in Vector Spaces for functions and you will see that functions can form a valid vector space where the functions serve as the abstract vector object. Thinking of functions as vectors at first seems somewhat confusing. We’re used to vector-valued functions like $\vec f(x,y,z)=x^2\hat i+(y+z)\hat j$ for instance but otherwise we think of vectors and functions and separate ideas. But we will come to see that we can think of functions as a continuous analogue to vectors.

Functions as Continuous Dimensional Vectors

Let’s first consider the infinite dimensional Hilbert Space where we have a countable infinite basis, indexed by integers $n$, that allows us to write vectors as described above with an infinite sum.

Consider a map of integers, $n$, onto an interval $I$ of the real line:

$$ n\to x_n~~~n\in \Z,x\in\R $$

$I$ can be a finite interval $[a,b]$ or infinite $(-\infin,\infin),(-\infin,a],[a,\infin)$

Let us assume that the spacing between $x_n$ values is equal, $\Delta x$

Now consider a vector in our hilbert space, $\ket f$: