3.3 Hilbert space essentials

The previous two sections have leaned on “states are vectors” throughout, but a few things were stepped around:

Does the infinite series $\ket{\psi}=\sum_n c_n\ket{n}$ converge? Converge to what?
$\ket{x}$ cannot be normalised — what entitles it to serve as a basis?
What exactly does “complete” mean?

This section fills those in. Only the parts that affect physical conclusions — for rigorous functional analysis, see a mathematics text.

What the space of states must satisfy

The picture

Condition one: it is a linear space.

States can be added and multiplied by complex numbers, and the result is again a state. This comes straight from the linearity of the Schrödinger equation, and is the mathematical statement of the superposition principle.

Condition two: it has an inner product.

$\braket{\phi}{\psi}$ is defined and satisfies the three properties of the previous section. Without an inner product there is no normalisation, no orthogonality, no probability — the Born rule would have nowhere to live.

Condition three: it is complete (in the analyst’s sense).

Every Cauchy sequence converges to an element of the space.

The mathematics

Definition

A space satisfying all three is a Hilbert space, usually written $\mathcal H$ .

The first postulate of quantum mechanics (section 3.7) reads:

The state of a physical system is described by a normalised vector in some Hilbert space.

\ket{\psi}\in\mathcal H,\qquad \braket{\psi}{\psi}=1\tag{3.3.1}

Why the third condition matters: a concrete hole

The first two conditions feel natural; the third looks like mathematical fussiness. It is not.

Consider the space of all continuous functions, with the inner product $\int f^*g\,\dd x$ . Take a sequence of ever-steeper continuous functions:

f_n(x)=\begin{cases}0,&x<0\\ nx,&0\le x\le 1/n\\ 1,&x>1/n\end{cases}

In the distance induced by the inner product, $\{f_n\}$ is a Cauchy sequence — its members get ever closer to one another. But the limit is a step function, discontinuous, and not in the space.

The space has a hole. And in quantum mechanics we take limits constantly: infinite series expansions, continuum limits of wave packets, time evolution. If the space has holes, a perfectly legitimate limiting process may produce something that “is not a state”.

$L^2$ plugs those holes, at the cost of containing many discontinuous — even nowhere-differentiable — functions. Physically that costs nothing: real wavefunctions are always smooth, and the extra elements simply guarantee that limits always have somewhere to land.

L²: square-integrable functions

The state space for a single particle in one dimension is

L^2(\mathbb R)=\left\{\psi:\mathbb R\to\mathbb C\ \middle|\ \int_{-\infty}^{\infty}|\psi(x)|^2\dd x<\infty\right\}\tag{3.3.2}

equipped with the inner product $\braket{\phi}{\psi}=\int\phi^*\psi\,\dd x$ .

The picture

What is in $L^2$ , and what is not

In: every bound state, every wave packet, every state that can physically be prepared.

Not in:

plane waves $\ee^{\ii kx}$ ( $\int 1\,\dd x=\infty$ )
$\delta(x-x_0)$ ( $\int\delta^2$ diverges)
constants, polynomials, anything that fails to decay

Note that $\psi\in L^2$ does not require $\psi(x)\to0$ pointwise, nor continuity — only that the integral converges. Physical wavefunctions are additionally required to be smooth enough, but that comes from the Schrödinger equation (which contains second derivatives), not from $L^2$ .

The mathematics

Separability: why the basis can be countable

$L^2$ is separable: it admits a countable orthonormal complete basis.

This property is enormously practical — it guarantees that the “list of numbers” expansion

\ket{\psi}=\sum_{n=1}^{\infty}c_n\ket{n}\tag{3.3.3}

always works, with no need for uncountably many coefficients.

The infinite well’s $\{\psi_n\}$ and the harmonic oscillator’s $\{\psi_n\}$ are both bases of this kind. Any $L^2$ function can be expanded in harmonic-oscillator eigenstates, even one that has nothing to do with the oscillator.

The continuous spectrum: the “eigenstates” that live outside the space

Now for the question we have been dodging.

The solutions of the position operator’s eigenvalue equation $\hat x\ket{x_0}=x_0\ket{x_0}$ are, in the position basis, $\delta(x-x_0)$ . The momentum operator’s eigenstates are plane waves. Neither belongs to $L^2$ .

'Not in the space' does not mean 'useless'

A common misconception: since $\ket{x}$ is not a legitimate state, using it must be wrong.

The right attitude: $\ket{x}$ and $\ket{p}$ are generalised eigenstates — extraordinarily useful bookkeeping devices that represent no preparable physical state.

Analogy: the $\delta$ function is not a function, yet under an integral sign it works beautifully. In the same way, $\ket{x}$ only means something when it gets “integrated away”:

\ket{\psi}=\int\psi(x)\ket{x}\,\dd x\tag{3.3.4}

The left-hand side is a legitimate state; no individual term on the right is. There is no contradiction — just as a finite integral can be built by accumulating infinitely many infinitesimals.

The picture

Normalisation becomes a δ function

Discrete spectrum:

\braket{m}{n}=\delta_{mn}\tag{3.3.5}

Continuous spectrum:

\braket{x}{x'}=\delta(x-x'),\qquad \braket{p}{p'}=\delta(p-p')\tag{3.3.6}

This is called δ normalisation. Formally it parallels the discrete case (Kronecker δ replaced by Dirac δ): every formula carries over, with sums traded for integrals.

The mathematics

Mixed spectra

Some Hamiltonians have both a discrete and a continuous spectrum — the finite well of section 2.8 is one: finitely many bound states at negative energy, plus a continuum of scattering states at positive energy.

The completeness relation then reads

\sum_n\ket{n}\bra{n}+\int\ket{E}\bra{E}\,\dd E=\hat 1\tag{3.3.7}

Hydrogen works the same way: the discrete ladder $-13.6/n^2$ , plus the ionised continuum at $E>0$ .

The rigorous version: rigged Hilbert space

Mathematics has a clean framework for all this, the rigged Hilbert space (also known as a Gelfand triple):

\Phi\subset\mathcal H\subset\Phi'\tag{3.3.8}

$\Phi$ is a smaller space of “very well-behaved functions” (say, rapidly decreasing ones), and $\Phi'$ is its dual, which contains generalised objects like $\delta$ functions and plane waves. The generalised eigenstates live in $\Phi'$ .

This framework makes every formal manipulation above rigorous. You do not need it for physics — the δ-normalisation rules work perfectly well — but it is reassuring to know the whole thing can be made rigorous. Von Neumann sorted out the mathematical foundations of quantum mechanics back in 1932; the physicist’s formal manipulations are not a castle in the air.

Key formulas

Hilbert space
$\text{linear space + inner product + complete in the analyst's sense}$
Where quantum states live; the first postulate
L² space
$L^2=\{\psi:\int|\psi|^2\dd x<\infty\}$
"Normalisable" = "square-integrable", singled out by the Born rule
δ normalisation
$\braket{x}{x'}=\delta(x-x'),\quad\braket{p}{p'}=\delta(p-p')$
Orthonormality, continuous-spectrum edition
Mixed-spectrum completeness
$\sum_n\ket{n}\bra{n}+\int\ket{E}\bra{E}\dd E=\hat 1$
The finite well and hydrogen both need this form

Self-check3 questions

What comes next

The space is ready, and the states live inside it. The next question: what operations can be performed on this space?

The answer is linear operators. And among all linear operators, only a very small class earns the title of “observable” — chapter 2’s unexplained decree begins, from the next section on, to become a derivable conclusion.

Hilbert space essentials

Recommended first

After this section you should be able to

What the space of states must satisfy

The picture

The mathematics

L²: square-integrable functions

The picture

The mathematics

The continuous spectrum: the “eigenstates” that live outside the space

The picture

The mathematics

Key formulas

Self-check3 questions

What comes next

What the space of states must satisfy

◑The picture

∑The mathematics

L²: square-integrable functions

◑The picture

∑The mathematics

The continuous spectrum: the “eigenstates” that live outside the space

◑The picture

∑The mathematics

∎Key formulas

What comes next

The picture

The mathematics

The picture

The mathematics

The picture

The mathematics

Key formulas