The Spectral Theorem in Quantum Mechanics

Patrick Reany

26 August 2025

Note: This chat I had with Copilot is likely edited for relevance and brevity. (Naturally, I get to correct my spelling and grammar errors for free.)

Patrick:
I'm reading through my notes on advanced quantum mechanics from a xxx lecture. He wrote the following \begin{equation} \frac{\partial\psi(t)}{\partial t} = - i H\psi(t)\,,\label{eq:SchrodingEq} \end{equation} which is the time-dependent Schrodinger Equation and $H$ is recognized as the Hamiltonian of the system.

Copilot:
You're absolutely right --- that equation is the time-dependent Schrodinger equation in natural units ($\hbar$ = 1), and it's the cornerstone of non-relativistic quantum mechanics. Let's unpack it a bit to highlight its significance and structure:

What the Equation Means \begin{equation} \frac{\partial\psi(t)}{\partial t} = - i H\psi(t)\,,\label{eq:SchrodingEq1} \end{equation}

• $\psi(t)$: The state vector (or wavefunction) of the quantum system at time $t$. It lives in a Hilbert space.
• H: The Hamiltonian operator, which encodes the total energy (kinetic + potential) of the system.
• $-i$: The imaginary unit, ensuring unitary time evolution (preservation of probability).
• $\partial\psi/\partial t$: The rate of change of the quantum state over time.

Time Evolution and Unitarity

This equation tells us how a quantum state evolves in time. If the Hamiltonian is time-independent, the solution is: \begin{equation} \psi(𝑡)=𝑒^{-𝑖𝐻𝑡}\psi(0) \end{equation} Here, $𝑒^{-𝑖𝐻𝑡}$ is a unitary operator, which guarantees that the norm of the wavefunction remains constant --- a reflection of probability conservation.

Connection to Observables

Since $𝐻$ is self-adjoint (Hermitian), its eigenvalues are real --- which is essential because they correspond to measurable energy levels. The eigenstates of $𝐻$ form a basis in which any state $\psi(𝑡)$ can be expressed as a superposition: \begin{equation} \psi(𝑡)= \sum_𝑛 𝑐_𝑛 𝑒^{-𝑖𝐸_n𝑡}\phi_𝑛 \end{equation} where $\phi_𝑛$ are the eigenstates of $H$ and $𝐸_𝑛$ are the corresponding eigenvalues.

Philosophical Footnote

This equation is more than just a dynamical law --- it encodes the entire logic of quantum evolution. Unlike classical mechanics, where trajectories are deterministic paths in phase space, quantum mechanics describes evolution in terms of probability amplitudes that interfere and evolve according to this linear, unitary rule. Would you like to explore how this equation changes in the Heisenberg picture, or how it generalizes in quantum field theory or curved spacetime? Or perhaps how it connects to the path integral formulation?

Patrick:
Let's call the first equation I presented as 'Eq 1'. Then xxx transitioned to the time-independent condition, and presented a 'plot hole' for me to fill: Okay, what about the time-independent Schrodinger Equation? In this case, let $|\,\psi\,\rangle$ be an eigenvector of $H$ then (let's call the following 'Eq 2') \begin{equation} H\,|\,\psi\,\rangle = E\,|\,\psi\,\rangle\,, \end{equation} where $E$ is the corresponding eigenvalue of $H$.......Why do I say this? Well, if we set $\frac{\partial\psi(t)}{\partial t}=0$ from Eq 1, we don't get it, right? Or if we do, I didn't follow the 'logic'.

Copilot:
Right --- simply setting \begin{equation} \frac{\partial\psi(t)}{\partial t} = 0 \end{equation} in Eq 1 is not how we get Eq 2. That would force the state to be completely time‑independent in the Schrodinger picture, and the only way that happens in Eq 1 is if $H\psi =0$, i.e. $E=0$ --- a much stronger and less interesting condition than the general time‑independent Schrodinger equation.

The actual logical bridge from Eq 1 to Eq 2 uses a stationary‑state ansatz, not the derivative‑equals‑zero shortcut. Here's the clean sequence:

1. Start from Eq 1 in natural units ($\hbar=1$) \begin{equation} \frac{\partial}{\partial t}\psi(t) = -iH\psi(t) \end{equation}

2. Look for separable solutions

Assume $\psi(t)$ has the form of a fixed spatial state multiplied by a pure phase in time: \begin{equation} \psi = e^{-iEt} \phi \end{equation}  Here $E$ is a constant, and $\phi$ is time‑independent in this picture.

3. Plug this ansatz into Eq 1 \begin{equation} \frac{\partial}{\partial t}\psi(t) = (-iE)e^{-iEt} \phi \end{equation} The right‑hand side of Eq 1 is: \begin{equation} -iH(e^{-iEt} \phi) = -ie^{-iEt} (H\phi) \end{equation}

4. Cancel the common factor $ -ie^{-iEt}$

You get: \begin{equation} E \phi = H\phi \end{equation}

5. Recognize the eigenvalue equation (Eq 2) \begin{equation} H \phi = E\phi \end{equation} So Eq 2 emerges from assuming a stationary state with a single definite energy $E$. That's what's meant by "time‑independent Schrodinger equation": it's the spatial (or configuration‑space) part of a solution whose only time dependence is a phase $𝑒^{-𝑖𝐸𝑡}$.

Where the "plot hole" comes from

If you literally set $\partial_𝑡 \psi(t) = 0$ in Eq 1, you're asserting $𝐻\psi = 0$, i.e. a zero‑energy eigenstate --- hardly the general case. The usual move is instead:

1. Propose $\psi(𝑡)=𝑒^{-iEt}\phi$ (energy eigenstate form).
2. Substitute and deduce $𝐻\phi=𝐸\phi$.

If you'd like, I can also diagram the parallel roles here:

• Eq 1: Full time dynamics for any state.
• Ansatz: "Stationary" means probability densities are time‑independent $\rightarrow$ only a phase evolves.
• Eq 2: The constraint $H\phi = E\phi$ that selects those stationary states.

That visual map makes it hard to fall into the "just set the derivative to zero" trap. Want me to sketch that diagram for you?

Patrick:
In other words, the time-independent 'equation' has been in Eq1 all along, and one just needs to assume a separation of variables, as you showed, to reveal it?

Copilot:
Exactly --- the "time‑independent Schrodinger equation" isn't some separate law you bolt on later, it's already latent within Eq 1's structure.

What xxx (and most derivations) are doing is:

• Eq 1 is the universal time‑evolution law.
• If you assume the state is a stationary energy eigenstate --- i.e. its time dependence is only a global phase --- you're effectively doing a separation of variables: \begin{equation} \psi(𝑡)=𝑒^{-iEt}\phi \end{equation} Plugging that form back into Eq 1 extracts the purely spatial/eigenvalue condition $H\phi=E\phi$ without any extra physics being added.

So the time‑independent form is just the "eigenvalue slice" of the more general time‑dependent law, revealed once you make that phase‑factor assumption. In spectral terms, you're resolving the dynamics into the Hamiltonian's eigenmodes, each of which evolves trivially in time by picking up its own phase.

Patrick:
Okay, but then this 'reasonable' separation of variables equation $\psi(t)=𝑒^{-𝑖𝐸𝑡 }\phi$ has to be justified. Is this a postulate of quantum mechanics?

Copilot:
Not quite --- it isn't a postulate of quantum mechanics in the same sense as, say, "observables are represented by Hermitian operators" or "measurement outcomes are eigenvalues."

Instead, it's a mathematical consequence of two ingredients that are postulates:

1. The Schrodinger equation itself (Eq 1) --- the postulate about how states evolve in time.
2. The spectral nature of the Hamiltonian --- another postulate says that physical observables (including the Hamiltonian) are represented by Hermitian operators, whose eigenvalues are real and whose eigenvectors form a basis.

Where the separation ansatz comes from

From those postulates, you can reason:

If the system is in an energy eigenstate $H\phi=E\phi$, then $\phi$ is constant in the Schrodinger picture except for an overall phase.
• The time evolution law, $\frac{\partial}{\partial t} \psi(t) = -iH\psi(t)$, applied to such an eigenstate tells you exactly what that phase must be: $𝑒^{-iEt}$.

So, the "separation" $\psi(𝑡)=𝑒^{-iEt}\phi$ is not an independent axiom --- it's the explicit form of the solution when the state is an eigenvector of $H$.

Why it's "reasonable"

In many problems we look for stationary states, meaning states whose probability densities $|\psi|^2$ are time‑independent. That requirement forces the time dependence into a pure phase factor. And that's exactly what the separation ansatz expresses.

For a time‑independent $H$, the full solution can then be built from superpositions of these stationary modes, each carrying its own phase.

If you'd like, I could walk you through how this same separation logic still works --- but in a more subtle way --- when $H$ depends explicitly on time, such as in driven or periodic systems. That's where Floquet theory and quasi‑energies enter the stage.

Patrick:
Okay, great! The 'separation ansatz' works just fine to express stationary states (i.e., the results are consistent with experiment). Now, can we advance from this 'reasonable ansatz' to a necessary requirement of quantum mechanics? That would really clear things up for me.

Copilot:
Yes --- you can promote the "reasonable ansatz" to something necessary once you reframe it as:

Any solution of the time-dependent Schrodinger equation with a time‑independent Hamiltonian can be expanded in a basis of its energy eigenstates, and each such eigenstate evolves by a pure phase factor.

1. Start from the core postulates

• Dynamics postulate: States evolve via \begin{equation} i\hbar\frac{\partial}{\partial t}| \psi(𝑡)\rangle= H|\psi(t)\rangle \end{equation}
• Observable postulate: $𝐻$ is Hermitian, so it has a complete orthonormal set of eigenvectors $\{|𝐸_𝑛\rangle\}$ with real eigenvalues $\{𝐸_𝑛\}$.

2. Solve in the Hamiltonian's eigenbasis

Expand any state as \begin{equation} | \psi(t)\rangle= \sum_n c_n(t) |E_n\rangle \end{equation} Plug into the Schrodinger equation. Orthogonality of the $|𝐸_𝑛\rangle$ decouples the coefficients: \begin{equation} i\hbar\frac{d}{d t} c_n(t) = E_n c_n(t) \end{equation}

3. Integrate exactly

This is a first‑order ODE with solution: \begin{equation} 𝑐_𝑛(𝑡)=𝑐_𝑛(0) 𝑒^{- 𝑖𝐸_𝑛𝑡/\hbar} \end{equation}

4. Recognize the "ansatz" as the special case

If the system is already in a single eigenstate $|𝐸\rangle$, then only one term survives: \begin{equation} |\psi(𝑡)\rangle= 𝑒^{- 𝑖𝐸𝑡/\hbar}|E\rangle \end{equation} No freedom left --- the phase form is forced by the postulates and linear algebra, not guessed.

Why this elevates the ansatz

• The separation of variables you started with isn't an independent hypothesis; it's the only possible form consistent with:
  • Linearity of the Schrodinger equation
  • Completeness of $H$'s eigenbasis
  • Time‑independence of $H$
• Experiment doesn't just validate it --- the whole Hilbert‑space machinery demands it.

If you like, I can also show you the parallel structure in the Heisenberg picture, where the necessity shows up as a simple harmonic evolution of operators' matrix elements between energy eigenstates. That perspective makes the "it must be a phase" point even more transparent. Would you like me to walk through that lens next?

Patrick:
So, your rephrasing claim above is actually a mathematical theorem.

Copilot:
Exactly --- it's not just a nicely phrased hunch, it's a rigorously provable statement within the standard axioms of nonrelativistic quantum mechanics.

The "separation ansatz" for stationary states is really the spectral decomposition theorem in action, applied to the Hamiltonian as a self‑adjoint operator on a Hilbert space, combined with the uniqueness of solutions to the time‑dependent Schrodinger equation.

Here's the skeleton of the theorem in functional‑analytic terms:

• Given: $H$ self‑adjoint, time‑independent on a Hilbert space $H$.
• Then: The spectral theorem ensures there is an orthonormal basis (discrete spectrum) or an orthogonal resolution of the identity (continuous/mixed spectrum) in terms of eigenkets $|𝐸\rangle$ of $H$.
• Moreover: For each $|𝐸\rangle$, the Schrodinger equation reduces to $\dot\psi=-(i/\hbar)𝐸\psi$, whose unique solution is a global phase $e^{-iEt/\hbar}$.
• Thus: Any $|\psi(t)\rangle$ evolves as a superposition of such phase‑rotating eigencomponents.

No physical guesswork survives this derivation --- the ansatz form $\psi(𝑟,𝑡)=\phi_E(𝑟)𝑒^{-iEt/\hbar}$ is an invariant consequence of the Hilbert‑space postulates and the time‑independence of $H$.

If you like, I can also outline how the proof adapts seamlessly to the continuous spectrum using projection‑valued measures. That generalization makes it absolutely clear that "stationary state = pure phase time evolution" holds for all legitimate energy eigenstates, bound or scattering alike. Would you like me to expand it that way?

Patrick:
I think that one of the problems of approaching QM with the mindset to "shut-up and calculate," if taken to an extreme, is the likelihood of retaining a lot of 'plot holes' in the logic of one's conception of QM.

Copilot:
That's a perceptive diagnosis.

The "shut up and calculate" posture works brilliantly if your only goal is to generate predictions --- it's pragmatism distilled to its most austere form. But when taken as a full philosophy, it can leave whole stretches of the conceptual scaffolding in shadow. You still get the numbers right, but the story your mind builds around those numbers can be riddled with gaps, contradictions, or vague placeholders you never press on.

In physics, those gaps are a bit like unobserved but consequential variables:

• Unexamined assumptions --- e.g., why separation of variables "just works" in certain cases --- can hide the fact it's a theorem, not a guess.
• Conflations of concepts --- "wavefunction" as both calculational tool and ontological entity --- slip by unchallenged.
• Missed generalizations --- you might master hydrogen‑atom stationary states but never see how the same reasoning structure runs through scattering theory or field quantisation.

One productive antidote is to make internal completeness part of your personal standard: never just accept a calculational method until you've located its anchoring postulate, proved its necessity where possible, and mapped its domain of validity. That way the "calculation" and the "story" reinforce each other --- no dangling threads in the narrative.

Note: We continue the next day. 27 August 2025.

Patrick:
I would now like to pursue the direction you indicated when you said, "If you like, I can also outline how the proof adapts seamlessly to the continuous spectrum using projection‑valued measures."

Copilot:
Statement

For a time-independent, self-adjoint Hamiltonian \(H\) on a Hilbert space \(\mathcal{H}\), every solution of the Schrodinger equation \(i\hbar\,\partial_t |\psi(t)\rangle =H|\psi(t)\rangle\) evolves by spectral phases: \[ |\psi(t)\rangle \;=\; e^{-\,\tfrac{i}{\hbar}Ht}\,|\psi(0)\rangle \;=\; \int_{\mathbb{R}} e^{-\,\tfrac{i}{\hbar}E t}\, dP(E)\,|\psi(0)\rangle. \] Here \(P\) is the projection‑valued measure (PVM) of \(H\). In the energy representation (a direct-integral decomposition), time evolution acts as pointwise multiplication by \(e^{-iEt/\hbar}\). The "stationary-state phase" is thus necessary and exact for both discrete and continuous spectra.

Spectral theorem and PVM essentials

• Self-adjointness and PVM. A self-adjoint \(H\) admits a unique PVM \(P:\mathcal{B}(\mathbb{R})\to \mathrm{Proj}(\mathcal{H})\) such that \[ H \;=\; \int_{\mathbb{R}} E\, dP(E), \] where the integral is understood in the strong sense (via the Borel functional calculus).
• Functional calculus. For any bounded Borel \(f:\mathbb{R}\to\mathbb{C}\), \[ f(H) \;=\; \int_{\mathbb{R}} f(E)\, dP(E). \] In particular, choosing \(f(E)=e^{-iEt/\hbar}\) defines the unitary group \(U(t)=e^{-\,\tfrac{i}{\hbar}Ht}\).
• Spectral measure of a vector. For \(|\psi\rangle\in\mathcal{H}\), define \(\mu_\psi(B)=\langle \psi,\, P(B)\psi\rangle\). Then for bounded \(f\), \[ \langle \psi,\, f(H)\psi\rangle \;=\; \int_{\mathbb{R}} f(E)\, d\mu_\psi(E). \]

Time evolution as a spectral integral

• Unitary group. \[ U(t) \;=\; e^{-\,\tfrac{i}{\hbar}Ht} \;=\; \int_{\mathbb{R}} e^{-\,\tfrac{i}{\hbar}E t}\, dP(E). \]
• Evolved state. \[ |\psi(t)\rangle \;=\; U(t)|\psi(0)\rangle \;=\; \int_{\mathbb{R}} e^{-\,\tfrac{i}{\hbar}E t}\, dP(E)\,|\psi(0)\rangle. \] This is a Bochner (strong) integral in \(\mathcal{H}\) and is well defined for all \(t\in\mathbb{R}\).
• Energy probabilities are constants of motion. Using \(U(t)P(B)U(t)^\dagger=P(B)\), \[ \langle \psi(t),\, P(B)\psi(t)\rangle \;=\; \langle \psi(0),\, P(B)\psi(0)\rangle, \] so the energy distribution \(\mu_{\psi(t)}\) is time-invariant.

Energy representation via direct integrals

• Decomposition. There exists a measure space \((\mathbb{R},\mu)\) and a measurable field of multiplicity spaces \(\{\mathcal{K}_E\}_{E\in\mathbb{R}}\) such that \[ \mathcal{H} \;\cong\; \int_{\mathbb{R}}^{\oplus} \mathcal{K}_E\, d\mu(E), \quad (H\psi)(E)= E\,\psi(E). \]
• Time evolution is fibrewise phase. \[ (U(t)\psi)(E) \;=\; e^{-\,\tfrac{i}{\hbar}E t}\,\psi(E), \quad \text{for almost every } E. \] Degeneracy is carried by the fibre \(\mathcal{K}_E\); the phase multiplies the entire fibre trivially.

• Discrete vs. continuous spectrum. Discrete eigenvalues appear as atoms of \(\mu\) (finite-dimensional fibres). Absolutely continuous parts give \(L^2\)-fibres; singular continuous parts are also handled without change in the formula.

Generalized eigenstates and rigged Hilbert spaces

• Generalized eigenvectors. For continuous spectrum points \(E\), honest eigenvectors need not lie in \(\mathcal{H}\). In a rigged Hilbert space \(\Phi \subset \mathcal{H} \subset \Phi^\times\), one can admit generalized kets \(|E,\alpha\rangle\in \Phi^\times\) with \[ H|E,\alpha\rangle \;=\; E|E,\alpha\rangle \] in the distributional sense; \(\alpha\) labels degeneracy.
• Spectral resolution with generalized kets. Heuristically, \[ I \;=\; \sum_{n}|E_n\rangle\!\langle E_n| \;+\; \int_{\sigma_c(H)} \sum_{\alpha}|E,\alpha\rangle\!\langle E,\alpha|\, dE, \] turning the PVM into the familiar "sum + integral" form. Time evolution multiplies each generalized eigencomponent by \(e^{-iEt/\hbar}\).
• Stationary vs. spreading. A pure energy component is "stationary" in the spectral sense (only a phase changes). In position space, continuous-spectrum wave packets spread, but their energy distribution and each spectral fibre’s norm remain invariant.

Consequences and examples

• Necessity of the phase law. In any spectral representation of \(H\), the Schrodinger equation reduces fibrewise to \[ i\hbar\, \partial_t \psi(E,t) \;=\; E\, \psi(E,t) \quad\Rightarrow\quad \psi(E,t) \;=\; e^{-\,\tfrac{i}{\hbar}E t}\,\psi(E,0). \] The "ansatz" is thus the only possible evolution compatible with self-adjoint \(H\).
• Free particle in 1D. With \(H=\tfrac{\hat p^2}{2m}\), the energy spectrum is continuous, labelled by \(k\) with \(E=\tfrac{\hbar^2 k^2}{2m}\). In momentum space, \[ \tilde\psi(k,t) \;=\; e^{-\,i \tfrac{\hbar k^2}{2m} t}\,\tilde\psi(k,0), \] i.e., phase multiplication per energy (or momentum) mode; position-space spreading follows from Fourier inversion.
• Degeneracy. If \(\dim\mathcal{K}_E>1\), then \(U(t)\) acts as \(e^{-iEt/\hbar} I_{\mathcal{K}_E}\) on that fibre. Internal rotations in \(\mathcal{K}_E\) do not occur unless symmetry-breaking terms are added to \(H\).
• Singular continuous spectrum. Even in exotic cases (e.g., quasi-periodic potentials), the PVM still yields \(U(t)=\int e^{-iEt/\hbar} dP(E)\); the same phase law holds pointwise \(\mu\)-a.e.
• Stone's theorem (backstop). The strongly continuous unitary group \(U(t)\) determines a unique self-adjoint generator \(H\) with \(U(t)=e^{-iHt/\hbar}\). The spectral calculus then enforces the phase evolution.

If you'd like, we can make this concrete by constructing the direct-integral decomposition for a specific scattering Hamiltonian (e.g., short-range potential in 1D), show how the Moller wave operators intertwine the spectral measures, and watch the phase evolution ride along the absolutely continuous spectrum.

Patrick:
Can you explain your statment: Self-adjointness and PVM (projection valued measure). A self-adjoint \(H\) admits a unique PVM \(P:\mathcal{B}(\mathbb{R})\to \mathrm{Proj}(\mathcal{H})\) such that \[ H \;=\; \int_{\mathbb{R}} E\, dP(E), \]

Copilot:
Let's unpack that statement and connect it to the precise functional‑analytic machinery.

Self‑Adjointness as the Starting Point

In quantum mechanics, observables --- in particular the Hamiltonian \(H\) --- are represented by self‑adjoint operators on a Hilbert space \(\mathcal{H}\). Self‑adjoint means:

• \(H\) is densely defined and \(H = H^\dagger\) (equal to its adjoint).
• Domain issues are handled so that \(\langle \psi, H\phi \rangle = \langle H\psi, \phi \rangle\) for all \(\psi,\phi\) in the domain of \(H\), and the adjoint has the same domain.
• Physically: eigenvalues are real; mathematically: the spectral theorem applies in its strongest form.

The Projection‑Valued Measure (PVM) The spectral theorem tells us:

Every self‑adjoint operator \(H\) has a unique projection‑valued measure \(P : \mathcal{B}(\mathbb{R}) \to \mathrm{Proj}(\mathcal{H})\) where \(\mathcal{B}(\mathbb{R})\) = Borel sets in \(\mathbb{R}\).

• Countable additivity in the strong operator topology: \(P\left(\bigcup_n B_n\right) = \sum_n P(B_n)\) (orthogonal sum).
• Projection property: \(P(B)^2 = P(B) = P(B)^\dagger\).
• Orthogonality: \(P(B_1)P(B_2) = 0\) if \(B_1 \cap B_2 = \varnothing\).

The Spectral Integral

The theorem says: \[ H \;=\; \int_{\mathbb{R}} E \, dP(E) \]

* The integral is defined in the strong sense (Bochner integral), via the Borel functional calculus: \[ f(H) = \int_{\mathbb{R}} f(E) \, dP(E) \] for any bounded Borel function \(f\).

* This is not a Riemann‑style integral but an operator‑valued integral where integration "against \(dP(E)\)" means projecting onto the spectral subspaces associated with energies in infinitesimal neighborhoods of \(E\).

Physical Meaning

• \(P(B)\) = projection onto the subspace of states whose energy lies in the set \(B \subset \mathbb{R}\).
• Given a normalized state \(\psi\), the scalar measure \[ \mu_\psi(B) = \langle \psi, P(B) \psi \rangle \] is the probability that a measurement of \(H\) yields a value in \(B\).
• The decomposition \(H = \int E\, dP(E)\) generalizes "sum over eigenvalues" to continuous and mixed spectra:
• Pure point spectrum: becomes \(\sum_n E_n \, |\varphi_n\rangle\langle\varphi_n|\)
• Continuous spectrum: becomes \(\int E\, |E\rangle\langle E|\,dE\) (rigorously: generalized eigenstates in a rigged Hilbert space).

Time Evolution from the PVM For a time‑independent \(H\), \[ U(t) = e^{-iHt} = \int_{\mathbb{R}} e^{-iEt} \, dP(E). \] This makes the "pure phase per energy component" evolution exact:

• Each spectral subspace of energy \(E\) gets multiplied by \(e^{-iEt}\).
• This is the rigorous underpinning for the "stationary state" idea.

Why "Unique" Matters

The PVM \(P\) is uniquely determined by \(H\):

• If two PVMs \(P\) and \(Q\) give \(H = \int E\, dP(E) = \int E\, dQ(E)\), then \(P(B) = Q(B)\) for all Borel \(B\).
• This ensures that the "energy measurement statistics" are an invariant property of \(H\).

If you like, I can walk through the proof sketch of why self‑adjointness guarantees the existence and uniqueness of this \(P\) --- via Stone's theorem for one‑parameter unitary groups plus the spectral theorem for commuting self‑adjoint operators --- or make it visual with a discrete‑vs‑continuous spectrum diagram showing how sums turn into integrals.