Superposition Revisited

Superposition is a linear operation on state representatives, followed by passage to the nonzero result’s ray. In infinite dimensions, an infinite superposition is also a convergence claim. Square-summable coefficients suffice for an orthonormal expansion to define a state, but they need not put that state in the domain of its Hamiltonian. This page isolates those mathematical qualifications; the physical meaning of relative phase and interference is developed in Superposition and Relative Phase.

Required background. Superposition and Relative Phase separates coherent sums from mixtures. Hilbert Space as State Space supplies norm convergence and orthonormal expansions.

Finite sums require chosen representatives

For vectors $\psi,\phi\in\mathcal H$, the vector $a\psi+b\phi$ exists for every $a,b\in\mathbb C$. It defines a state only when it is nonzero. Its squared norm is

$$\|a\psi+b\phi\|^2 =|a|^2\|\psi\|^2+|b|^2\|\phi\|^2 +2\operatorname{Re}\bigl(\overline a b\langle\psi,\phi\rangle\bigr).$$

The coefficient squares are therefore not automatically probabilities of exclusive alternatives. That interpretation requires an orthonormal decomposition or an explicitly specified measurement.

The input rays alone do not specify the output ray. Replacing $\phi$ by $e^{i\theta}\phi$ while keeping $a,b$ fixed generally changes the sum. To express the same vector in the rephased representatives, one must also replace $b$ by $e^{-i\theta}b$. This is the coordinate consistency condition behind a coherent preparation, not an exception to global-phase invariance.

Even the existence of a mathematical sum does not establish that a particular restricted observable algebra can detect its coherence. That separate issue is treated in Direct Sums and Superselection.

Infinite orthonormal expansions

Let $\{e_n\}_{n\geq1}$ be an orthonormal sequence, and let

$$s_N=\sum_{n=1}^{N}c_ne_n.$$

For $M>N$, orthogonality gives

$$\|s_M-s_N\|^2=\sum_{n=N+1}^{M}|c_n|^2.$$

Hence $s_N$ converges in Hilbert norm exactly when $(c_n)\in\ell^2$. Completeness supplies the limit, and orthogonality gives its norm. If the sequence is an orthonormal basis, every state admits such an expansion. If it is not complete, only the closed span is represented.

These series converge unconditionally: every rearrangement has the same norm limit, because the squared tail can be made arbitrarily small outside a sufficiently large finite set. This statement depends on orthogonality. For a general sequence of normalized, nonorthogonal vectors, square-summable coefficients alone need not make the vector series converge. Taking every vector equal to a fixed unit vector and $c_n=1/n$ gives an immediate counterexample.

A formal continuous-basis integral is different again. Position and momentum kets are generally generalized vectors, not members of an uncountable orthonormal basis in $\mathcal H$. The Schrödinger Representation explains the Hilbert-space operators underlying that notation.

An admissible state outside the energy domain

On $\ell^2(\mathbb N)$ define the self-adjoint diagonal Hamiltonian

$$He_n=E_0n\,e_n, \qquad D(H)=\left\{\sum_n c_ne_n:\sum_n n^2|c_n|^2<\infty\right\},$$

with fixed energy scale $E_0>0$. Every finite sum belongs to $D(H)$. Nevertheless,

$$\psi=C\sum_{n\geq1}\frac1n e_n, \qquad C=\frac{\sqrt6}{\pi},$$

is normalized and lies outside $D(H)$. Its finite truncations converge in norm, while the norms of their Hamiltonian images diverge. Even the mean energy diverges, since the corresponding sum is $E_0C^2\sum_n1/n$.

There is no inconsistency: a Hilbert-space state need not have every energy moment. The unitary $e^{-itH/\hbar}$ acts on $\psi$ for every $t$, but its orbit is not differentiable in Hilbert norm at zero. Treating the formal termwise derivative as an $\ell^2$ vector would be the invalid step.

A finite mean energy does not itself restore membership in $D(H)$. The moment example in Measure Theory Minimum constructs a state in the form domain $D(H^{1/2})$ but outside $D(H)$, separating first-moment control from the stronger second-moment condition.

Graph norms preserve operator information

For a closed operator $A$, the graph norm is

$$\|\psi\|_A^2=\|\psi\|^2+\|A\psi\|^2, \qquad \psi\in D(A).$$

When units matter, scale the second term by an appropriate fixed quantity. Convergence in this norm means convergence both of the state and of its operator image. Closedness then ensures that the limit remains in $D(A)$ and that the image limit is $A\psi$.

For the diagonal Hamiltonian, coefficients proportional to $n^{-2}$ produce graph-norm-convergent truncations: the Hamiltonian-image tail is controlled by $\sum_{n>N}n^{-2}$. This gives a practical lesson for expansions and numerical approximations. State convergence, energy-form convergence, and convergence of the Hamiltonian action are distinct targets. The observable being approximated determines which one is needed.

Exercise: a bounded test and an unbounded test

For the normalized $c_n=C/n$ state above, let $P_N$ project onto the first $N$ basis vectors. Explain why every bounded observable has a convergent expectation along normalized $P_N\psi$, even though the energy expectation diverges.

Solution

The normalized projections converge in Hilbert norm to $\psi$. For any bounded $B$, the difference of expectations is at most $2\|B\|$ times the vector-norm error, so it tends to zero. For energy, the normalized truncated expectation is

$$E_0\frac{\sum_{n=1}^{N}n^{-1}}{\sum_{n=1}^{N}n^{-2}},$$

which diverges logarithmically. The denominator approaches a finite nonzero limit. There is no bounded operator norm $\|H\|$ with which to apply the preceding estimate.

References

• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013 — unbounded observables and unitary dynamics.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, Chapters 1–3 — orthonormal series, closed operators, and spectral domains.