Norms and Metrics

A norm measures the length of a vector. A metric measures the distance between two points. In quantum mechanics these ideas appear whenever a state is normalized, a wavefunction is declared square-integrable, a sequence of approximations is said to converge, or a numerical eigenvector is checked by a residual.

The Hilbert-space norm used for state vectors comes from an inner product. The physics of unit-norm states is treated in Normalization; here we develop lengths, distances, and convergence. For the function-space language around integrability and differentiability, see Real Analysis Essentials.

Helpful background. inner products for norms, conjugation, and orthogonality.

Norms

Let $V$ be a vector space over $\mathbb R$ or $\mathbb C$. A norm is a function

$$\lVert\cdot\rVert:V\to\mathbb R$$

such that, for all $u,v\in V$ and scalars $a$,

$$\lVert v\rVert\ge 0, \qquad \lVert v\rVert=0 \text{ if and only if } v=0,$$ $$\lVert av\rVert = \lvert a\rvert\,\lVert v\rVert,$$

and

$$\lVert u+v\rVert \le \lVert u\rVert+\lVert v\rVert.$$

The last property is the triangle inequality. It is the algebraic statement behind the geometric fact that going from the origin directly to $u+v$ is no longer than going first along $u$ and then along $v$.

Metrics

A norm induces a metric by

$$d(u,v)=\lVert u-v\rVert.$$

This distance is nonnegative, symmetric, zero exactly when $u=v$, and satisfies

$$d(u,w)\le d(u,v)+d(v,w).$$

Not every metric comes from a norm, and not every norm comes from an inner product. The Hilbert-space case is special: the inner product gives both lengths and angles.

Norms from Inner Products

If $V$ has an inner product, the associated norm is

$$\lVert v\rVert = \sqrt{\langle v\vert v\rangle}.$$

For the standard inner product on $\mathbb C^n$,

$$\lVert v\rVert_2 = \left( \sum_{j=1}^n \lvert v_j\rvert^2 \right)^{1/2}.$$

In an orthonormal basis this is just the square root of the sum of squared moduli of the coordinates. In a nonorthonormal basis, that shortcut is wrong: the Gram matrix of the basis enters the inner product.

An inner-product norm satisfies the Cauchy–Schwarz inequality,

$$\lvert\langle u\vert v\rangle\rvert \le \lVert u\rVert\,\lVert v\rVert,$$

and the parallelogram identity,

$$\lVert u+v\rVert^2+\lVert u-v\rVert^2 = 2\lVert u\rVert^2+2\lVert v\rVert^2.$$

The parallelogram identity is one way to recognize when a norm really comes from an inner product.

Convergence and Completeness

A sequence $v_n$ converges to $v$ in norm when

$$\lVert v_n-v\rVert\to 0.$$

It is Cauchy when, for every tolerance $\epsilon>0$, there is an $N$ such that

$$\lVert v_n-v_m\rVert\lt\epsilon \qquad \text{for all } n,m\ge N.$$

A normed vector space is complete if every Cauchy sequence has a limit in the space. A complete normed vector space is a Banach space. A complete inner-product space is a Hilbert space.

Completeness is not cosmetic. Wave packets, Fourier expansions, variational approximations, and numerical discretizations often produce sequences. The formal calculation is physically meaningful only after the limiting object still belongs to the intended state space.

L2 Norms for Wavefunctions

For a measure space $X$, the $L^2$ norm is

$$\lVert\psi\rVert_2 = \left( \int_X \lvert\psi(x)\rvert^2\,d\mu(x) \right)^{1/2}.$$

On the real line this becomes

$$\lVert\psi\rVert_2^2 = \int_{-\infty}^{\infty} \lvert\psi(x)\rvert^2\,dx.$$

Normalizable wavefunctions have finite $L^2$ norm; normalized wavefunctions have $L^2$ norm one. The functional-analytic setting is developed in L2 Spaces, while the probability interpretation is developed in Wavefunctions as Representations.

There are other useful function norms, for example $L^1$ and $L^\infty$ norms, but the ordinary pure-state Hilbert-space norm in nonrelativistic wave mechanics is the $L^2$ norm. Different norms can give different notions of convergence in infinite-dimensional spaces. Equivalent norms give the same topology; in a fixed finite-dimensional space all norms are equivalent.

Distances Between Quantum States

For vectors in a Hilbert space,

$$d(\psi,\phi) = \lVert\psi-\phi\rVert$$

is the natural norm distance between chosen representatives. This is useful when comparing wavefunctions or checking a calculation, but a physical pure state is a ray, not a single vector. Multiplying a normalized representative by a global phase does not change the pure state.

For normalized representatives, one phase-insensitive chordal distance between rays is

$$d_{\mathrm{ray}}([\psi],[\phi]) = \min_{\alpha\in\mathbb R} \lVert\psi-e^{i\alpha}\phi\rVert = \sqrt{ 2-2\lvert\langle\psi\vert\phi\rangle\rvert }.$$

To obtain the minimum, expand the squared vector distance:

$$\lVert\psi-e^{i\alpha}\phi\rVert^2 =2-2\operatorname{Re}\!\left(e^{i\alpha}\langle\psi\vert\phi\rangle\right).$$

If the overlap is $r e^{i\theta}$ with $r>0$, choosing $\alpha=-\theta$ makes the real part as large as possible, namely $r$. For orthogonal states $r=0$, every phase gives the same distance $\sqrt2$. The minimization removes only the arbitrary overall phase; it does not erase relative phases within either state.

For instance, $\psi$ and $-\psi$ have vector distance $2$ but ray distance $0$. The Hilbert-space vector distance depends on phase choices, while ray geometry does not. The more geometric version of this idea is introduced in Projective Hilbert Space.

Approximation and Error Estimates

Norms let one state what it means for an approximation to be good. For a bounded operator $A$ with operator norm $\lVert A\rVert_{\mathrm{op}}$, normalized vectors satisfy

$$\left\lvert \langle\psi\vert A\vert\psi\rangle - \langle\phi\vert A\vert\phi\rangle \right\rvert \le 2\lVert A\rVert_{\mathrm{op}}\, \lVert\psi-\phi\rVert.$$

To see where the factor of two comes from, put $\delta=\psi-\phi$ and add and subtract $\langle\phi\vert A\vert\psi\rangle$:

$$\langle\psi\vert A\vert\psi\rangle-\langle\phi\vert A\vert\phi\rangle =\langle\delta\vert A\vert\psi\rangle+\langle\phi\vert A\vert\delta\rangle.$$

Cauchy–Schwarz and $\lVert Av\rVert\le\lVert A\rVert_{\mathrm{op}}\lVert v\rVert$ bound each term by $\lVert A\rVert_{\mathrm{op}}\lVert\delta\rVert$. The two expectation values are unchanged by rephasing either state, so the bound also holds with $d_{\mathrm{ray}}$ on its right-hand side.

For an unbounded observable there is no finite uniform operator norm, and norm convergence of states alone need not imply convergence of its expectation values. For bounded observables, however, this calculation turns a state error into a quantitative prediction error.

In numerical diagonalization, a computed eigenpair $(\lambda,v)$ is checked by the residual

$$r=Hv-\lambda v.$$

The residual norm $\lVert r\rVert$ measures how closely the pair satisfies the eigenvalue equation. Normalize $v$, or use $\lVert r\rVert/\lVert v\rVert$ for $v\ne0$: otherwise shrinking $v$ makes the residual arbitrarily small without improving its direction. Even with this scaling, a small residual does not identify an individual eigenvector without information about spectral separation. The relevant numerical page is Matrix Diagonalization.

Worked Example

Let

$$\chi = \begin{pmatrix} 1\\ 1+i \end{pmatrix} \in\mathbb C^2.$$

With the standard inner product,

$$\lVert\chi\rVert_2^2 = \lvert1\rvert^2+\lvert1+i\rvert^2 = 1+2 = 3.$$

The corresponding unit vector is

$$\psi = \frac{1}{\sqrt3} \begin{pmatrix} 1\\ 1+i \end{pmatrix}.$$

If

$$\phi = \begin{pmatrix} 1\\ 0 \end{pmatrix},$$

then

$$\lVert\psi-\phi\rVert_2^2 = \left\lvert \frac{1}{\sqrt3}-1 \right\rvert^2 + \left\lvert \frac{1+i}{\sqrt3} \right\rvert^2 = 2-\frac{2}{\sqrt3}.$$

This is a distance between the chosen representatives. If only rays matter, the possible global phase of $\psi$ should be minimized over.

Common Mistakes

• Writing $\lvert\psi\rvert$ when the intended object is the Hilbert-space norm $\lVert\psi\rVert$.
• Normalizing by coordinate squares in a basis that is not orthonormal.
• Treating a phase-dependent vector distance as a distance between physical pure states.
• Assuming a Cauchy sequence converges before checking completeness.
• Using a residual norm in a numerical calculation without saying which norm and scaling are being used.
• Applying finite-dimensional norm equivalence intuition to infinite-dimensional function spaces without checking the topology.

References

• J. B. Conway, A Course in Functional Analysis, 2nd ed., Springer, 1990.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, Academic Press, 1980.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• L. N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM, 1997.

Exercises

1. Let $v=(1,2i,-2)\in\mathbb C^3$. Compute $\lVert v\rVert_2$.

Solution

Use the standard norm:

$$\lVert v\rVert_2^2 = \lvert1\rvert^2+\lvert2i\rvert^2+\lvert-2\rvert^2 = 1+4+4 = 9.$$

Thus $\lVert v\rVert_2=3$.

2. Suppose $\psi$ and $e^{i\theta}\psi$ are both normalized. What is their ray distance according to the phase-minimized formula?

Solution

They represent the same ray. Choosing $\alpha=-\theta$ gives

$$\lVert\psi-e^{i\alpha}e^{i\theta}\psi\rVert = \lVert\psi-\psi\rVert = 0.$$

The ray distance is zero even though the distance between the two written vectors may be nonzero.

3. Let $H=\operatorname{diag}(1,2)$, $v=(1,\epsilon)^T$, and $\lambda=1$. Compute the residual norm and its scale-independent ratio $\lVert r\rVert_2/\lVert v\rVert_2$. What happens if the same candidate direction is represented by $10^{-8}v$?

Solution

The residual is

$$r = Hv-\lambda v = \begin{pmatrix} 1\\ 2\epsilon \end{pmatrix} - \begin{pmatrix} 1\\ \epsilon \end{pmatrix} = \begin{pmatrix} 0\\ \epsilon \end{pmatrix}.$$

Therefore

$$\lVert r\rVert_2 = \lvert\epsilon\rvert.$$

Since $\lVert v\rVert_2=\sqrt{1+\lvert\epsilon\rvert^2}$, the ratio is

$$\frac{\lVert r\rVert_2}{\lVert v\rVert_2} =\frac{\lvert\epsilon\rvert}{\sqrt{1+\lvert\epsilon\rvert^2}}.$$

Replacing $v$ by $10^{-8}v$ multiplies the raw residual by $10^{-8}$ but leaves the ratio unchanged. Rescaling has not made the proposed eigendirection more accurate.