📂Quantum Mechanics

Continuous Spectra and Dirac Orthonormality

Definition¹

In quantum mechanics, an observable is represented by a Hermitian operator, and the values obtainable through measurement are the eigenvalues of that operator. The spectrum, which is the set of all eigenvalues, is broadly divided into two kinds.

• discrete spectra: the case where the eigenvalues are quantized and separated from one another. The eigenfunctions are normalizable and thus represent physically realizable states. The Hamiltonians of the infinite potential well and the harmonic oscillator are representative examples.

• continuous spectra: the case where the eigenvalues continuously fill some range. The eigenfunctions cannot be normalized, so none of them can be a wave function. The Hamiltonian of a free particle, the position operator, and the momentum operator are representative examples.

Explanation

In the case of a discrete spectrum, the eigenfunctions lie within the Hilbert space, so the existence of the inner product is guaranteed. From Hermiticity one can therefore easily prove that the eigenvalues are real and that eigenfunctions corresponding to distinct eigenvalues are orthogonal. In the case of a continuous spectrum, on the other hand, the eigenfunctions are not elements of the Hilbert space, so this proof, which relies on the inner product, does not go through. Even for a Hermitian operator, a formal eigenfunction may correspond to a non-real eigenvalue; indeed, the momentum operator has solutions of eigenfunction form for any complex number $p$. Nevertheless, if we collect only the eigenfunctions corresponding to real eigenvalues, orthogonality and completeness hold, and the Dirac delta function plays a crucial role here.

Suppose the operator $\hat{Q}$ satisfies the following eigenvalue equation.

$$ \hat{Q} f = q f $$

When the set of eigenvalues $\left\{ q_{n} \right\}$ forms a discrete spectrum, the eigenfunctions can be chosen, through normalization and Gram–Schmidt orthogonalization, so as to satisfy the orthonormality condition $\braket{f_{m} | f_{n}} = \delta_{mn}$ expressed via the Kronecker delta. Then, because the set of eigenfunctions has completeness $\sum_{n} \ket{f_{n}}\bra{f_{n}} = \hat{I}$, any wave function $\psi$ can be expressed as follows.

$$ \psi = \sum\limits_{n} c_{n}\ket{f_{n}}, \qquad c_{n} = \braket{f_{n} | \psi} $$

When the eigenvalues form a continuous spectrum, on the other hand, the eigenfunctions are not in the Hilbert space; they are not square-integrable functions. For instance, the eigenfunction $f_{p}$ corresponding to a real eigenvalue $p$ of the momentum operator is the plane wave below, and the eigenfunction $g_{y}$ corresponding to the eigenvalue $y$ of the position operator is the Dirac delta function itself.

$$ f_{p}(x) = \frac{1}{\sqrt{2\pi \hbar}} e^{\i p x / \hbar}, \qquad g_{y}(x) = \delta (x - y) $$

However, since these functions do not belong to the Hilbert space, the inner product $\braket{f_{p^{\prime}} | f_{p}}$ is not well defined. For example, looking at the eigenfunctions of the momentum operator, they organize as below: when $p = p^{\prime}$ the integrand is $1$, so the integral diverges, and when $p \ne p^{\prime}$ the integrand oscillates, so the integral does not converge in the usual sense.

$$ \braket{f_{p^{\prime}} | f_{p}} = \dfrac{1}{2\pi\hbar} \int e^{\i (p - p^{\prime}) x/\hbar} dx $$

Using somewhat difficult mathematics to resolve this, it can be expressed as below, and this is called Dirac orthonormality.

$$ \braket{f_{p^{\prime}} | f_{p}} = \delta (p - p^{\prime}) $$

Thus, in a continuous spectrum, the orthonormality condition, the expansion, and completeness appear in integral form as follows.

	Discrete Spectrum	Continuous Spectrum
Orthonormality Condition	$$\braket{f_{m}\vert f_{n}} = \delta_{mn}$$	$$\braket{f_{p^{\prime}}\vert f_{p}} = \delta (p - p^{\prime})$$
Expansion	$$ \psi = \sum\limits_{n}c_{n} \ket{n}$$	$$ \psi = \int_{-\infty}^{\infty} c(z) \ket{z} dz$$
Completeness	$$\hat{I} = \sum\limits_{n} \ket{n}\bra{n}$$	$$\hat{I} = \int_{-\infty}^{\infty} \ket{z}\bra{z} dz$$

See Also

• Completeness of Eigenfunctions
• Dirac Delta Function
• Momentum Operator
  • Eigenfunctions of the Momentum Operator
• Position Operator
  • Eigenfunctions of the Position Operator