Continuous Spectra and Dirac Orthonormality 📂Quantum Mechanics

Continuous Spectra and Dirac Orthonormality

Overview¹

In quantum mechanics, observables are represented by Hermitian operators, and the values obtainable by measurement are the eigenvalues of those operators. 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 admit normalization 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.

In the case of a discrete spectrum, the eigenfunctions live inside the Hilbert space, so the existence of an inner product is guaranteed. Hence, from Hermiticity, it is easy to prove that the eigenvalues are real and that eigenfunctions corresponding to distinct eigenvalues are orthogonal. By contrast, in the case of a continuous spectrum, 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, 🔒(26/08/05)the momentum operator has solutions of the eigenfunction form for any complex number $p$. Nonetheless, if we collect only the eigenfunctions corresponding to real eigenvalues, orthogonality and 🔒(26/07/18)completeness hold, and the Dirac delta function plays a crucial role here.

Explanation

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

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

When the set of eigenvalues $\left\{ q_{n} \right\}$ is a discrete spectrum, the eigenfunctions can be chosen, via normalization and Gram–Schmidt orthogonalization, so as to satisfy the orthonormality condition $\braket{f_{m} | f_{n}} = \delta_{mn}$ expressed by the Kronecker delta. Then, since the set of eigenfunctions has 🔒(26/07/18)completeness $\sum_{n} \ket{f_{n}}\bra{f_{n}} = \hat{I}$, any wave function $\psi$ can be expressed as follows.

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

On the other hand, when the eigenvalues form a continuous spectrum, the eigenfunctions are functions that do not lie in the Hilbert space and are not square-integrable. For instance, the eigenfunction $f_{p}$ of the momentum operator corresponding to the real eigenvalue $p$ is the plane wave below, and the eigenfunction $g_{y}$ of the position operator corresponding to the eigenvalue $y$ 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 instance, looking at the eigenfunctions of the momentum operator, one arrives at the expression 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.

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

To resolve this, using somewhat difficult mathematics, it can be expressed as follows, which is called Dirac orthonormality.

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

Thus, for a continuous spectrum, the orthonormality condition, the expansion, and completeness take the following integral forms.

discrete spectrum	continuous spectrum
$$\braket{f_{m}\vert f_{n}} = \delta_{mn}$$	$$\braket{f_{p^{\prime}}\vert f_{p}} = \delta (p - p^{\prime})$$
$$ \psi = \sum\limits_{n}c_{n} \ket{n}$$	$$ \psi = \int_{-\infty}^{\infty} c(z) \ket{z} dz$$
$$\hat{I} = \sum\limits_{n} \ket{n}\bra{n}$$	$$\hat{I} = \int_{-\infty}^{\infty} \ket{z}\bra{z} dz$$

Continuous Spectra and Dirac Orthonormality

Overview1

Explanation

See Also

Overview¹