📂Quantum Mechanics

Completeness of Eigenfunctions in Quantum Mechanics

Definition

Suppose we are given a sequence (set) of wave functions $S = \left\{ \psi_{0}, \psi_{1}, \psi_{2}, \dots \right\}$. When an arbitrary wave function $\phi$ can be expressed as a series of $S$ as follows, we say that this sequence (set) has completeness.

$$ \phi(x) = \sum\limits_{n = 0}^{\infty} c_{n}\psi_{n}(x) $$

In particular, if $S$ is an orthonormal set, that is, if each $\psi_{n}$ is normalized and mutually orthogonal so that $\braket{\psi_{m} | \psi_{n}} = \delta_{mn}$ holds, then the coefficients $c_{n}$ are given by the inner product as follows.

$$ c_{n} = \braket{\psi_{n} | \phi} $$

In the finite-dimensional case, we say it is complete when it can be expressed as a finite linear combination as follows.

$$ \phi(x) = \sum\limits_{n = 0}^{N} c_{n}\psi_{n}(x) $$

When the eigenvalues have a continuous spectrum, it is expressed not as a sum but as an integral.

$$ \phi(x) = \int_{-\infty}^{\infty} c_{z}\psi_{z}(x) dz $$

Explanation¹

The word completeness is translated into Korean as 완비성 or 완전성. In mathematics, 완비성 is the standard translation, but in physics textbooks the expressions '완전하다' and '완전성' are also used. In this article, we unify to 완비성.

The reason completeness is important is that the theoretical framework of quantum mechanics stands on the following premise.

The eigenfunctions of a Hermitian operator representing an observable form a complete set.

In quantum mechanics, the act of measuring a physical quantity is represented by an operator, and the value one can obtain from a measurement is an eigenvalue of that operator. That the eigenfunctions obtained by solving the eigenvalue equation $A \psi_{n} = a_{n} \psi_{n}$ of a Hermitian operator $A$ form a complete set means that no matter what state $\phi$ the system is in, that state can be expanded as a series of the eigenfunctions. Here, if $\phi$ is normalized, the coefficients satisfy $\sum_{n} \left| c_{n} \right|^{2} = 1$, and $\left| c_{n} \right|^{2}$ is interpreted as the probability of obtaining the eigenvalue (= energy) $a_{n}$ when measuring $A$ in the state $\phi$. In other words, if completeness is not guaranteed, then $\sum_{n} |c_{n}|^{2} \lt 1$ may hold, and the probabilistic interpretation of measurement itself does not hold.

Discrete Spectrum

When the eigenvalues have a discrete spectrum, it is mathematically guaranteed that the corresponding eigenfunctions form a complete set. That is, if we know the set of eigenfunctions, we can express an arbitrary wave function $\phi$ as follows. This means that even without knowing $\phi$ concretely, we can handle it as a sum of eigenfunctions.

$$ \phi = \sum\limits_{n = 0}^{\infty} c_{n}\psi_{n}, \qquad c_{n} = \braket{\psi_{n} | \phi} $$

For example, in the case of the infinite potential well or the harmonic oscillator, the eigenfunctions can be obtained as follows. That is, in each system, an arbitrary wave function can be expressed as a linear combination of these eigenfunctions.

$$ \psi_{n}{(x)} = \textstyle \sqrt{\frac{2}{a}}\sin \left( \frac{n\pi}{a}x \right),\qquad \psi_{n}(x) = \dfrac{1}{\sqrt{2^{n}n!}} \alpha^{-n/2} \left( \dfrac{\alpha}{\pi} \right)^{1/4} \left( \alpha x - \dfrac{\d }{\d x} \right)^{n} e^{-\frac{\alpha}{2} x^{2}} $$

Continuous Spectrum

On the other hand, when the eigenvalues have a continuous spectrum, there is a problem. This is because the corresponding eigenfunctions do not belong to the Hilbert space. In a vector space, the sum of its elements must again be an element of the vector space. That is, it must be closed under addition. However, since the eigenfunctions of a continuous spectrum are not elements of the Hilbert space, simply adding them like ordinary vectors cannot represent an arbitrary wave function in the Hilbert space.

Instead, by continuously superposing these eigenfunctions through integration, we can construct a wave function that belongs to the Hilbert space. No matter how much we add them, we cannot represent any wave function. Fortunately, if we superpose the eigenfunctions by integration rather than addition, this belongs to the Hilbert space. Thus, an arbitrary wave function $\phi$ can be expressed as an integral of the continuous-spectrum eigenfunctions $\psi_{p}$ as follows.

$$ \phi = \int_{-\infty}^{\infty} c_{p} \psi_{p} \d p, \qquad c_{p} = \braket{\psi_{p} | \phi} $$

Here, although $\psi_{p}$ itself is not an element of the Hilbert space, an integral superposition made up of appropriate coefficients $c_{p}$ can be an element of the Hilbert space.