Self-Adjoint Operators and Eigenfunction Expansions
The relevance of waves in quantum mechanics naturally implies that the decomposition of arbitrary wave packets in terms of monochromatic waves, commonly known as Fourier decomposition after Jean-Baptiste Fourier’s Théorie analytique de la Chaleur (1822),
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Self-Adjoint Operators and Eigenfunction Expansions
The relevance of waves in quantum mechanics naturally implies that the decomposition of arbitrary wave packets in terms of monochromatic waves, commonly known as Fourier decomposition after Jean-Baptiste Fourier’s Théorie analytique de la Chaleur (1822), plays an important role in applications of the theory. Dirac’s δ function, on the other hand, gained prominence primarily through its use in quantum mechanics, although today it is also commonly used in mechanics and electrodynamics to describe sudden impulses, mass points, or point charges. Both concepts are intimately connected to the completeness of eigenfunctions of self-adjoint operators. From the quantum mechanics perspective, the problem of completeness of sets of functions concerns the problem of enumeration of all possible states of a quantum system.
2.1 The δ Function and Fourier Transforms Let f (x) be a continuous function in the interval [a, b]. Dirichlet’s equation [28, 29] lim
κ→∞ a
b
dx
sin[κ(x − x )] f (x ) = π(x − x )
0, x ∈ / [a, b], f (x), x ∈ (a, b),
(2.1)
motivates the formal definition sin(κx) 1 = lim δ(x) = lim κ→∞ κ→∞ πx 2π ∞ 1 = dk exp(ikx), 2π −∞
κ
−κ
dk exp(ikx)
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 R. Dick, Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_2
(2.2)
25
26
2 Self-Adjoint Operators and Eigenfunction Expansions
such that Eq. (2.1) can (in)formally be written as
b
dx δ(x − x )f (x ) =
a
0, x ∈ / [a, b], f (x), x ∈ (a, b).
(2.3)
A justification for Dirichlet’s equation is given below in the derivation of Eq. (2.19). The generalization to three dimensions follows immediately from Dirichlet’s formula in a three-dimensional cube, and exhaustion of an arbitrary three-dimensional volume V by increasingly finer cubes. This yields δ(x) =
3 i=1
sin(κi xi ) 1 lim = κi →∞ π xi (2π )3
d 3 x δ(x − x )f (x ) = V
d 3 k exp(ik · x), 0, x ∈ / V, f (x), x inside V .
(2.4)
(2.5)
The case x ∈ ∂V (x on the boundary of V ) must be analyzed on a case-by-case basis. Equation (2.4) implies ψ(x, t) = =
d 3 x δ(x − x )ψ(x , t)
1 (2π )3
d 3 k exp(ik · x)
d 3 x exp(− ik · x )ψ(x , t).
(2.6)
This can be used to introduce Fourier transforms by splitting the previous equation into two equations, 1 ψ(x, t) = √ 3 2π
d 3 k exp(ik · x)ψ(k, t),
(2.7)
d 3 x exp(− ik · x)ψ(x, t).
(2.8)
with 1 ψ(k, t) = √ 3 2π
Use of ψ(x, t) corresponds to the x-representation of quantum mechanics. Use of ψ(k, t) corresponds to the k-representation or momentum-representation of quantum mechanics. The notation above for Fourier transforms is a little sloppy, but convenient and common in quantum mechanics. From a mathematical perspective, the Fourier ˜ transformed function ψ(k, t) should actually be denoted by ψ(k, t) to make it clear that it is not the same function as ψ(x, t) with different symbol
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