Making sense of the divergent series for reconstructing a Hamiltonian from its eigenstates and eigenvalues

TL;DR

This paper demonstrates how to use Euler summation to assign meaning to divergent series in quantum mechanics, enabling the reconstruction of Hamiltonians from eigenstates and eigenvalues in infinite-dimensional spaces.

Contribution

It introduces a method to sum divergent series of eigenstates and eigenvalues, clarifying the formal completeness relation in quantum mechanics.

Findings

Euler summation successfully sums divergent series for specific potentials

Reconstruction of Hamiltonians becomes well-defined using summation techniques

Provides a practical approach to handle divergence in quantum completeness relations

Abstract

In quantum mechanics the eigenstates of the Hamiltonian form a complete basis. However, physicists conventionally express completeness as a formal sum over the eigenstates, and this sum is typically a divergent series if the Hilbert space is infinite dimensional. Furthermore, while the Hamiltonian can be reconstructed formally as a sum over its eigenvalues and eigenstates, this series is typically even more divergent. For the simple cases of the square-well and the harmonic-oscillator potentials this paper explains how to use the elementary procedure of Euler summation to sum these divergent series and thereby to make sense of the formal statement of the completeness of the formal sum that represents the reconstruction of the Hamiltonian.

Equations

$$\hat{H}=\sum _n{E}_n|{\varphi }_n⟩⟨{\varphi }_n|.$$

$$\sum _{n=1}^{\infty }{E}_n{\varphi }_n(x){\varphi }_n(y)=\frac{1}{\pi }\sum _{n=1}^{\infty }{n}^2\sin (nx)\sin (ny)$$

$$\delta (x-y)=\sum _n⟨x|n⟩⟨n|y⟩=\sum _n{\bar{\varphi }}_n(x){\varphi }_n(y),$$

$$E\equiv \underset{t\to 1^{-}}{\lim }f(t)$$

$$\zeta (s)=\sum _{n=1}^{\infty }\frac{1}{n^s},$$

$$\zeta(s)=\frac{\Gamma(1-s)}{2\pi i}\oint_{C}\mbox{$\rm d$}t\frac{t^{s-1}}{\mbox{$\rm e$}^{-t}-1},$$

$$\zeta (0)=-\frac{1}{2},\zeta (-1)=-\frac{1}{12}.$$

$$f(t)=\frac{1}{1-t}(|t|<1).$$

$$\zeta (s)=\frac{1}{1-2^{1-s}}\sum _{n=1}^{\infty }\frac{(-1{)}^{n+1}}{n^s}.$$

$$f(t)=-\frac{1}{3}+\frac{2}{3}t-\frac{3}{3}{t}^2+\frac{4}{3}{t}^3-\cdots =-\frac{1}{3(1+t{)}^2}.$$

$$\displaystyle{\hat{H}}=-\frac{1}{2}\frac{\mbox{$\rm d$}^{2}}{\mbox{$\rm d$}x^{2}}$$

$$\displaystyle\phi_{n}(x)=\sqrt{\frac{2}{\pi}}\sin(nx)\quad{\rm and}\quad E_{n}=\mbox{$\textstyle\frac{1}{2}$}n^{2},$$

$$\frac{2}{\pi }\sum _{n=1}^{\infty }\sin (nx)\sin (ny)=\delta (x-y)$$

$$H(x,y)=\frac{1}{\pi }\sum _{n=1}^{\infty }{n}^2\sin (nx)\sin (ny)$$

$$H(x,y)=-\frac{1}{2}\frac{{\partial }^2}{\partial y^2}\left(\frac{2}{\pi }\sum _{n=1}^{\infty }\sin (nx)\sin (ny)\right).$$

$$H(x,y)=-\frac{1}{2}\frac{{\partial }^2}{\partial y^2}\delta (x-y).$$

$$K(x,y,t)$$

$$\sum_{n=1}^{\infty}t^{n}\cos(nz)=\frac{1}{2-2t\,\mbox{$\rm e$}^{{\rm i}z}}+\frac{1}{2-2t\,\mbox{$\rm e$}^{-{\rm i}z}}-1.$$

$$K(x,y,t)=D(x-y,t)-D(x+y,t),$$

$$D(z,t)=\frac{1-t\cos z}{\pi (1-2t\cos z+t^2)}.$$

$$\lim_{t\to t_{0}}\int_{a}^{b}f_{t}(x)\mbox{$\rm d$}x=1$$

$$H(x,y,t)=\frac{1}{\pi }\sum _{n=1}^{\infty }{n}^2{t}^n\sin (nx)\sin (ny),$$

$$H(x,y,t)=-\frac{1}{2}\frac{{\partial }^2}{\partial y^2}K(x,y,t).$$

$$\displaystyle{\hat{H}}=-\frac{1}{2}\frac{\mbox{$\rm d$}^{2}}{\mbox{$\rm d$}x^{2}}+\frac{1}{2}x^{2}.$$

$$\phi_{n}(x)=\frac{1}{\sqrt{2^{n}n!\surd{\pi}}}\mbox{$\rm e$}^{-\frac{1}{2}x^{2}}\!H_{n}(x)\quad\!{\rm and}\!\quad E_{n}=n+\mbox{$\textstyle\frac{1}{2}$}.$$

$$\displaystyle H_{n}(x)=(-1)^{n}\mbox{$\rm e$}^{x^{2}}\frac{\mbox{$\rm d$}^{n}}{\mbox{$\rm d$}x^{n}}\,\mbox{$\rm e$}^{-x^{2}}.$$

$$H(x,y)=\frac{1}{2\sqrt{\pi}}\,\mbox{$\rm e$}^{-\frac{1}{2}(x^{2}+y^{2})}\sum_{n=0}^{\infty}\frac{2n+1}{2^{n}n!}H_{n}(x)\,H_{n}(y).$$

$$K(x,y,t)=\sum _{n=0}^{\infty }{t}^n{\varphi }_n(x){\varphi }_n(y).$$

$$\mbox{$\rm e$}^{-x^{2}}=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}\mbox{$\rm e$}^{-u^{2}+2{\rm i}xu}\mbox{$\rm d$}u.$$

$$H_{n}(x)=\frac{(-2\mbox{$\rm i$})^{n}\mbox{$\rm e$}^{x^{2}}}{\sqrt{\pi}}\int_{-\infty}^{\infty}u^{n}\mbox{$\rm e$}^{-u^{2}+2{\rm i}xu}\mbox{$\rm d$}u.$$

Full text

In physics courses quick formal arguments are often presented to reach correct conclusions that may be quite difficult to justify at a mathematical level. An example of such a formal argument is the derivation of the spectral decomposition of a Hamiltonian (or of any other observable). If $\{|\phi_{n}\rangle\}$ are the normalized eigenvectors of a Hermitian Hamiltonian ${\hat{H}}$ and $\{E_{n}\}$ are the corresponding eigenvalues, then the Hamiltonian is usually represented (reconstructed) formally as the sum

[TABLE]

There is no problem with (1) if the Hilbert space is finite-dimensional; that is, if ${\hat{H}}$ is an $N\times N$ matrix. In such a case $\hat{H}$ is given explicitly by the finite sum $\sum_{n=1}^{N}E_{n}|\phi_{n}\rangle\langle\phi_{n}|$. For $N=2$ or $N=3$, an explicit calculation of this sum is a useful exercise in an introductory quantum mechanics course.

However, a thoughtful student might ask, For a square-well potential on the interval $[0,\pi]$ the normalized eigenfunctions of the Hamiltonian are $\phi_{n}(x)=\sqrt{2/\pi}\sin(nx)$ and the corresponding eigenvalues are E_{n}=\mbox{\textstyle\frac{1}{2}}n^{2}. How can I evaluate the infinite sum

[TABLE]

to obtain the coordinate-space representation H(x,y)=\langle x|{\hat{H}}|y\rangle=-\mbox{\textstyle\frac{1}{2}}\delta^{\prime\prime}(x-y) of the Hamiltonian {\hat{H}}=\mbox{\textstyle\frac{1}{2}}{\hat{p}}^{2}? The problem here is that the series (2) diverges. A mathematically precise treatment for the reconstruction of the Hamiltonian may be found in mathematical literature rs but not in standard textbooks on quantum mechanics texts . Furthermore, the formal series

[TABLE]

which is conventionally used to express the completeness of the eigenstates, is also divergent for the square-well even though it diverges less rapidly than the series (2).

The series analogous to (2) for reconstructing the Hamiltonian of a quantum harmonic oscillator is also divergent. The $n$th term in this series grows for large $n$ like $\sqrt{n}$ multiplied by an oscillatory term. This sum is better behaved than that in (2), but it still diverges.

In what sense can we interpret the divergent series (2) as the Hamiltonian for the square-well potential? Is it possible to sum the divergent series in (3)? In this paper we propose to use the procedure of Euler summation as a simple and sensible way to regulate these divergent series and thereby to obtain closed-form coordinate-space representations $H(x,y)$ of the Hamiltonian and $\delta(x-y)$ of the identity operator.

Euler summation is easy to perform: Suppose that the series $\sum_{n=0}^{\infty}a_{n}$ diverges but that $f(t)=\sum_{n=0}^{\infty}t^{n}a_{n}$ converges for $|t|<1$. The Euler sum $E$ of the series $\sum_{n=0}^{\infty}a_{n}$ is then defined as the limit

[TABLE]

if this limit exists euler .

To illustrate Euler summation and demonstrate its effectiveness we use it to evaluate a sum that represents the Riemann zeta function $\zeta(s)$. The zeta function is conventionally defined in terms of the series

[TABLE]

which converges for ${\rm Re}(s)>1$, but diverges for ${\rm Re}(s)\leq 1$. Analytic continuation may be used to construct a complex integral representation of $\zeta(s)$ that is valid for all values of $s\neq 1$:

[TABLE]

where $C$ is a Hankel contour that encircles the negative-$t$ axis in the positive direction. One can use this integral representation to calculate $\zeta(s)$ at values of $s$ for which the sum (4) does not converge. For example, when $s=0$ and when $s=-1$ we get

[TABLE]

These results are remarkable because they suggest in some formal sense that the divergent sum $1+1+1+\cdots$ has the value $-\frac{1}{2}$ and that the divergent sum $1+2+3+\cdots$ has the value $-\frac{1}{12}$.

Summation of divergent series such as these is not just a formal mathematical procedure. It may be used to solve physical problems involving divergent sums over vibrational modes. For example, calculating the Casimir force, which has been verified and measured in laboratory experiments, requires that divergent sums over physical modes be evaluated casimir .

What happens if we attempt to use Euler summation to evaluate the series $1+1+1+\cdots$? The series for the Euler function $f(t)=1+t+t^{2}+\cdots$ converges for $|t|<1$ and we get

[TABLE]

However, $\lim_{t\to 1}f(t)$ does not exist. Thus, Euler summation is not powerful enough to assign a value to the sum of the divergent series $1+1+1+\cdots$. Euler summation also fails to assign a value to the sum of the divergent series $1+2+3+\cdots$.

However, there is a clever way to use Euler summation to evaluate $\zeta(0)$: We rewrite the series representation (4) for $\zeta(s)$ as an alternating series by subtracting the even-$n$ terms from the odd-$n$ terms in (4):

[TABLE]

This alternating series converges for ${\rm Re}(s)>0$ but diverges for ${\rm Re}(s)\leq 0$. Let us use Euler summation to evaluate the series obtained by setting $s=0$, which is $-1+1-1+1-\cdots$. For this divergent series $f(t)=-1/(1+t)$, and thus the Euler sum $E$ of the series is $f(1)=-\frac{1}{2}$, which is the correct value for $\zeta(0)$.

Similarly, if we set $s=-1$ in (5), we obtain the divergent series $-\frac{1}{3}+\frac{2}{3}-\frac{3}{3}+\frac{4}{3}-\cdots$. For this case

[TABLE]

Thus, Euler summation gives $\zeta(-1)=f(1)=-\frac{1}{12}$. Evidently, Euler summation is quite impressive; it implicitly performs an analytic continuation of the sums in (4) and (5) into the complex plane without the appearance of complex numbers!

Having demonstrated the power of Euler summation we now use it to regulate and evaluate divergent quantum-mechanical sums like that in (2) and thus make sense of this formal series. A rigorous discussion of completeness requires advanced mathematical techniques used in Hilbert-space theory, but here we show how to perform the sum in (1) explicitly for the special case of a square-well potential and we repeat the process for a harmonic-oscillator potential. We use only elementary techniques that are within reach of undergraduate physics students.

Square-well potential. A unit-mass particle trapped in an infinite square-well potential on the interval $[0,\pi]$ is described by the coordinate-space Hamiltonian $\hbar=1$)

[TABLE]

(in units such that Planck’s constant is unity). The normalized solutions to the time-independent Schrödinger equation that satisfy vanishing boundary conditions at $x=0$ and $x=\pi$ are

[TABLE]

and thus from (1) we obtain the divergent series (2).

A naive summation of (2) that one might encounter in an elementary quantum-mechanics course consists of using the completeness condition

[TABLE]

to verify the coordinate-space representation of the square-well Hamiltonian. The argument begins by replacing the factor of $n^{2}$ in the formal series

[TABLE]

with a second derivative acting on the full series:

[TABLE]

Then (6) is used to write

[TABLE]

The problem with this argument is that the interchange of differentiation and summation is justified only if the sum in (6) is absolutely and uniformly convergent and this is not so because the formal series (6) diverges.

Let us now use Euler summation to make sense of the formal statement of completeness in (6). We do so at a freshman-calculus level and without appealing to advanced theorems about Fourier series. The usual approach in Fourier analysis wiener relies on regulating the infinite sum (6) by replacing it with an $N$-term finite sum and then taking the limit $N\to\infty$ by using the Riemann-Lebesgue lemma to argue that this sum converges to a delta function. Instead, here we regulate the infinite sum (6) by inserting the geometrical convergence (Euler) factor $t^{n}$:

[TABLE]

This sum converges absolutely and uniformly for $|t|<1$. We then use the exponential form of the cosine function, \cos(nz)=\mbox{\textstyle\frac{1}{2}}(\exp({\rm i}nz)+\exp(-{\rm i}nz)), and sum the infinite geometric series:

[TABLE]

Thus, we find that

[TABLE]

where

[TABLE]

It is easy to show that $D(z,t)$ has the following properties: If $z=n\pi$ with $n$ even, then $D(z,t)=[\pi(1-t)]^{-1}$, whereas for other values of $z$, $D(z,t)\to(2\pi)^{-1}$ as $t\to 1$. Therefore, as $t\to 1$ and for $0<x,y<\pi$, the value of $K(x,y,t)$ becomes infinite along the line $x=y$ and is zero otherwise. These properties suggest that $K(x,y,t)\to\delta(x-y)$ as $t\to 1$ and we must show that this is so. In general, a parametric family of functions $\{f_{t}(x)\}$ having the property that $f_{t}(x)\to\delta(x)$ as $t\to t_{0}$ is said to be delta convergent. (For a precise definition of a delta-convergent function see Ref. gelfand .) In practice, to show that $\{f_{t}(x)\}$ is delta convergent, it suffices to show that for any $a<0<b$

[TABLE]

and that if $a<b<0$ or if $0<a<b$, the limit $t\to t_{0}$ of the integral vanishes. Thus, to establish completeness we integrate $K(x,y,t)$ in the variable $y$ and then take the limit $t\to 1^{-}$. This calculation is straightforward but lengthy, so we have relegated it to the Appendix. It is shown there that $K(x,y,t)$ in (8) is a delta-convergent series. This verifies the completeness condition (6).

Our next task is to use Euler summation to make sense of the divergent series (7) for the reconstruction of the Hamiltonian. As before, we introduce the convergence factor $t^{n}$ into (7):

[TABLE]

where $|t|<1$. The Euler-summation factor $t^{n}$ regulates the divergent coordinate-space representation (7) for the Hamiltonian; the regulated series $H(x,y,t)$ is absolutely and uniformly convergent. Therefore, term-by-term differentiation of $H(x,y,t)$ with respect to the variables $x$, $y$, and $t$ can be performed. We differentiate term-by-term with respect to $y$ to obtain

[TABLE]

Therefore, in the limit $t\to 1^{-}$, $H(x,y,t)$ converges to the coordinate-space representation H(x,y)=-\mbox{\textstyle\frac{1}{2}}\delta^{\prime\prime}(x-y) of the Hamiltonian for the square-well potential.

Quantum harmonic oscillator. The coordinate-space Hamiltonian that describes a particle of unit mass trapped in a harmonic potential is

[TABLE]

The stationary states and associated eigenvalues are

[TABLE]

Here, $H_{n}(x)$ denotes the $n$th Hermite polynomial, which can be obtained from the standard identity

[TABLE]

Thus, from (1) the formal coordinate-space representation for the Hamiltonian takes the form of the infinite sum

[TABLE]

While this series is divergent, we show below how to use Euler summation to sum the series to obtain the operator ${\hat{H}}$ in (10).

As in the case of the square-well potential, the procedure is first to use Euler summation to sum the formal series that represents the completeness of the harmonic-oscillator eigenstates. To do so, we need a formula known as the Mehler generating function:

[TABLE]

Perhaps the simplest derivation of the Mehler formula for $K(x,y,t)$ was given by Hardy in his lectures on orthogonal polynomials delivered in the Lent Term, 1933 (see watson ). We reproduce Hardy’s derivation of the Mehler formula here for completeness. To begin, we remark that the Fourier transform of a Gaussian is a Gaussian:

[TABLE]

Differentiating (12) $n$ times in $x$ and using (11), we find that

[TABLE]

It follows that

[TABLE]

When $|t|<1$ we can interchange the order of summation and integration because the series is uniformly and absolutely convergent and we get the Mehler formula

[TABLE]

The last expression in (13) is not explicitly symmetric in the variables $x$ and $y$, but it can be symmetrized by means of an elementary manipulation (see Ref. wiener , §8):

[TABLE]

Using (14), we observe that in the limit $t\to 1^{-}$ the expression in (13) assumes a Gaussian form with vanishing standard deviation (a Dirac delta function). Indeed, we can easily integrate a Gaussian density explicitly to show that in the limit of vanishing standard deviation a Gaussian density is delta convergent (see Ref gelfand , §2.5). Hence, we deduce the completeness condition for the eigenstates $\{\phi_{n}(x)\}$ of the quantum harmonic oscillator:

[TABLE]

To go from this statement of completeness to the series reconstruction of the Hamiltonian we again observe that differentiation of $K(x,y,t)$ in $t$ or in $y$ can be performed under the summation because the sum for $K(x,y,t)$ is uniformly and absolutely convergent for $|t|<1$. Exploiting the relation

[TABLE]

we show that

[TABLE]

Note that in going from the first to the second line above we use the identity

[TABLE]

Alternatively, making use of the eigenvalue equation

[TABLE]

satisfied by the harmonic oscillator eigenfunctions in the second line of (15) we arrive at the same conclusion more expediently. This completes the analysis for the quantum-harmonic-oscillator Hamiltonian.

Summary. We have used Euler summation to make sense of the formal divergent series that express the completeness of the square-well and the harmonic-oscillator eigenstates. We then showed how to interpret the divergent series representing the coordinate-space reconstruction of the corresponding Hamiltonian operators. The advantage of Euler summation is that it allows us to use term-by-term differentiation on simple-looking but divergent series. Thus, Euler summation (and more generally other techniques such as Borel summation and Padé summation) can be used to make sense of the divergent series that physicists often encounter in their work.

Appendix. Here we show that the function $K(x,y,t)$ defined in (8) satisfies the conditions of a delta-convergent series. For fixed $x$ and $0\leq a<b\leq\pi$ we evaluate the integral

[TABLE]

where f(u)=1-t\,{\mbox{\rm e}}^{{\rm i}u} for $u\in(-\pi,\pi)$, and ${\bar{f}}$ denotes the complex conjugate of the function $f$. Note that

[TABLE]

In the limit $t\to 1^{-}$, $\arg f(u)$ can be found by geometric or trigonometric methods. In either case we deduce that

[TABLE]

Thus,

[TABLE]

We conclude that if $x<a$, then

[TABLE]

so the integral vanishes. Similarly, if $x>b$, then

[TABLE]

so the integral again vanishes. However, if $a<x<b$, we find that

[TABLE]

from which we deduce that the value of the integral is unity. This establishes that $K(x,y,t)$ is indeed a delta-convergent series.