Infinitesimal and Infinite Numbers As an Approach to Quantum Mechanics

Inﬁnitesimal and Inﬁnite Numbers as an Approach to Quan- tum Mechanics

Vieri Benci1, Lorenzo Luperi Baglini2, and Kyrylo Simonov2

1Dipartimento di Matematica, Università degli Studi di Pisa, Via F. Buonarroti 1/c, 56127 Pisa, Italy 2Fakultät fur¨ Mathematik, Universität Wien, Oskar-Morgenstern Platz 1, 1090 Vienna, Austria April 29, 2019

Non-Archimedean mathematics is an ap- Non-Archimedean mathematics (particularly, proach based on fields which contain in- nonstandard analysis) is a framework that treats finitesimal and infinite elements. Within the infinitesimal and infinite quantities as num- this approach, we construct a space of a bers. Since the introduction of nonstandard anal- particular class of generalized functions, ysis by Robinson [2] non-Archimedean mathe- ultrafunctions. The space of ultrafunctions matics has found a plethora of applications in can be used as a richer framework for a physics [3,5,4,6,7], particularly in quantum me- description of a physical system in quan- chanical problems with singular potentials, such tum mechanics. In this paper, we pro- as δ-potentials [8,9, 10, 11, 12]. vide a discussion of the space of ultrafunc- In this paper, we build a non-Archimedean tions and its advantages in the applica- approach to quantum mechanics in a simpler tions of quantum mechanics, particularly way through a new space, which can be used for the Schrödinger equation for a Hamil- as a basic construction in the description of a tonian with the delta function potential. physical system, by analogy with the Hilbert space in the standard approach. This space 1 Introduction is called the space of ultrafunctions, a particular class of non-Archimedean generalized func- Quantum mechanics is a highly successful physi- tions [13, 14, 15, 16, 17, 18, 19]. The ultrafunc- ∗ cal theory, which provides a counter-intuitive but tions are defined on the hyperreal field R , which accurate description of our world. During more extends the reals R by including infinitesimal and than 80 years of its history, there were devel- infinite elements into it. Such a construction al- oped various formalisms of quantum mechanics lows studying the problems which are difficult that use the mathematical notions of different to solve and formalize within the standard ap- complexity to derive its basic principles. The proach. For example, variational problems, that standard approach to quantum mechanics han- have no solutions in standard analysis, can be dles linear operators, representing the observ- solved in the space of ultrafunctions [18]. In this ables of the quantum system, that act on the way, non-Archimedean mathematics as a whole vectors of a Hilbert space representing the phys- and the ultrafunctions as a particular propose a ical states. However, the existing formalisms in- richer framework, which highlights the notions clude not only the standard approach but as well hidden in the standard approach and paves the arXiv:1901.10945v3 [math-ph] 26 Apr 2019 some more abstract approaches that go beyond way to a better understanding of quantum me- Hilbert space. A notable example of such an chanics. abstract approach is the algebraic quantum me- The paper is organized as follows. In Section1 chanics, which considers the observables of the we introduce the needed notations and the notion quantum system as a non-abelian C∗-algebra, of a non-Archimedean field. In Section2 we in- and the physical states as positive functionals on troduce a particular non-Archimedean field, the it [1]. field of Euclidean numbers E, through the notion of Λ-limit, which is a useful, straightforward ap- Vieri Benci: [email protected] proach to the nonstandard analysis. In Section3 Lorenzo Luperi Baglini: [email protected] we introduce the space of ultrafunctions, which Kyrylo Simonov: [email protected] are a particular class of generalized functions. In

Accepted in Quantum 2019-04-25, click title to verify 1 Section4 we apply the ultrafunctions approach to 1.2 Non-Archimedean fields quantum mechanics and discuss its advantages in Our approach to quantum mechanics makes mul- contrast to the standard approach. In Section5 tiple uses of the notions of infinite and infinites- we provide a discussion of a quantum system with imal numbers. A natural framework to intro- a delta function potential for the standard and duce these numbers suitably is provided by non- ultrafunctions approaches. Last but not least, in Archimedean mathematics (see, e.g., [20]). This Section6 we provide the conclusions. framework operates with the infinite and infinitesimal numbers as the elements of the new 1.1 Notations non-Archimedean field. N Let Ω be an open subset of , then 1 R Definition 1. Let K be an infinite ordered field . • C (Ω) denotes the set of continuous functions An element ξ ∈ K is: N defined on Ω ⊂ R , • infinitesimal if, for all positive n ∈ N, |ξ| < 1 • Cc (Ω) denotes the set of continuous func- n , tions in C (Ω) having compact support in Ω, • finite if there exists n ∈ N such that |ξ| < n, • Ck (Ω) denotes the set of functions defined on • infinite if, for all n ∈ , |ξ| > n (equiva- Ω ⊂ N which have continuous derivatives N R lently, if ξ is not finite). up to the order k, We say that is non-Archimedean if it contains • Ck (Ω) denotes the set of functions in K c an infinitesimal ξ 6= 0, and that is superreal if Ck (Ω) having compact support, K it properly extends R. • D (Ω) denotes the space of infinitely differ- Notice that, trivially, every superreal field is entiable functions with compact support de- non-Archimedean. Infinitesimals allow introduc- fined almost everywhere in Ω, ing the following equivalence relation, which is • L2 (Ω) denotes the space of square integrable fundamental in all non-Archimedean settings. functions defined almost everywhere in Ω, Definition 2. We say that two numbers ξ, ζ ∈ K 1 are infinitely close if ξ−ζ is infinitesimal. In this • Lloc (Ω) denotes the space of locally integrable functions defined almost everywhere case we write ξ ∼ ζ. in Ω, In the superreal case, ∼ can be used to intro- N duce the fundamental notion of “standard part”2. • mon(x) = {y ∈ E | x ∼ y} (see Def.4), Theorem 3. If K is a superreal field, every finite • given any set E ⊂ X, χE : X → R denotes the characteristic function of E, namely, number ξ ∈ K is infinitely close to a unique real number r ∼ ξ, called the the standard part of  1 if x ∈ E,  ξ. χE(x) :=  0 if x∈ / E, Following the literature, we will always denote by st(ξ) the standard part of any finite number

• with some abuse of notation, we set χa(x) := ξ. Moreover, with a small abuse of notation, we also put st(ξ) = +∞ (resp. st(ξ) = −∞) if ξ ∈ χ{a}(x), K is a positive (resp. negative) infinite number. ∂ • ∂i = ∂x denotes the usual partial deriva- i Definition 4. Let K be a superreal field, and ξ ∈ tive, Di denotes the generalized derivative a number. The monad of ξ is the set of all (see Section 3.1), K numbers that are infinitely close to it, • R denotes the usual Lebesgue integral, H denotes the pointwise integral (see Section mon(ξ) = {ζ ∈ K : ξ ∼ ζ}. 3.1), 1 Without loss of generality, we assume that Q ⊆ K. • if E is any set, then |E| denotes its cardinal- 2For a proof of the following simple theorem, the inter- ity. ested reader can check, e.g., [21].

Accepted in Quantum 2019-04-25, click title to verify 2 Notice that, by definition, the set of infinitesi- Let Λ be an infinite set containing R and let L mals is mon(0) precisely. Finally, superreal fields be the family of finite subsets of Λ. A function can be easily complexified by considering ϕ : L → R will be called net (with values in R). The set of such nets is denoted by F (L, R) and K + iK, equipped with the natural operations namely, a field of numbers of the form (ϕ + ψ)(λ) = ϕ(λ) + ψ(λ), (ϕ · ψ)(λ) = ϕ(λ) · ψ(λ), a + ib, a, b ∈ K. and the partial order relation In this way, the complexification of non- Archimedean fields shows no particular difficulty ϕ ≥ ψ ⇐⇒ ∀λ ∈ L, ϕ(λ) ≥ ψ(λ). and is straightforward. In this way, F (L, R) is a partially ordered real algebra.

2 The field of Euclidean numbers Definition 5. We say that a superreal field E is a field of Euclidean numbers if there is a surjective Nonstandard analysis plays one of the most map prominent roles between various approaches to J : F (L, R) → E, non-Archimedean mathematics. One reason is that nonstandard analysis provides a handy tool which satisfies the following properties, to study and model the problems which come • J (ϕ + ψ) = J (ϕ) + J (ψ), from many different areas. However, the classical representations of nonstandard analysis can feel • J (ϕ · ψ) = J (ϕ) · J (ψ), overwhelming sometimes, as they require a good knowledge of the objects and methods of mathe- • if ϕ(λ) > r, then J (ϕ) > r. matical logic. This stands in contrast to the ac- J will be called the realization of . tual use of nonstandard objects in the mathemat- E ical practice, which is almost always extremely The proof of the existence of such a field is an close to the usual mathematical practice. easy consequence of the Krull – Zorn theorem. For these reasons, we believe that it is worth It can be found, e.g., in [24, 13, 14, 26]. In this to present nonstandard analysis avoiding most of paper, we also use the complexification of E, de- the usual logic machinery. We will introduce the noted5 by nonstandard approach to the formalism of quan- ∗ C = E + iE. (1) tum mechanics via the notion of Λ-limit3 and the Euclidean numbers. These numbers are the Definition 6. Let E be a field of Euclidean num- underlying object of Λ-theory, and can be intro- bers, let J be its realization and let ϕ ∈ F (L, R). duced via a purely algebraic approach, as done We say that J (ϕ) is the Λ-limit of the net ϕ. in [24]4. We will also denote it by The basic idea of the construction of Euclidean numbers is the following: as the real numbers J (ϕ) := lim ϕ(λ). λ↑Λ can be constructed by completion of the rational numbers using the Cauchy notion of limit, the The name “limit” has been chosen because the Euclidean numbers can be constructed by com- operation pletion of the reals with respect to a new notion ϕ 7→ lim ϕ(λ) of limit, called Λ-limit, that we are now going to λ↑Λ introduce. satisfies the following properties,

3 In [22] and [23], the reader can ﬁnd several other ap- • (Λ-1) Existence. Every net ϕ : L → R has proaches to nonstandard analysis and an analysis of them. a unique limit L ∈ E. 4An elementary presentation of (part of) this theory can be found in [25] and [26]. 5The choice of the notation will become clear later on.

Accepted in Quantum 2019-04-25, click title to verify 3 • (Λ-2) Constant. If ϕ(λ) is eventually con- n (E) = |E|. On the other hand, if E is an stant, namely, ∃λ0 ∈ L, r ∈ R such that inﬁnite set, the above limit gives an inﬁnite 7 ∀λ ⊃ λ0, ϕ(λ) = r, then number (called the numerosity of E ).

lim ϕ(λ) = r. Definition 7. A mathematical entity is called λ↑Λ internal if it is a Λ-limit of some other entities. • (Λ-3) Sum and product. ∀ϕ, ψ : L → R 2.1 Extension of functions, hyperfinite sets and lim ϕ(λ) + lim ψ(λ) = lim(ϕ(λ) +ψ(λ)), grid functions λ↑Λ λ↑Λ λ↑Λ lim ϕ(λ) · lim ψ(λ) = lim(ϕ(λ) · ψ(λ)). The Λ-limit allows extending the field of real λ↑Λ λ↑Λ λ↑Λ numbers to the field of Euclidean numbers. Sim- ilarly, we can use it to extend sets and functions The Cauchy limit of a sequence (as formalized in arbitrary dimensions, by Weierstraß) satisfies the second and the third6 N condition above. The main difference between • if N ∈ N, then, in order to extend R to N the Cauchy and the Λ-limit is given by the prop- E , we proceed in a trivial way: if ϕ(λ) := N erty (Λ − 1), namely, by the fact that the Λ-limit (ϕ11(λ), ..., ϕN (λ)) ∈ R , we set always exists. Notice that it implies that E must N be larger than since it must contain the limit lim ϕ(λ) = lim ϕ1(λ), ..., lim ϕN (λ) ∈ E , R λ↑Λ λ↑Λ λ↑Λ of diverging nets as well. For those nets that have a Cauchy limit, the • if A ⊂ N , we let the natural extension relationship between the Cauchy and the Λ-limit R of A to be is expressed by the following identity, A∗ = lim ϕ(λ) | ∀λ, ϕ(λ) ∈ A , lim ϕ(λ) = st lim ϕ(λ) . (2) λ↑Λ λ→Λ λ↑Λ With the introduction of the Euclidean num- • if N bers there arises a natural question: What do f : A → R,A ⊂ R , they look like? Let us give a few examples. we let the natural extension of f to A∗ 1 to be a function such that, for every x = 1. Let ϕ(λ) := |λ| for every λ 6= ∅, and ε = ∗ limλ↑Λ xλ ∈ A , limλ↑Λ ϕ(λ). Then ε > 0, because ϕ(λ) > 0 for every λ ∈ L. However, ε < r for every ∗ ∗ f lim xλ = lim f (xλ) , positive r ∈ R since ϕ(λ) < r eventually in λ↑Λ λ↑Λ λ. Therefore, ε is a non-zero infinitesimal in E. • the above case can be generalized, in order to define directly the functions between subsets 2. Analogously, let ϕ(λ) := |λ| for every λ ∈ L. N of E , if Let σ = limλ↑Λ ϕ(λ). Straightforwardly, σ is N a positive infinite element in E, and σ·ε = 1. uλ : A → R,A ⊂ R 3. Let us generalize the previous example. If is a net of functions, its Λ-limit E ⊂ R ⊂ Λ, we set ∗ u = lim uλ : A → E n (E) = lim |E ∩ λ|, λ↑Λ λ↑Λ is a function such that, for any x = where |F | denotes the number of elements N limλ↑Λ xλ ∈ E , of a finite set F . Since λ is a finite set, |E ∩λ| ∈ for every λ. If E is finite, then E u(x) = lim uλ (xλ) , N λ↑Λ belongs to L, and the net |E ∩ λ| is eventually constant (in λ) and equal to |E|, so that 7The reader interested in the details and the de- velopments of the theory of numerosities is referred to 6When both sequences have a limit. [27, 28, 29, 30].

Accepted in Quantum 2019-04-25, click title to verify 4 • finally, if V is a function space, we let its Definition 9. A hyperfinite set Γ such that N N natural extension to be R ⊂ Γ ⊂ E is called hyperfinite grid. ∗ If {Γ } is a family of finite subsets of N , V := lim uλ | uλ ∈ V . λ λ∈L R λ↑Λ which satisfies the property

N N Notice that if, for all x ∈ R , R ∩ λ ⊂ Γλ,

v(x) = lim uλ (x) then it is not difficult to prove that the set λ→Λ by Eq. (2), it follows that Γ = lim xλ | xλ ∈ Γλ λ↑Λ ∀x ∈ N , v(x) = st [u(x)] . R is a hyperfinite grid. Below Γ denotes a hyperfinite grid extending Moreover, it is clear that E is the natural ex- ∗ N , which is fixed once and forever. tension of R, namely, E = R . Notice that this R choice justifies also our notation (1) for the com- Definition 10. The space of grid functions on Γ plexification of E, is the family G(Γ) of functions ∗ = + i = ∗ + i ∗. C E E R R u :Γ → R Now we introduce a fundamental notion for all such that, for every x = limλ↑Λ xλ ∈ Γ, our applications, the “hyperfinite” set. u(x) = lim uλ(xλ). Definition 8. We say that a set F ⊂ E is hy- λ↑Λ perfinite if there is a net {Fλ} of finite sets λ∈Λ N such that If f ∈ F(R ), and x = limλ↑Λ xλ ∈ Γ, we set ◦ f (x) := lim f(xλ), (3) F = lim xλ | xλ ∈ Fλ . λ↑Λ λ↑Λ namely, f ◦ is a restriction to Γ of the natural Hyperfinite sets play the role of finite sets in ∗ N the non-Archimedean framework since they share extension f which is defined on the whole E . many properties with finite sets. We will often It is easy to check that, for every a ∈ Γ, the use the following feature provided by the hyper- characteristic function χa(x) of {a} is a grid func- finite sets: it is possible to “add” the elements of tion. In perfect analogy with the classical finite a hyperfinite set of numbers. If F is a hyperfi- case, every grid function can be represented by nite set of numbers, the hyperfinite sum of the the following hyperfinite sum, elements of F is defined in the following way, X u(x) = u(a)χa(x), (4) X x = lim X x. a∈Γ λ↑Λ x∈F x∈F λ namely, {χa(x)}a∈Γ is a set of generators for N Let us illustrate the hyperfinite sets with a G(R ). Moreover, from Definition 10 it follows that very simple example. Let Fλ = {n ∈ N | n ≤ |λ|} for every λ ∈ L. Further, let F = actually {χa(x)}a∈Γ is a basis for the space of lim x | x ∈ F . Then F is hyperfinite grid functions on Γ. λ↑Λ λ λ λ N ∗ In general, if E is a subset of and f is and F ⊆ N . Moreover, n ∈ F , for every n ∈ N, R defined only on E, we set as n ∈ Fλ, for every λ, so that |λ| ≥ n. There- fore, we have constructed a hyperfinite subset of f ◦(x) = X f ∗(a)χ (x), ∗ 8 a N which contains the infinite set N. a∈Γ∩E∗ The kind of hyperfinite sets that we will use are the so-called “hyperfinite grids”. namely, we set it to be 0 on every grid point that does not belong to the natural extension of its 8At first sight, it might seem absurd to have a set that 1 ◦ domain. For example, if f(x) = |x| , then f (0) = extends N but still behaves like a finite set. This is one of 0. the peculiar and key properties of hyperfinite sets.

Accepted in Quantum 2019-04-25, click title to verify 5 3 Ultrafunctions Definition 11. A space of ultrafunctions V ◦(Ω) generated by V (Ω) and modelled on the In this section, we introduce the key ingredient family of its finite subspaces {Vλ}λ∈L is a space needed for the description of a quantum system, of grid functions the space of ultrafunctions. An explicit technical construction of (several) ultrafunctions spaces u :Γ → E has been provided in Ref. [13, 16, 17, 14, 15, 19, 18] by various reformulations of nonstandard equipped with an internal functional analysis. However, we prefer to pursue an ax- I ◦ iomatic approach to underscore the key proper- : V (Ω) → E ties of ultrafunctions needed for our aims since such technicalities are not important in the ap- (called pointwise integral) and N internal op- plications of the ultrafunctions spaces. For an erators explicit construction, we refer to [31]. The basic D : V ◦(Ω) → V ◦(Ω) idea one has to keep in mind is that, in order to i build a space of ultrafunctions, we start with a (called generalized partial derivative), which classical space of functions V (Ω) and extend it to satisfy the axioms below. a space of grid functions V ◦(Ω) with several ad ◦ hoc properties. Axiom 1. ∀u ∈ V (Ω), there exists a net uλ such that ∀λ ∈ L uλ ∈ Vλ, and 3.1 Axiomatic definition of ultrafunctions u = lim uλ. λ↑Λ In order to build a space of ultrafunctions that suites for an adequate description of a quantum Axiom 2. ∀u = limλ↑Λ uλ, uλ ∈ Vλ, its point- system, we have to fix an appropriate function wise integral is defined as space V (Ω). In that way, we choose V (Ω) ⊇ C0(Ω) to be a function space that includes in- I Z u(x) dx = lim uλ dx, (6) finitely differentiable real functions and is a sub- λ↑Λ Qλ space of the space of locally integrable functions (which includes real p-integrable functions with where p ≥ 1), n o N Qλ = x ∈ R |xi| ≤ |λ| . 0 1 C (Ω) ⊂ D(Ω) ⊂ V (Ω) ⊂ Lloc(Ω). Axiom 3. ∀a ∈ Γ, there exists a positive in- In the further step, we build a family of all finite- finitesimal number d(a) such that dimensional subspaces of the chosen function I space V (Ω). We label this family by {Vλ}λ∈L, d(a) = χa(x)dx > 0. so that it has the following property,

Axiom 4. If x = limλ↑Λ xλ ∈ Γ and u = ∀λ1, λ2 ∈ L λ1 ⊆ λ2 ⇒ Vλ1 ⊆ Vλ2 . 1 N limλ↑Λ uλ such that ∀λ ∈ L uλ ∈ Vλ ∩ Cc (R ), then its generalized partial derivative at the point Such a family provides a net {Vλ}λ∈L of the finite subspaces of V (Ω). Hence it allows us to x is defined as perform a Λ-limit of this net resulting in the re- Diu(x) = lim ∂iuλ(xλ). (7) quired hyperfinite function space, the space of λ↑Λ ultrafunctions, Axiom 5. If D : V ◦(Ω) → (V ◦(Ω))N is the gen- ◦ V (Ω) := lim Vλ. (5) λ↑Λ eralized gradient, i.e.,

This leads to a clear deﬁnition of the space of Du := (D1u ... DN u), ultrafunctions, which we equip with the axioms providing some properties needed in quantum then mechanics. Du = 0 ⇔ u = const .

Accepted in Quantum 2019-04-25, click title to verify 6 Axiom 6. If we set the support of an ultrafunc- integrable functions. This is the reason why the tion u as new notation H has been introduced. A key example (being very important when dealing with supp (u) = {x ∈ Γ | u(x) 6= 0} , singular potentials) is the characteristic function of a singleton. In fact, if a ∈ N ∩ Γ, then then R Z supp (Diχa(x)) ⊂ mon(a). χa(x)dx = 0 Axiom 7. For every u, v ∈ V ◦(Ω), I 6= χa(x)dx = ε ∼ 0, (9) I I (Diu(x))v(x)dx = − u(x)(Div(x))dx. (8) with ε > 0, so that it represents the “weight” of Let us notice that, in most previous papers, the point a. At the first sight, this Axiom might ultrafunctions spaces were assumed to witness seem unnatural. However, we work in a non- only some of the above axioms. In fact, con- Archimedean approach, thus, the infinitesimals structing a space of ultrafunctions that witnesses cannot be forgotten as is done in the Riemann all these axioms presents several technical chal- integration. lenges. However, these properties are fundamen- Axiom 4 shows that the generalized deriva- tal to develop many applications better, as we are tive extends the usual derivative. In fact, if 1 N N going to show. For more details on the construc- f ∈ C (R ) and x ∈ R , then tion of such a space, as well as for the relevance ◦ Dif (x) = lim ∂if(x) = ∂if(x). of these axioms, we refer to [31]. λ↑Λ However, the less intuitive fact is that the op- 3.2 Discussion of the axioms of the space of erator Di is defined on all ultrafunctions. The ultrafunctions last three axioms have been introduced to high- Axiom 1 means that every ultrafunction is a light that all the most useful properties of the grid function (based on V (Ω)), which has the fol- usual derivative are also satisfied by Di. In fact, lowing peculiar property: at every step λ the cor- Axiom 5 says that the ultrafunctions behave as responding finite-dimensional vector space which compactly supported C1 functions. contains all uλ is Vλ. Hence, every ultrafunction Axiom 6 states that the derivative is a local 9 is a Λ-limit of functions in Vλ. operator , which is a fundamental fact in the ap- Axiom 2 is a definition of the pointwise inte- plications of ultrafunctions to quantum mechan- gral. This integral extends the usual Riemann in- ics. 0 N Axiom 7 provides a weak form of Leibniz rule, tegral from functions in Cc (R ) to ultrafunctions ◦ 0 N which is of primary importance in the theory of in V (Ω). In fact, by definition, ∀f ∈ Cc (R ),

I Z 9 ◦ We do not want to enter too much into details here, f (x) dx = f(x) dx, as “locality” is all we need to develop the applications we have in mind. That said, using some basic tools of non- since for the functions of a compact support the standard analysis (underspill and saturation), it would be R simple to prove that, actually, there exists an infinitesi- net limλ↑Λ Q u dx becomes constantly equal to λ mal number η such that, for every a ∈ Γ, the expansion R u(x)dx. Moreover, if f ∈ L1( N ) ∩ V , then R of the derivative Dχa(x) in the base {χb}b∈Γ would in- I Z volve only the points b ∈ [a − η, a + η]. Therefore, the f(x) dx ∼ f(x) dx, second derivative of χa(x) would involve only the points b ∈ [a−2η, a+2η] and, more in general, the n-th derivative would involve the points in [a − nη, a + nη]. This shows as the quantity ∗ n that, for every n ∈ N such that nη ∼ 0, the operator D would still be local. This includes some infinite n, but not Z Z all infinite n, e.g., let n > 1 . In particular, the infinite η f(x) dx − f(x) dx matrix MD that corresponds to the operator D in the base N Qλ R {χb}b∈Γ is close to be diagonal, in the sense that if we let N = max |{[a − η] ∩ Γ|, |[a + η] ∩ Γ| | a ∈ Γ}, then in every becomes arbitrarily small for large λ. row of MD the only non-zero elements are the Mn,m with Axiom 3 shows that the equality H f ◦(x) dx = m ∈ [n − N, n + N]. This is similar to the computational R f(x) dx cannot hold for all arbitrary Riemann approximations of the derivative.

Accepted in Quantum 2019-04-25, click title to verify 7 weak derivatives, distribution, calculus of varia- where z is a complex conjugation of z. Due to tions, etc. We express this rule in its weak form Axiom 3 this form is a scalar product. More- since the Leibniz rule over, if u, v, u · v ∈ V ∩ L2 ∩ C0, then Z I D(fg) = (Df)g + fDg u(x) · v(x) dx ∼ u(x)v(x) dx X cannot be satisfied by every ultrafunction be- = u(x)v(x)d(x), (11) cause of the Schwartz impossibility theorem (see x∈Γ [32, 15]). In fact, if Leibniz rule held for all ul- with ∼ substituted by equality when u, v, u · v ∈ 2 0 N trafunctions, we would construct a differential L ∩ Cc R . This means that this sesquilinear algebra extending C0, that is not possible. In form extends the usual L2 scalar product. some sense, ultrafunctions provide a “solution” The norm of an ultrafunction is given by of Schwartz impossibility theorem at the cost of   I 1 2 2 X 2 using the weak formulation for Leibniz rule in- kuk = |u(x)| dx =  |u(a)| d(a) . 10 stead of the full one . a∈Γ In the theory of ultrafunctions, the Dirac delta 3.3 The structure of the space of ultrafunc- function has a simpler interpretation due to the tions pointwise integral. In fact, we can define the Since the space of ultrafunctions is generated by delta ultrafunction (called also the Dirac ultrafunction) as follows. For every a ∈ Γ, {χa}a∈Γ, it follows from Axiom 3 that the ultrafunctions integral H is actually a hyperfinite χ (x) δ (x) = a . sum. In fact, for every u ∈ V ◦, its integral can a d(a) be expanded as Our choice can be easily motivated, since, for ev- I ery u ∈ V ◦, u(x) dx = X u(a)d(a). I X a∈Γ δa(x)u(x) dx = u(x)δa(x)d(x) x∈Γ In order to analyze the applications of the space χ (x) = X u(x) a d(x) of ultrafunctions to quantum mechanics we con- d(a) sider the complex-valued ultrafunctions, so that x∈Γ we provide a complexification of the space of ul- = u(a). (12) ∗ trafunctions, In particular, if u = f |Γ for some f ∈ D(Ω), this shows that the scalar product between δ ◦ ◦ ◦ a H := V ⊕ iV . ∗ and u equals f (a). In particular, if a ∈ R then one recovers the classical expected property of a Coherently with this notation, we let delta function. H := V ⊕ iV Moreover, as in the ultrafunctions framework delta functions are actual functions (and not and functionals, like in distributions theory), we can perform on them all the classical operations that Hλ := Vλ ⊕ iVλ. do not have sense in the standard analysis like, The pointwise integral allows to define the follow- for example, δ2(x). ∗ ◦ ing sesquilinear form with values in C on H , As D ⊆ V , this shows that the delta ultrafunction behaves as the delta distribution when I X u(x)v(x) dx = u(x)v(x)d(x), (10) tested against functions in D. Moreover, delta x∈Γ functions are mutually orthogonal with respect to the scalar product (10). Hence, being normal- 10The reader interested in a deeper discussion of this fact is referred to [32]. Let us also mention that ized they provide an orthonormal basis, called Colombeau functions provide an alternative “solution” of delta-basis, given by the Schwartz theorem, see [33]. In the Colombeau ap- ( ) proach, Leibniz rule is preserved. However, we have to np o χa ∞ k δa = . (13) take C functions instead of C functions. a∈Γ p d(a) a∈Γ

Accepted in Quantum 2019-04-25, click title to verify 8 Hence, every ultrafunction can also be expanded Proposition 12 ensures that the above defini- in the following way, tion is well posed, as the map Z X I u(x) = u(ξ)δa(ξ)dξ χa(x). (14) Φ: v 7→ ψv dx a∈Γ is a functional on the space H◦. The scalar product allows the following ultrafunctions version of the Riesz representation the- 3.4 Ultrafunctions and distributions orem. Distributions can be easily embedded into ultra- Proposition 12. If functions spaces by identifying them with equiv- Φ: H → C alence classes [18]. is a linear internal functional, then there exists Definition 14. The space of generalized dis- ◦ tributions on Ω is defined as follows, uΦ such that, for all v = limλ↑Λ vλ ∈ H , I D0(Ω) = V ◦(Ω)/N, uΦv dx = lim Φ(vλ), λ↑Λ where and, for every f ∈ V , I N = τ ∈ V ◦(Ω) | ∀ϕ ∈ D(Ω), τϕ dx ∼ 0 . I u f ◦ dx = Φ(f). Φ The equivalence class of an ultrafunction u ∈ V ◦(Ω) is denoted by [u] . It contains all ultra- Proof. If v ∈ H , then the map D(Ω) λ functions, whose action on the functions in D(Ω) v 7→ Φ(v) differs from the action of u by at most an infinitesimal quantity. An obvious idea is to iden- is a linear functional over Hλ and hence, since tify this action with the action of a distribution. there exists uλ ∈ Hλ such that ∀v ∈ Hλ, However, it is not directly possible since there are Z ultrafunctions, whose action does not correspond u v dx = Φ (v) . λ to the action of any distribution. For example, If we set if u = τδ0, where τ is an infinite number, then ∀ϕ ∈ D(Ω) H uϕ∗ dx = τϕ(0), which is an in- uΦ = lim uλ, λ↑Λ finite quantity, whenever ϕ(0) is different from the conclusion follows. 0. This issue can be overcome by considering the As already stated, our goal is to apply the so-called “bounded” ultrafunctions. theory of ultrafunctions to quantum mechanics. In this way, we are interested in understanding Definition 15. Let [u]D(Ω) be a generalized dis- the relationship between ultrafunctions and L2- tribution. We say that [u]D(Ω) is a bounded gen- functions. Even though the scalar product in V ◦ eralized distribution if ∀ϕ ∈ D(Ω) the integral H ∗ 0 can be seen as an extension of the L2-scalar prod- uϕ dx is finite. We will denote by DB(Ω) the uct, it is still not clear whether L2 functions can set of bounded generalized distributions. be embedded into V ◦. The basic idea is to use In turn, the spaces of generalized distributions Eq. (3) to do such an association. However, this and bounded generalized distributions can be does not work since the L2-functions are not de- identified by an isomorphism, as shows the fol- fined pointwise. For this reason, we introduce lowing theorem. the following definition, which uses a weak form Theorem 16. There is a linear isomorphism of association. Φ: D0 (Ω) → D0(Ω) Definition 13. Given a function ψ ∈ L2(Ω), we B denote by ψ◦ the unique ultrafunction such that, 0 such that, for every [u]D(Ω) ∈ DB(Ω) and for ev- ◦ for every v = limλ↑Λ vλ(x) ∈ H , ery ϕ ∈ D(Ω), I Z I ◦ D E ∗ ψ v dx = lim ψvλ dx. Φ [u]D(Ω) , ϕ = st uϕ dx . λ↑Λ Ω D(Ω)

Accepted in Quantum 2019-04-25, click title to verify 9 Proof. For the proof see, e.g., [15]. that way, the ultrafunctions approach resembles the finite-dimensional vector spaces approach, in In this way, any equivalence class [u]D(Ω) ∈ the sense that the distinction between self-adjoint 0 0 DB(Ω) can be substituted by Φ ([u]D) ∈ D (Ω), operators and Hermitian operators is not needed. so that we can define its action on the functions We have proven the following in D(Ω), Theorem 17. Every Hermitian operator L : D E D E ◦ ◦ [u]D(Ω) , ϕ := Φ([u]D(Ω)), ϕ H → H is self-adjoint. D(Ω) D(Ω) I In [31], it is possible to find a detailed analysis = st uϕ∗ dx . of the ultrafunctions formalization of the position and the momentum operators. For our applica- In particular, if f ∈ C0(Ω) and f ∗ ∈ [u] , c D(Ω) tions to the Schrödingerequation, let us notice then, for all ϕ ∈ D(Ω), that the Laplacian operator D E I [u] , ϕ = st u ϕ∗ dx 2 N 0 N D(Ω) D(Ω) ∆ : C R → C R I Z = st f ∗ϕ∗ dx = fϕ dx. has the following expression as its ultrafunctions formulation, 3.5 Self-adjoint operators on ultrafunctions N 2 X 2 ◦ ◦ D = Dj : V → V . Let an operator j=1

L : H◦ → H◦ For applications to quantum mechanics, the following Hamiltonian operator is fundamental, be an internal linear operator on the space of ul- 1 trafunctions, which is a hyperfinite-dimensional Huˆ (x) = − ∆u(x) + V(x)u(x), (15) space by definition. In this way, L can be repre- 2 sented by a hyperfinite matrix (viewed as an in- where we assume that the mass of the i-th parti- finite matrix by the standard analysis), because cle mi = 1 and ~ = 1. we can build a Λ-limit There is a deep and important difference between the standard and the ultrafunctions ap- Lu := lim Lλuλ, proach to the study of Hˆ . In the classical L2- λ↑Λ theory, a fundamental problem is a choice of an where the operators appropriate potential V such that (15) makes sense. Moreover, it is fundamental to define an Lλ : Hλ → Hλ appropriate self-adjoint realization of Hˆ . On the other hand, in the theory of ultrafunctions any can be represented by finite matrices (as every internal function V :Γ → provides a self- space V is finite-dimensional). In particular, if E λ adjoint operator on H◦, given by L is a self-adjoint operator, 1 I I Hˆ ◦u(x) := − D2u(x) + V(x)u(x), (16) Luv dx = uLv dx, 2 which always has a discrete (in the ∗ sense) then the matrices L are Hermitian. Hence, L R λ spectrum that consists of eigenvalues only. Of can be represented by a hyperfinite-dimensional course, this spectrum will be hyperfinite, of car- Hermitian matrix. Therefore, the spectrum σ(L) dinality equal to the cardinality of Γ (once the of L consists of eigenvalues only, more precisely, multiplicities of the eigenvalues are taken into account). For these reasons, we believe that the σ(L) = lim µλ ∈ E | ∀λ, µλ ∈ σ(Lλ) , λ↑Λ ultrafunctions approach allows a much simpler study of “very singular potentials” such as and its corresponding normalized eigenfunctions (being Λ-limits of the corresponding eigenfunc- V(x) = τδa(x), τ ∈ E, (17) ◦ tions of Lλ) form an orthonormal basis of H . In V(x) = ΩχE(x), Ω ∈ E, (18)

Accepted in Quantum 2019-04-25, click title to verify 10 where τ and Ω might be inﬁnite numbers. The will reformulate the basis of quantum mechanics use of non-Archimedean methods is fundamental — its system of axioms — through the ultrafunc- to give a reasonable model of these potentials, tions. which have an interesting physical meaning. Par- ticularly, the delta function potential (17), which represents a very short-ranged interaction, appears in many physical problems. For example, 4 Ultrafunctions and quantum me- it can serve as a reasonable model for the inter- chanics actions between atoms and electromagnetic ﬁelds (particularly, dipole-dipole interactions) and in- 4.1 Axioms of QM based on ultrafunctions teratomic potential in a many-body system (particularly, Bose – Einstein condensate). In this In the following table, we provide a set of axioms way, in the following, we will take a look at the of quantum mechanics formulated within the ul- quantum system with a delta potential within trafunctions approach and compare it with the the ultrafunctions framework. Before to start, we standard axioms11.

Standard QM Ultrafunctions QM

Axiom 1 A physical state of a quantum system A physical state of a quantum system is is described by a unit vector |ψi in a described by a unit complex-valued complex Hilbert space H. ultrafunction ψ ∈ H◦.

Axiom 2 A physical observable A is represented A physical observable A is represented by a linear self-adjoint operator Aˆ act- by a linear Hermitian operator Aˆ acting ing in H. in H◦.

Axiom 3 The only possible outcomes of a mea- The only possible outcomes of a measurement of an observable A form a set surement of an observable A form a {µj}, where µj ∈ σ(Aˆ) are the (gener- set {st(µj)}, where µj ∈ σ(A) are the alized) eigenvalues of Aˆ. eigenvalues of the operator Aˆ.

Axiom 4 An outcome µj of a measurement of an An outcome st(µj) of a measurement of observable A can be obtained with a an observable A can be obtained with probability a probability

2 2 Pj = |hψj|ψi| , Pj = |hψ, ψji| ,

where |ψji is the (generalized) eigen- where ψj is the eigenstate associated state associated with the observed with the observed eigenvalue µj. After (generalized) eigenvalue µj. After the the measurement, the quantum system measurement, the quantum system is is left in the state ψj. left in the state |ψji.

11For the sake of simplicity, we focus on the system of axioms based on pure states of the quantum system. How- ever, it can be straightforwardly generalized to the case of mixed states, which describe a statistical mixture of the pure states.

Accepted in Quantum 2019-04-25, click title to verify 11 Axiom 5 The time evolution of the state of the The time evolution of the state of the quantum system is described by the quantum system is described by the Schr¨odingerequation Schr¨odingerequation

∂|ψi ∂ψ i = Hˆ |ψi, i = Hˆ ◦ψ, ∂t ∂t

where Hˆ is the Hamiltonian of the where Hˆ ◦ is the Hamiltonian of the system. system.

Axiom 6 — In a laboratory only the states associated to a ﬁnite expectation value of the physically relevant quantities can be re- alized. These states are called physical states, the rest of the states is called ideal states.

4.2 Discussion of the axioms self-adjoint (Aˆ = Aˆ†) so that it can represent a physical observable12. Axiom 1: States In the ultrafunctions formalism, the observ- In the standard formalism, a physical system is ables of a quantum system can be represented described by a unit vector in Hilbert space. The by internal Hermitian operators, which are triv- ultrafunctions formulation of Axiom 1 guar- ially self-adjoint due to the Theorem 17. Hence, antees that we use the ultrafunctions space H◦ for observables, “Hermitian” and “self-adjoint” (hence, non-Archimedean mathematics) instead become equivalent. of the Hilbert one H. In particular, working within wave functions, the state |ψi can be repre- Axioms 3, 4: Measurement sented by a normalized function ψ ∈ L2(Ω), Ω ⊂ In the standard formalism, the possible measure- N . Since L2 can be embedded in V ◦, there ex- R ment outcomes of an observable A belong to the ists a canonical embedding due to the Def. 13, spectrum of the corresponding self-adjoint operator Aˆ, which, generally speaking, is decomposed ◦ : H → H◦, into two sets, discrete spectrum, and continuous given by spectrum. While the discrete spectrum contains the eigenvalues of Aˆ, the continuous one is a set of ψ 7→ ψ◦. the so-called generalized eigenvalues, which cor- Since H◦ is a space much richer than H, the ultra- respond to the generalized eigenstates, the eigenfunctions framework recovers all standard states states which do not belong to H. A typical ex- and provides more possible states, particularly, ample is given by the position operator qˆ on the 2 the ideal states of Axiom 6. Hilbert space L (R), whose spectrum is purely

12For example, if we consider the system of a parti- Axiom 2: Observables cle in an infinite potential well, the momentum operator ˆ d P = −i dx is Hermitian but not self-adjoint: the domain The standard and ultrafunctions formulations of of Pˆ is only a subset of the domain of its adjoint. This Axiom 2 highlight many similarities as well as fact leads to problems with a physical interpretation of some differences. The main difference is the the spectrum of an Hermitian but not self-adjoint operator, which can turn out to be the so-called residual spec- fact that the ultrafunctions formalism needs von trum (being not interpretable physically). The reader in- Neumann’s notion of the self-adjoint operator no terested in a deeper discussion of the difference between more. In the standard formalism, it is not enough Hermitian and self-adjoint operators, and its consequences for an operator Aˆ to be Hermitian – it has to be in the standard quantum mechanics is referred to [36].

Accepted in Quantum 2019-04-25, click title to verify 12 continuous (the whole real line R), and the cor- Axiom 5: Evolution responding eigenfunctions ψ (x) = δ(x − q) do q The ultrafunctions version of this axiom is very not belong to L2( ). In that way, the manner R similar to the standard one. Since Hˆ ◦ is an inter- the standard formalism treats the discrete spec- nal operator defined on a hyperfinite-dimensional trum in does not fit for the continuous spectrum vector space H◦, it can be represented by an Her- leading to numerous misunderstandings: an in- mitian hyperfinite matrix due to the Theorem 17. troduction of some additional constructions such Hence, the evolution operator of the quantum as rigged Hilbert spaces is needed [36]. system is described by the exponential matrix In the ultrafunctions formalism, due to the Uˆ ◦(t) = e−iHˆ ◦t. self-adjointness of internal Hermitian operators, it follows that any observable has exactly κ = dim∗(H◦) = |Γ| eigenvalues (taking into account Axiom 6: Physical and ideal states their multiplicity). In this way, no more essential This is the most peculiar axiom in the ultra- distinction between eigenvalues and a continuous functions approach. In the ultrafunctions theory, spectrum is needed since a continuous spectrum the mathematical distinction between the phys- can be considered as a discrete spectrum contain- ical eigenstates and the ideal eigenstates is in- ing eigenvalues infinitely close to each other. trinsic. It does not correspond to anything in For example, if we consider the eigenvalues of the standard formalism. Hence, it opens a very the position operator q of a free particle, then the interesting problem of the physical relevance of eigenfunction relative to the eigenvalue q ∈ R is such states. Basically, we can intuitively say the Dirac ultrafunction δq. Therefore, it can be that the physical states correspond to the states trivially seen that its spectrum is Γ, which is not which can be prepared and measured in a labora- ∗ a continuous spectrum in R . However, the stan- tory, while the ideal states represent “extreme” dard continuous spectrum can be recovered since states useful in the foundations of quantum me- the eigenvalues of an internal Hermitian operator chanics, thought experiments (Gedankenexperi- ˆ A are Euclidean numbers. Hence, assuming that mente) and computations. measurement gives a real number, we have im- For example, a Dirac ultrafunction is not a posed in the Axiom 3 that its outcome is st(µ). physical state but an ideal state. It represents In the case of the spectrum of the position oper- a state in which the position of the particle is ator, we can show that perfectly determined, which is precisely what one has in mind considering the Dirac delta distribu- {st(µ) | µ ∈ σ(ˆq)} = , R tion. Clearly, this state cannot be produced in a because every real number lies in Γ. laboratory since it requires infinite energy. How- Working in a non-Archimedean framework, we ever, it is useful in our description of the physi- have set in a natural way that the transition cal world since such a state makes more explicit probabilities should be non-Archimedean. In the standard approach. Therefore, Axiom 6 fact, we assume that the probability is better highlights a concept that is already present (but 2 somehow hidden) in the standard approach. described by the Euclidean number |hψ, ψji| , rather than the real number st(|hψ, ψ i|2). For For example, in the Schrödingerrepresentation j 3 example, let ψ ∈ H◦ be the state of a system. of a free particle in R , consider the state The probability of an observation of the particle ϕ(x) in a position q is given by ψ(x) = , ϕ ∈ D( 3), ϕ(0) > 0. |x| R I 2 q 2 2 3 ψ(x)δq(x) d(q) dx = |ψ(q)| d(q), We see that ψ(x) ∈ L (R ) but this state cannot be produced in a laboratory, since the expected which is an infinitesimal number. The standard value of its energy probability can be recovered by the means of the 1 Z standard part, in the case of the position operator hHψ,ˆ ψi = |∇ψ|2 dx 2 it would be zero (as is expected). We refer to [34, 35] for a presentation and discussion of the is infinite (even if the result of a single experiment non-Archimedean probability. is a finite number).

Accepted in Quantum 2019-04-25, click title to verify 13 5 Example: Hamiltonian with a δ- and, in the obtained solution, takes the limit ε → potential 0 and expands the box by the limit L → ∞, in order to recover the original problem. To compare the standard and the ultrafunctions approaches, in this section, we want to study the Potential barrier (τ > 0) Schr¨odingerequation We start with the case of a potential barrier with 1 τ > 0, which is illustrated on the Fig.1. Hψˆ ≡ − ∆ψ + τδ (x)ψ = Eψ, (19) 2 0 which corresponds to the problem of a particle V0 moving in the Dirac delta potential of transparency τ at the point x = 0. As we are going to show, there are two main diﬀerences between the standard and the ultrafunctions approaches,

1) the standard approach changes if we change −L 0 L −L −ε ε L the dimension of the space (as in the space of dimension D > 1 one of the main prob- Figure 1: Delta potential as a limit of a finite barrier. lems is to find a self-adjoint representation of Hˆ ), while in the ultrafunctions approach we always have a self-adjoint representation 2 2 of Hˆ , independently of D, due to Theorem Using the abbreviations k = 2E and κ = 17, 2(V0 − E) we obtain the following solution of the Schrödingerequation, 2) in the ultrafunctions approach the constant  τ can be an infinitesimal or an infinite num-  ±  ∓A sin(k(x + L)), x ∈ [−L, −ε], ber, which allows us to construct the mod-   els which cannot be considered within the ±  ψ (x) = B±f ±(x), x ∈ [−ε, ε], standard framework (for example, a poten-   tial equal to the characteristic function of a   A± sin(k(x − L)), x ∈ [ε, L], point13).  where A± and B± are the normalization con- 5.1 The standard approach + − stants, f (x) = cosh(κx), f (x) = sinh(κx), Let us review the solution of the problem within “+” corresponds to the even function and “−” the standard approach (namely, τ ∈ R)[37, 38]. corresponds to the odd function. We seek the At first, it is assumed that the particle moves in energies of the eigenstates, which can be found a box of length 2L. In turn, the delta potential with use of the continuity condition, is approximated as a square wall (well) of a finite ψ0(x) ψ0(x) length 2ε and a finite height V0, = . (22) ψ(x) ψ(x) V0→∞, ε→0 x→ε−0 x→ε+0 Vε(x) = V0θ(|ε − x|) −−−−−−−−→ τδ0(x),(20) In turn, the original problem is recovered by where the transparency of the potential is re- taking the limit (20), which leads to the following lated to the parameters of the approximation as solution, τ = 2εV0. In that way, within the standard ap-  proach, one solves the Schrödingerequation with   ∓A± sin(k±(x + L)), x ∈ [−L, 0], the approximated potential, ±  n n ψn (x) =  1  A± sin(k±(x − L)), x ∈ [0,L], − ∆ψ + V (x)ψ = Eψ, (21)  n n 2 ε 2k±L−sin(2k±L) 13 ± −2 n n In fact, within standard approach such a potential with the normalization |An | = ± 2kn would be indistinguishable from 0. ± and the quantities kn , which are defined by the

Accepted in Quantum 2019-04-25, click title to verify 14 following equations due to Eq. (22), limits as done above one obtains the following solution, k+ cot(k+L) = −τ, (23) n n  π + 2πn − − −  ± ± kn cot(kn L) = −∞ ⇒ kn = ,(24)  ∓A sin(k (x + L)), x ∈ [−L, 0], L ±  n n ψn (x) = 2  kn  A± sin(k±(x − L)), x ∈ [0,L], and, in turn, deﬁne the energies En = 2 of the  n n eigenstates. At the singularity point, we obtain ± ± ± −2 2kn L−sin(2kn L) + + + with the normalization |An | = ± ψ (0) = −A sin(k L), (25) 2kn n n n and the quantities k±, which are deﬁned by the ψ−(0) = 0. (26) n n equations

+ + Potential well (τ < 0) kn cot(kn L) = τ, (27) 2πn Let us now consider the case of a potential well k− cot(k−L) = ∞ ⇒ k− = , (28) n n n L (illustrated on the Fig.2), namely, the negative 2 kn transparency τ < 0. In this case, we have two and, in turn, deﬁne the energies En = 2 of the diﬀerent situations with respect to energy, E > 0 eigenstates. At the singularity point, we obtain

(scattering states) and E < 0 (bound states). + + + ψn (0) = −An sin(kn L), (29) − ψn (0) = 0. (30)

Negative energies (E < 0): bound states. 2 2 Using the abbreviations k = 2|E| and κ = 2(V0 − |E|) we obtain the following solution of the Schrödinger equation, 0 −ε ε   ± −L L −L L  ∓A sinh(k(x + L)), x ∈ [−L, −ε],   ±  ψ (x) = B±f ±(x), x ∈ [−ε, ε],     A± sinh(k(x − L)), x ∈ [ε, L], −V0 where A± and B± are the normalization con- + − Figure 2: Delta potential as a limit of a finite well. stants, f (x) = cos(κx), f (x) = sin(κx),“+” corresponds to the even function and “−” corresponds to the odd function. By performing the limits as done above one obtains the following Positive energies (E > 0): scattering solution, 2 states. Using the abbreviations k = 2E and  2  κ = 2(V0 + E) we obtain the following solution  ∓A± sinh(k±(x + L)), x ∈ [−L, 0], ±  n n of the Schrödingerequation, ψn (x) =  ± ±   An sinh(kn (x − L)), x ∈ [0,L],  ±  ∓A sin(k(x + L)), x ∈ [−L, −ε],  ± ±  ± −2 2kn L−sinh(2kn L) ±  with the normalization |An | = ± ψ (x) = ± ± 2kn B f (x), x ∈ [−ε, ε], ±  and the quantities kn , which are defined by the   equations  A± sin(k(x − L)), x ∈ [ε, L], + + kn coth(kn L) = τ, (31) where A± and B± are the normalization con- − − kn coth(kn L) = ∞, (32) stants, f +(x) = cos( x), f −(x) = sin( x),“+” κ κ 2 kn corresponds to the even function and “−” corre- and, in turn, define the energies En = − 2 of sponds to the odd function. By performing the the eigenstates. In this way, we find that, for

Accepted in Quantum 2019-04-25, click title to verify 15 E < 0, there exists a unique value k+ defined by in this case. However, since the Bessel func- Eq. (31). With expanding the box by taking the tion J0(r) with an azimuthal symmetry asymp- limit L → ∞, it corresponds to the energy of the totically behaves as a cosine, the number of the unique bound state bound states can be approximately estimated as 2 N = εκmax/π, where κmax = 2V0. Plugging in τ 2 E = − . (33) the transparency, we find 1D 2 1 r2τ At the singularity point, we obtain N2D ' , (36) π π ψ+(0) = −A+ sin(k+L). (34) which is a finite number. The solution for the three-dimensional potential well can be found in Multidimensional δ-potentials the form of trigonometric functions, that simpli- fies the calculations. The number of bound states In the analysis of the delta potential, we have fo- is given by [40, 41] cused on the 1-dimensional case. In this case, one can easily prove that there exists a self-adjoint re- 1 r 3τ N ' , (37) alization of the Hamiltonian Hˆ bounded below. 3D π 2πε As shown above, there exists a unique bound which is already an infinite number. state for a Hamiltonian with a one-dimensional The energy of a bound state for the delta po- delta potential well. tential can be found by solving the following The delta potentials of a higher dimensional- equation [48, 49], ity D provide not only a pedagogical model but find their applications in nuclear and condensed 1 1 Z dD k + = 0, (38) matter physics [39, 40, 41, 42]. However, solving τ (2π)D k2 − E a Schrödingerequation with these potentials is a more cumbersome problem since there is no rig- where D is the number of dimensions. The inte- orous construction of a self-adjoint realization of gral in this equation is divergent for D ≥ 2, that the Hamiltonian in Eq. (19) as long as it is not results in infinite energies of the bound states. In extended to a non-interacting Hamiltonian on a particular, in the two-dimensional case, it can be space with a removed point [11, 43, 44, 45, 46, shown that [40] 47]. Let us illustrate it by the case of a two- 1 − 2π dimensional and three-dimensional delta poten- E2D ' − e τ , 2ε2 tial well. As in the one-dimensional case, the delta po- which is infinite. In this way, we conclude that tential well of dimensionality D can be approx- multidimensional delta potential wells provide, imated by a finite square well described by the generally speaking, an infinite number of bound Heaviside function θ(r), states with the energies diverging to −∞. In order to avoid the problems with the in- V0→∞, ε→0 finite quantities (diverging energies) within the Vε(r) = −V0θ(ε − r) −−−−−−−−→ −τδ0(r), (35) standard approach, one usually applies the so- where its depth V0 and length ε are related to called regularization and renormalization proce- the transparency of the delta potential well as dures to the calculations. The regularization pro- 2 14 τ2D = πε V0 in the two-dimensional case and vides a cut-off σ for the divergent integral, and 4π 3 the renormalization re-defines the transparency τ3D = 3 ε V0 in the three-dimensional case, re- spectively [40, 41]. τ via τR (where R stands for “renormalized”), The most interesting part of the problem with which absorbs the dependence on the cut-off in a delta potential well is the estimation of the order to absorb the divergence of the integral. bound states. After solving the corresponding 14In the renormalization theory, as well as in the ref- Schrödingerequation, the bound states can be erences, the cut-off is denoted by Λ. However, to avoid found from the continuity equation. In the two- confusion with the Λ-limit we prefer to change the nota- dimensional case, it is more difficult to do, be- tion here, as we will not discuss renormalization in detail cause the wave functions are Bessel functions in this paper.

Accepted in Quantum 2019-04-25, click title to verify 16 P χ0(x) In particular, in the two-dimensional case, the Therefore, since a∈Γ u(a)δ(a)χa(x) = u(0) d(0) renormalized transparency is deﬁned by the fol- we can rewrite Eq. (42) as lowing equation [48, 49],   X X χ0(x) 2 ! − u(a) Da,bχb(x) + τu(0) 1 1 1 σ d(0) = + ln , (39) a∈Γ b∈Γ∩mon(a) τ τ 4π ω2 R X = E u(a)χa(x). (43) where ω is an arbitrary parameter, which repre- a∈Γ sents the renormalization scale. Then, one takes Let us explicitly note that the above discus- a limit σ → ∞ and varies the “bare” trans- sion does not depend on the dimensionality of parency τ in such a way that the renormalized the space. In fact, for any number of dimensions, transparency τR remains ﬁnite [48, 49]. Such Eq. (42) reveals the self-adjoint representation of a renormalized transparency leads to a unique the Hamiltonian in Eq. (19) in the ultrafunctions bound state with the binding energy setting.

4π Outside the monad of 0, the equation actually 2 − τ E2D = −ω e R . (40) reads

X 2 Finally, this procedure means that one introduces − u(a)Da(x) = Eu(x), (44) an additional scale in order to “forget” about the a∈mon(x)∩Γ unboundedness of the Hamiltonian from below. which is formally the same that one gets for the equation 5.2 The ultrafunctions approach − D2u = Eu. (45) Solution in the singularity point On the other hand, at the point 0 ∈ Γ, we obtain In the ultrafunctions approach, the Schr¨odinger the equation equation reads τ − X u(a)D2(0) + u(0) = Eu(0), a d(0) 2 a∈mon(0)∩Γ − D u + τδ0u = Eu, (41)

∗ ◦ which corresponds to where τ, E ∈ R . We use the linearity of V to ! write the solution u of Eq. (41) as P P u(a) Da,bχb(x) X a∈mon(0)∩Γ b∈Γ∩mon(0) u(x) = u(a)χa(x). u(0) = τ . a∈Γ E − d(0) Therefore, we can rewrite Eq. (41) as Notice that, for every other point γ ∈ mon(0)∩ Γ, we obtain − X u(a)D2χ (x) + τ X u(a)χ (x)δ (x) a a 0 ! a∈Γ a∈Γ P P X u(a) Da,bχb(x) = E u(a)χ (x). (42) a∈mon(0)∩Γ b∈Γ∩mon(0) a u(γ) = . a∈Γ E 2 P For every a ∈ Γ, let D χa(x) := b∈Γ Da,bχb(x). In particular, it shows that, in the ba- Notice that the locality of the ultrafunctions sis {χa}a∈Γ, the matrix representation Mδ of 15 derivative entails that, actually, we can write Eq. (41) and the matrix representation M∆ of Eq. (45) satisfy the relation Mδ = M∆+H, where 2 X D χa(x) = Da,bχb(x). H is a matrix with b∈Γ∩mon(a)  τ  d(0) , if a = b = 0, 15Readers familiar with nonstandard analysis will notice Ha,b = P 0, otherwise. that our notation b∈mon(a) is not proper, as mon(a) ∩ Γ is not an internal set (hence, in particular, it is not hyperfinite). However, by underspill it is trivial to prove Particularly, in the case of a one-dimensional po- that Da,b 6= 0 only for a hyperfinite amount of points tential well with an infinitesimal transparency, b ∈ mon(a) ∩ Γ, as we have already noticed in Section 3.2, there will still exist a unique bound state with hence ours is just a small abuse of notations. infinitesimal negative energy.

Accepted in Quantum 2019-04-25, click title to verify 17 ∗ Connection with the standard solutions τ and the energy E are finite numbers in R . Let w ∈ D0(Ω) be such that16 As usual, in order to introduce a new solution concept for a standard problem, it is important 1. [u]D(Ω) = w, to discuss the relationship between the standard and new solution. Our first result shows that the 2. [δ · u]D(Ω) = δ · w. approximation procedure used in most standard Then w is a solution of the standard Schrödinger approaches to the Schrödingerequation with a equation delta potential can be reproduced within the ultrafunctions approach, in the following sense. − ∆w + 2 st(τ)δ0w = st(E)w. (48)

Theorem 18. Let 0 ∼ aΛ = limλ↑Λ aλ and Proof. By the deﬁnition of [u]D(Ω), we can write τΛ = limλ↑Λ τλ. Then there exists a space of ul- ◦ ◦ u = w + ψ, (49) trafunctions V that contains a solution uΛ of the equation where ψ is such that for any f ∈ D(Ω),

2 ◦ − D uΛ + τΛδ0uΛ = EΛuΛ, (46) hψ, f i ∼ 0. (50) which is a Λ-limit of a net of solutions uλ ∈ V (Ω) If f ∈ D(Ω), we have of the standard Schr¨odingerequations hw, fi = hw◦, f ◦i = hu, f ◦i − hψ, f ◦i ∼ hu, f ◦i. − ∆u + τ A u = E u , (47) λ λ aλ λ λ λ Using Eq. (41) we can write

1 where Aaλ is an approximation of the delta dis- ◦ 2 ◦ ◦ hu, f i = hD u, f i + τhδ0u, f i . tribution by a square well of length aλ ∈ R, and, E for every λ, Eλ is an eigenvalue of the energy The first term can be rewritten in the following of the solution of the standard Schrödingerequa- way, tion with the approximated delta potential such 1 2 ◦ 1 2 ◦ that EΛ = limλ↑Λ Eλ. hD u, f i = hu, D f i E E 1 1 Proof. For every λ ∈ Λ, let uλ be a solution of ∼ hw, ∆fi = h∆w, fi. st(E) st(E) Eq. (47). Let {Vλ}λ be an increasing net of finite dimensional subspaces of V (Ω) as in Definition Using the hypothesis on [δ · u]D(Ω), the second 11. If ∀λ the solution uλ ∈ Vλ, then a Λ-limit term can be rewritten in the following way, uΛ = limλ↑Λ uλ is contained in the space of ul- τ 2 st(τ) 2 st(τ) ◦ hδ u, f ◦i ∼ hδ u, f ◦i ∼ hδ w, fi. trafunctions V generated by the net {Vλ}λ, and E 0 st(E) 0 st(E) 0 theorem is proven. This means that If uλ ∈/ Vλ, then we extend Vλ by 1 2 st(τ) 0 hw, fi = h∆w, fi + hδ0w, fi, Vλ := span(Vλ ∪ {uλ}), st(E) st(E)

0 or, equivalently, and build the Λ-limit with the net {V }λ. In this way, the Λ-limit of the net {uλ} gives an ∆w + 2 st(τ)δ0w = st(E)w, ultrafunction in V ◦ by construction. which is the corresponding standard Schrödinger The second result we want to show precises equation. the relationship between the ultrafunction and 16 standard solution in the case of finite energy and In condition 2, we have a distributional product δ · w, which is not defined, in general. We tacitly assume that transparency. w is such that this product makes sense as a distribution. This is the case, e.g., when w is smooth. Notice that, Theorem 19. Let u ∈ V ◦ be an ultrafunction in contrast, the product of ultrafunctions is always well solution of the Eq. (41), where the transparency defined.

Accepted in Quantum 2019-04-25, click title to verify 18 Bounds for the energy valid not for the one-dimensional potential only, but for the multidimensional ones as well. Some easy bounds on the energy of the solution Let us notice that this bound reveals a possibil- of Eq. (41) can be found using the scalar product ity to fix always an (infinitesimal) transparency τ of ultrafunctions. By Axiom 7 of the space of close to d(0), so that there exists a solution of the ultrafunctions, corresponding Schrödingerequation with finite energy. For example, we can fix τ = d(0). Notice h∆u(x), u(x)i = −hDu(x), Du(x)i ≤ 0. (51) that, in this case, the potential τδ0 is equal to the ultrafunction indicator χ0, that in the standard Using the Schrödingerequation we can rewrite approach would not differ from 0. the previous equation, in order to get

D E h∆u(x), u(x)i = −2 E−τδ (x) u(x), u(x) ≤ 0. 0 Splitting of the ultrafunction solution Calculating the scalar product and applying the Let us discuss a particular aspect of non- normalization of the wave function ||u(x)||2 = Archimedean mathematics in general, and ul- P 2 a∈Γ u (a)d(a) = 1 we obtain trafunctions in particular briefly. As we have highlighted above, the ultrafunction solution of E − τu2(0) ≥ 0. (52) Eq. (19) has the form of the sum of the “extended” standard solution ψ◦ and the non- Taking into account that the transparency can Archimedean contribution ϕ. Now, one might take both positive (potential barrier) and nega- ask whether it is possible for ϕ to be concen- tive (potential well) values we obtain inequalities trated in the monad of zero or have some similar for allowed energies additional property. In principle, it is possible if the delta distri- E ≥ u2(0), τ ≥ 0, (53) bution δ is represented within the ultrafunctions τ framework differently. In fact, we have used E 2 χ0 ≤ u (0), τ < 0. (54) the delta ultrafunction δ0 = d(0) to model the τ delta distribution, which appears in the standard The first inequality implies Hamiltonian in Eq. (19), because we believe this is a natural approach to follow in our setting. E ≥ 0, τ ≥ 0, (55) However, there are several ultrafunctions u in ◦ V (Ω) which correspond to δ0 as distributions, ◦ which means that the negative energies are not so that the scalar product hu, f i = f(0) for ev- 17 allowed in the case of a potential barrier. ery f ∈ D(Ω) . Of course, by changing the set In order to understand better the second in- of test functions, we obtain different conditions. ◦ equality, let us take a look to the following in- For example, if we use V instead of D(Ω), then equality, the unique “delta” is precisely the delta ultrafunction. In this sense, there are several different u2(0)d(0) ≥ X u2(a)d(a) = ||u(x)||2 = 1. (56) models of the Hamiltonian in Eq. (19) that could ◦ a∈Γ be constructed in V (Ω). This gives us a degree of freedom which is not present in the standard Implying this inequality results in approach.

2|τ| |E| ≤ , τ < 0. (57) Comparison of the approaches d(0) We summarize the discussion of the interpreta- In this way, we see that the value of transparency tion of the Schr¨odingerequation with a delta po- of the potential well establishes an upper bound tential within the ultrafunctions approach in the for the allowed negative energies. This bound is following table.

17This is the notion of “association”, which is often considered in non-Archimedean mathematics when dealing with distributions, see, e.g., [33].

Accepted in Quantum 2019-04-25, click title to verify 19 Standard approach Ultrafunction approach

Analysis of the solutions of the The solutions can be analyzed directly Schrödingerequation (19) needs an ap- with the use of the delta ultrafunction, proximation of the delta function by a which describes an infinite jump con- finite potential (for example, a square centrated in 0. However, the approxi- barrier/well) and performing a limit of mation procedure can be used as well its zero width. due to Theorem 18.

Multidimensional delta potentials are There is a unique framework for the difficult to interpret because of the di- delta potentials of any dimension based vergence of the corresponding integrals. on hyperfinite spaces and, thus, hyperfinite matrices. Infinite or infinitesimal numbers can be used as parameters of the equation. In this way, a model comes out which better describes the physical phenomenon.

The number of the eigenstates depends There always exists the number of on the number of physical dimensions eigenstates equal to the hyperﬁ- D. In particular, the number of bound nite dimensionality of the space of states is inﬁnite for D ≥ 3. ultrafunctions.

The energies of the eigenstates are fi- Natural bounds for the energies of the nite only for a one-dimensional delta eigenstates can be estimated indepen- potential. dently of the dimension of the system. Moreover, it is possible to obtain the bound states with finite energies by fix- ing suitable transparency.

— The splitting of the corresponding ultrafunction can recover the standard content of the solution.

6 Conclusions set of the axioms of quantum mechanics. Even though the new axioms are obviously related with In this paper, we have introduced a new approach the standard Dirac – von Neumann formulation to quantum mechanics using the advantages of of quantum mechanics, ultrafunctions make it non-Archimedean mathematics. It is based on ul- possible to build a framework which is closer trafunctions, non-Archimedean generalized func- to the matrix approach. Moreover, the pres- tions defined on a hyperreal field of Euclidean ence of infinite and infinitesimal elements allows ∗ numbers E = R , which is a natural extension constructing new simplified models of physical of the field of real numbers. Euclidean numbers problems. In particular, this framework is based have been chosen because they allow constructing on Hermitian operators, so that unbounded self- a non-Archimedean framework through a simple adjoint operators are not more needed. Further- notion of Λ-limit, which gives an advantage con- more, the difference between continuous and dis- cerning other nonstandard approaches to quan- crete spectra is not more present. tum mechanics [8,9, 10, 11, 12]. These aspects were illustrated by a compari- The ultrafunctions approach proposes a new son between standard and ultrafunction solutions

Accepted in Quantum 2019-04-25, click title to verify 20 of a Schrödingerequation with a delta potential. [10] F. Bagarello and S. Valenti, Int. We have shown that the ultrafunctions approach J. Theor. Phys. 27, 557 (1988), provides a more straightforward way to solve the DOI:10.1007/BF00668838. Schrödingerequation by unifying the cases of a [11] S. Albeverio, F. Gesztesy, R. Høegh- potential well and a potential barrier and provid- Krohn and H. Holden, Solvable Models ing the extreme cases of infinite and infinitesimal in Quantum Mechanics (Springer, 1988), transparency of the potential. Moreover, it gives DOI:10.1007/978-3-642-88201-2. a connection between energy and transparency, providing a bound for the allowed energies of the [12] A. Raab, J. Math. Phys. 45, 47 (2004), quantum system. DOI:10.1063/1.1812358. [13] V. Benci, Adv. Nonlinear Stud. 13, 461 Acknowledgements (2013), DOI:10.1515/ans-2013-0212. [14] V. Benci and L. Luperi Baglini, Discrete L. Luperi Baglini and K. Simonov acknowledge Contin. Dyn. Syst. Ser. S 7, 593 (2014), financial support by the Austrian Science Fund DOI:10.3934/dcdss.2014.7.593. (FWF): P30821. [15] V. Benci and L. Luperi Baglini, Monatsh. Math. 176, 503 (2014), References DOI:10.1007/s00605-014-0647-x. [1] F. Strocchi, An Introduction to the [16] V. Benci and L. Luperi Baglini, in Vari- Mathematical Structure of Quantum ational and Topological Methods: The- Mechanics (World Scientific, 2005), ory, Applications, Numerical Simulations, DOI:doi.org/10.1142/7038. and Open Problems, Flagstaff, Arizona, USA, 2012, edited by J. M. Neuberger, [2] A. Robinson, Non-standard Analysis M. Chherti, P. Girg and P. Takac, Electron. (North-Holland, 1974). J. Diff. Eqns., Conference 21 (2014), pp. 11– [3] L. O. Arkeryd, N. J. Cutland and 21. C. W. Henson (Eds.), Nonstandard Anal- ysis: Theory and Applications (Springer, [17] V. Benci and L. Luperi Baglini, in Anal- 1997), DOI:10.1007/978-94-011-5544-1. ysis and Topology in Nonlinear Differen- tial Equations, edited by D. G. Figueiredo, [4] S. Albeverio, in Mathematics+Physics: Lec- J. M. do O´ and C. Tomei (Birkhäuser,2014), tures on Recent Results, edited by L. Streit Vol. 85, pp. 61–86, DOI:10.1007/978-3-319- (World Scientific, 1986), Vol. 2, pp. 1–49, 04214-5 4. DOI:10.1142/9789814503068 0001. [18] V. Benci, L. Luperi Baglini and [5] S. Albeverio, in Nonstandard Anal- M. Squassina, Adv. Nonlinear Anal. 9, ysis and its Applications, edited 124 (2018), DOI:10.1515/anona-2018-0146. by N. Cutland (Cambridge Uni- versity Press, 1988), pp. 182–220, [19] V. Benci and L. Luperi Baglini, Arab. J. DOI:10.1017/CBO9781139172110.005. Math. 4, 231 (2015), DOI:10.1007/s40065- [6] J. Harthong, Adv. Appl. Math. 2, 24 (1981), 014-0114-5. DOI:10.1016/0196-8858(81)90038-5. [20] P. Ehrlich, Arch. Hist. Exact Sci. 60, 1 [7] J. Harthong, Etudes´ sur la mécanique (2006), DOI:10.1007/s00407-005-0102-4. quantique (Astérisque, Vol. 111, Société [21] R. Goldblatt, Lectures on the Hyperreals: Mathématiquede France, 1984), pp. 20–25. An Introduction to Nonstandard Analysis [8] M. O. Farrukh, J. Math. Phys. 16, 177 (Springer, 1998), DOI:10.1007/978-1-4612- (1975), DOI:10.1063/1.522525. 0615-6. [9] S. Albeverio, J. E. Fenstad and R. Høegh- [22] P. Fletcher, K. Hrbacek, V. Kanovei, Krohn, Trans. Amer. Math. Soc. 252, M. G. Katz, C. Lobry, and S. Sanders, 275 (1979), DOI:10.1090/S0002-9947-1979- Real Anal. Exch., 42, 193 (2017), 0534122-5. DOI:10.14321/realanalexch.42.2.0193.

Accepted in Quantum 2019-04-25, click title to verify 21 [23] V. Benci, M. Di Nasso and M. Forti, [38] M. Belloni and R. W. Robi- in Nonstandard Methods and Applications nett, Phys. Rep. 540, 25 (2014), in Mathematics, edited by N. J. Cut- DOI:10.1016/j.physrep.2014.02.005. land, M. Di Nasso and D. A. Ross [39] I. Mitra, A. DasGupta and B. Dutta- (A K Peters/CRC Press, 2006), pp. 3–44, Roy, Am. J. Phys. 66, 1101 (1998), DOI:10.1017/9781316755761.002. DOI:10.1119/1.19051. [24] V. Benci, in Calculus of Variations [40] M. de Llano, A. Salazar and M. A. Sol´ıs, and Partial Differential Equations, Rev. Mex. Phys. 51, 626 (2005). edited by G. Buttazzo, A. Marino and M. K. V. Murthy (Springer, 2000), pp. 285– [41] S. Geltman, J. Atom. Mol. Opt. Phys. 2011, 326, DOI:10.1007/978-3-642-57186-2 12. 573179 (2011), DOI:10.1155/2011/573179. [25] V. Benci, MatematicaMente 218–222 [42] A. Farrell and B. P. van Zyl, Can. J. Phys. (2016–2017). 88, 817 (2010), DOI:10.1139/P10-061. [26] V. Benci, Alla scoperta dei numeri infinites- [43] R. Jackiw, in M. A. B. Bég Memorial imi: Lezioni di analisi matematica esposte Volume, edited by A. Ali and P. Hood- in un campo non-archimedeo (Aracne ed- boy (World Scientific, 1991), pp. 25–42, itrice, 2018). DOI:10.1142/1447. [27] V. Benci, I numeri e gli insiemi etichet- [44] S. Albeverio, Z. Brze´zniak and tati (Conferenze del seminario di matemat- L. D¸abrowski, J. Funct. Anal. 130, ica dell’Universitàdi Bari, Vol. 261, Laterza, 220 (1995), DOI:10.1006/jfan.1995.1068. 1995). [45] S. Albeverio and L. Nizhnik, Ukr. Math. J. [28] V. Benci and M. Di Nasso, Adv. Math. 52, 664 (2000), DOI:10.1007/BF02487279. 173, 50 (2003), DOI:10.1016/S0001- [46] G. Dell’Antonio, A. Michelangeli, R. Scan- 8708(02)00012-9. done and K. Yajima, Ann. Henri Poincaré [29] V. Benci, M. Di Nasso and M. Forti, 19, 283 (2018), DOI:10.1007/s00023-017- Ann. Pure Appl. Logic 143, 43 (2006), 0628-4. DOI:10.1016/j.apal.2006.01.008. [47] R. Scandone, arXiv:1901.02449 (2019). [30] V. Benci and M. Forti, The Euclidean num- [48] R. M. Cavalcanti, Rev. Bras. Ensino Fis. 21, bers, in preparation. 336 (1999). [31] V. Benci, An improved setting for gener- [49] S.-L. Nyeo, Am. J. Phys. 68, 571 (2000), alized functions: robust ultrafunctions, in DOI:10.1119/1.19485. preparation. [32] L. Schwartz, C. R. Acad. Sci. Paris 239, 847 (1954). [33] J. F. Colombeau, Elementary Introduction to New Generalized Functions (North Hol- land, 1985). [34] V. Benci, L. Horsten and S. Wenmack- ers, Milan J. Math. 81, 121 (2013), DOI:10.1007/s00032-012-0191-x. [35] V. Benci, L. Horsten and S. Wenmack- ers, Brit. J. Phil. Sci. 69, 509 (2018), DOI:10.1093/bjps/axw013. [36] F. Gieres, Rep. Prog. Phys. 63, 1893 (2000), DOI:10.1088/0034-4885/63/12/201. [37] S. Flügge, Practical Quantum Me- chanics (Springer, 1999), pp. 35–40, DOI:10.1007/978-3-642-61995-3.

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