Fermi's ansatz and Bohm's quantum potential

Physics Letters A 378 (2014) 2363–2366

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Physics Letters A www.elsevier.com/locate/pla

Fermi’s ansatz and Bohm’s quantum potential Glen Dennis a , Maurice A. de Gosson b,∗ , Basil J. Hiley a a b

TPRU, Birkbeck College, University of London, London, WC1E 7HX, United Kingdom University of Vienna, Faculty of Mathematics, NuHAG, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

a r t i c l e

i n f o

Article history: Received 11 February 2014 Received in revised form 12 May 2014 Accepted 12 May 2014 Available online 17 June 2014 Communicated by P.R. Holland

a b s t r a c t In this paper we address the following simple question: Given a wavefunction ψ(x, t ) in a onedimensional configuration space, is it possible to give a unique two-dimensional representation ρ (x, p , t ) in a position-momentum phase plane? If so, can we find the general condition that makes this possible? We will show that this is indeed possible by using an idea introduced originally by Fermi provided the boundary of the phase space area is a closed curve satisfying a certain exact quantum condition. © 2014 Elsevier B.V. All rights reserved.

Keywords: Fermi function Uncertainty principle Quantum potential

1. Introduction Consider a particle with mass m moving along the x-axis. According to Louis de Broglie [1] one should associate the particle with a wavefunction ψ(x) = R (x)e i ϕ (x)/¯h (this wavefunction is, in general, time-dependent, but we will drop any explicit reference to the time t). In a largely forgotten paper [2] from 1930 Enrico Fermi showed that one could associate with every such wavefunction a certain curve γF in the phase space plane Rx × R p . Fermi’s work has recently been rediscovered by Benenti and Strini [3,4]. Strangely enough, Fermi’s curve consists of two branches γF± represented by the equations ±



p = ϕ (x) ±



2m Q (x)

(1)

(ϕ  the first x-derivative of the phase) where the “energy” Q (x) is Bohm’s quantum potential [5,6,36] (see [7] for a detailed account):

Q (x) = −

h¯ 2 R  (x) 2m R (x)

.

(2)

As we will see, when γF is a closed curve, the area of the surface ΩF it encloses is never less than 12 h, i.e. one half of the quantum of action. Elsewhere [22], we have called such surfaces (and their generalizations to higher dimensions) “quantum blobs”, and shown that they are closely related to the uncertainty principle. One possible interpretation of our result is that it is the quantum force

*

Corresponding author. Tel.: +43 (0) 6767877995. E-mail addresses: [email protected] (G. Dennis), [email protected], [email protected] (M.A. de Gosson), [email protected] (B.J. Hiley). http://dx.doi.org/10.1016/j.physleta.2014.05.020 0375-9601/© 2014 Elsevier B.V. All rights reserved.

(derived from the quantum potential) that prevents the quantum blob ΩF from collapsing to a particle-like point. This interpretation is consistent with Hiley’s observation [8] that in “...a process based approach, the notion of a ‘particle’ is not a sharply defined point object, instead it is a quasi-local invariant feature of the total process. Each individual process may be characterized by a mean position and a mean momentum...”. In fact, Eqs. (1) have the following precise meaning in terms of the quantum theory of motion. For instance, the positive branch γF+ is described by the equation

p + = p B (x) +



2m Q (x)

√

where p B (x) is Bohm’s momentum and the term 2m Q (x) can be viewed as an additional momentum coming from the quantum force. The motivation for our discussion of the relation between configuration and phase space descriptions arises from a detailed examination of the relation between the von Neumann–Moyal algebra and the Bohm approach [9,10]. The Moyal [11] algebra is exactly the algebra introduced by von Neumann [12] in his classic paper that forms the basis of the Stone–von Neumann theorem for the uniqueness of the Schrödinger representation. The von Neumann–Moyal algebra leads to a phase space (symplectic) representation of the quantum formalism in contrast to the usual representation in a configuration space. Although they appear to be different structures, it has been shown by Várilly and GraciaBondia [13] that one can recover the standard results by projecting from the phase space onto the configuration space. Within the Moyal structure, the Bohm momentum p B (x) appears as a conditional expectation value of the momentum obtained using the Wigner distribution F (x, p , t ) [11]. If we now

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proceed and evaluate the conditional mean square deviation, we find it leads to the relation shown in Eq. (1). All of the mathematics underlying these relations is quite involved but the underlying idea is actually surprisingly simple. 2. Fermi’s trick This simple approach, which we will now discuss, is reminiscent of that used in supersymmetric quantum mechanics (SUSY QM, see e.g. Cooper et al. [14]) where one notes that a given nodeless wavefunction, ψ(x), vanishing at infinity can always be viewed as a ground state with energy E of some time-independent Schrödinger equation, namely

 −

h¯ 2 d2 2m dx2

 + V (x) ψ(x) = E ψ(x)

(3)

where the potential is given by

V (x) =

h¯ 2 ψ  (x)

(4)

2m ψ(x)

(here ψ  is the second x-derivative of ψ ). If instead we apply this procedure to the modulus of the wavefunction; that is writing ψ(x) = R (x)e i ϕ (x)/ with R (x) ≥ 0 and ϕ (x) real, we find that

 −h¯ 2

d2 dx2

+ h¯ 2

R  (x)



R = 0.

R (x)

(5)

Next we notice that the gauge transformation −i h¯ (d/dx) −→ −ih¯ (d/dx) − ϕ  (ϕ  the first x-derivative) brings this equation into

 −ih¯

d dx

−ϕ



2 + h¯

2

R 

 (6)

This is exactly Fermi’s equation (2) if we use his Eq. (3) [2]. This is now a differential equation satisfied by ψ ; by introducing the operator

1  HF =

 −ih¯

2m

d dx

−ϕ



2 +

h¯

2

R 

2m R

,

(7)

Eq. (6) can be written in the form  H F ψ = 0; we now remark that the last term appearing in the right-hand side of (7) is (up to the sign) Bohm’s quantum potential (2) so that we can write

1  HF =

 −ih¯

2m

d dx

− ϕ

2 − Q.

(8)

1  2m

2

p − ϕ  (x)

− Q (x),

(9)

and the equation H F (x, p ) = 0 in general determines a curve γF in the phase plane Rx × R p , which Fermi ultimately identifies with the quantum state ψ itself. Notice that the Fermi function (9) (and hence γF ) only depends on the quantum state ψ in the sense that replacing ψ with λψ (λ = 0 a complex number) changes neither H F nor  H F . It follows that the mathematical objects  H F , H F , and γF intrinsically define the quantum state. 3. An exact quantization rule

h¯ 2 d2 2m

dx2

V (x) = E

for x = x A or x = x B

V (x) < E

for x A < x < x B .

Elementary Sturm–Liouville theory tells us that Eq. (10) has nonzero solutions only for a set of discrete values E 0 ≤ E 1 ≤ · · · of the energy E. Ma and Xu [15,16] have shown that these values can be explicitly calculated using an exact quantization rule which reduces to the semiclassical EBK quantization in the limit h¯ → 0 (Barclay [17] has obtained similar results using a different method). The Ma and Xu argument goes as follows: setting χ = ψ  /ψ , a straightforward calculation shows that Schrödinger’s equation (10) is equivalent to the Riccati equation

−χ  (x) = k(x)2 + χ (x)2

 ψ(x) = E − V (x) ψ(x).

(10)

(11)

for V (x) ≤ E, where k(x) = p (x)/¯h is determined by the relation

p (x) =







2m E − V (x) .

(12)

Supposing that the potential V (x) is continuous at the turning points x A and x B , the exact quantum condition of Ma and Xu is then

x B

x B

xA

xA

χ (x) k (x)  dx = N π χ (x)

(13)

where N − 1 is the number of nodes of ψ in that region V (x) ≤ E. In particular, for the ground state, we have

x B

x B k0 (x)dx =

xA

xA

χ (x) k0 (x)  dx + π . χ (x)

(14)

A crucial fact noted by Qiang and Dong [18] and Serrano et al. [19,20] is that Ma and Xu’s condition (13) can be put, for all exactly solvable systems, in the simple form

x B

x B k(x)dx −

k0 (x)dx = ( N − 1)π

(15)

xA

where the function k0 is defined by (12) with E = E 0 (the ground energy level). The exact quantization condition thus becomes

x B





2m E N − V (x) dx =

xA

x B





2m E 0 − V (x) dx +

Nh 2

(16)

xA

which shows that the energy levels E 1 , E 2 , ... are determined by the ground state energy E 0 . As demonstrated by Serrano et al. [19], the correction term in the quantum rule (13) of Ma and Xu satisfies

x B

Consider the one-dimensional time-independent Schrödinger equation

−

for −∞ < x < x A or x B < x < +∞

xA

The operator  H F represents the Hamiltonian function

H F (x, p ) =

V (x) > E

k(x)dx −

ψ = 0.

R

We assume that the potential V is piecewise continuous, and that there exist exactly two real values x A and x B (“turning points”) such that

xA

χ (x) π k (x)  dx ≥ − χ (x) 2

for all common Hamiltonians (among others, Morse, first and second Pöschl–Teller); moreover one has equality for all quadratic

G. Dennis et al. / Physics Letters A 378 (2014) 2363–2366

Hamiltonians, which reflects the fact that EBK quantization is exact for all such Hamiltonians. It follows, in view of the ground state quantum rule (14), that

x B k0 (x)dx ≥

π

(17)

2

xA

π h¯

p 0 (x)dx ≥

2

5. Higher dimensions Fermi’s trick is easily generalized to higher dimensions. Con3N sider an N-particle system with phase space R3N r × Rp ; everything can be extended mutatis mutandis to the case of multiparticle systems. Writing the wavefunction as ψ(r) = R (r)e i ϕ (r)/ with R (r) ≥ 0 and ϕ (r) real, R (r) satisfies

  ∇2 R −h¯ 2 ∇r2 + h¯ 2 r R =0

and hence, setting p 0 (x) = h¯ k0 (x),

x B

2365

(22)

R

(18)

.

xA

We conjecture that this inequality is true for Hamiltonians with arbitrary potentials. This guess is corroborated by the study in de Gosson [25] where EBK quantization is related to the uncertainty principle (UP).

for all r such that R (r) > 0. The gauge transformation −i h¯ ∇r −→ −ih¯ ∇r − ∇r ϕ leads to the equation  H F ψ = 0 where

1  HF = (−ih¯ ∇r − ∇r ϕ )2 − Q 2m

and where Q is the quantum potential:

Q (r) = −

4. The main result Let us return to the Fermi curve curves

γF . It is the union of the two



(23)

h¯ 2 ∇r2 R (r) 2m

(24)

R (r)

The operator  H F is the quantization of the Hamiltonian function

1 

2

γF± : p ± = ϕ  (x) ± 2m Q (x)

H F (r, p) =

which are defined for all x where Q (x) ≥ 0. We assume that these values form an interval [x A , x B ], the turning points x A , x B being determined by the condition Q (x) = 0; this condition is satisfied if − R is convex in the interval [x A , x B ] and concave outside (the values x A , x B are precisely then the only zeroes of R  ). The area of the surface ΩF enclosed by γF = γF+ − γF− is then given by the equation

and the equation H F (r, p) = 0 in general determines a hypersurface 3N ΣF in the phase space R3N r × Rp . While a general exact quantization condition extending the results of Section 3 to systems with several degrees of freedom is still lacking, the outlined procedure still applies to systems with spherical symmetry in which case V (x) is replaced by the effective potential V eff (r) (see [23,15, 16,18]). In particular this allows the hydrogen atom to be studied. The extension of the inequality (21) and of the notion of quantum blob to higher dimensions requires the use, not of volume, but that of the very subtle notion of symplectic capacity, related to Gromov’s [28] famous symplectic non-squeezing theorem (de Gosson [24,25,21,26,22], also see de Gosson and Luef [27] for a detailed review of these notions).

x B 

Area(ΩF ) = 2

2m Q (x)dx.

(19)

xA

Since the term ϕ  (x) does not appear in this equation, we may as well assume that ϕ = 0, reducing the equation  H F ψ = 0 to the trivial equation

  d2 R  −h¯ 2 2 + h¯ 2 R = 0. R

dx

This is of course just the stationary Schrödinger equation (10) with E = 0 and V = − Q , so we may use the quantization theory of Section 3; since we are assuming that ψ = R has no nodes, the Ma and Xu quantization rule (14) for the ground state applies, leading to

x B Area(ΩF ) = 2h¯

k0 (x)dx.

(20)

xA

In view of the inequality (17) we thus have

Area(ΩF ) ≥

1 2

h.

(21)

This shows that the surface bounded by a Fermi curve always contains a “quantum blob”, a minimal uncertainty quantum cell [21,22]. The inequality (21) can be viewed as a topological version of the UP. As we have shown in [22], this inequality shows that the UP leads to a coarse-graining of phase space by ellipses with area 12 h in one-to-one correspondence with generalized Gaussians (which are minimal uncertainty states); we have called these ellipses “quantum blobs”.

2m

p − ∇r ϕ (r)

− Q (r),

(25)

6. Discussion It has been shown that the Fermi curve associated with a quantum state cannot enclose an arbitrarily small area in the phase plane. Instead, the curve must contain a quantum blob i.e., it has to enclose an area representing an action of at least 12 h (or, in higher dimensional phase spaces: the symplectic capacity of a Fermi ellipsoid must be at least 12 h). The Fermi function therefore helps to illustrate how a quantum phase space (coarse-grained by ellipsoidal quantum blobs having a symplectic capacity of 12 h, in one-to-one correspondence with Gaussian wavefunctions [29,21, 22]) emerges as a mathematical necessity whenever there is a nonzero Q i.e., whenever a quantum process is unfolding. Quantum blobs provide a phase space geometric picture of minimum uncertainty: the Robertson–Schrödinger inequalities become saturated for the wavefunctions (i.e., squeezed coherent states) that are associated with quantum blobs. Hence these blobs represent “the smallest regions of phase space which make sense from a quantum mechanical perspective” [22]. It is also intriguing that in his classic work, Moyal derives the UP using the marginal momentum (i.e. the space-conditional mean momentum that is equal to the Bohm momentum) and, in effect, Q . For a quantum process, points in phase space therefore have no intrinsic meaning. This finding lends support to the following postulates proposed by Dragoman [30]: Axiom 10 ‘It is not possible to localize a quantum particle in phase space regions smaller than that of a quantum blob’. Axiom 11 ‘The phase space extent of a quantum particle is smaller

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than a quantum blob.’ Dragoman [30], de Gosson [22,37]. Moreover, as Hiley puts it in [8] “This gives mathematical form to Weyl’s [33] notion of a particle: ‘Hence a particle itself is not even a point in field space, it is nothing spatial (extended) at all. However, it is confined to a spatial neighborhood, from which its field effects originate.” It is intriguing that two of the main variables of the Bohm– Hiley approach to quantum theory (i.e., the Bohm momentum and the quantum potential) feature explicitly in the Fermi function. Moreover, Hiley has recently revealed that both variables also emerge in Moyal’s classic work [9]. There thus seems to be a fundamental connection between these two approaches to quantum mechanics which the Fermi function can perhaps help to elucidate. Utilizing Fermi’s trick we have shown that there is a oneto-one correspondence between wavefunctions and surfaces in the phase plane whose boundaries are closed curves satisfying a certain quantum condition. Moreover, de Gosson has previously related the Wigner transform of a squeezed coherent state to its Fermi function thereby corroborating an observation made by Benenti and Strini [4]. Hiley has demonstrated that the Bohm momentum and Q are weak values and as such are experimentally measurable [31]. We note that in recent work Kocsis et al. [32] have made weak measurements of the momentum of photons. Yves Couder and his coworkers [34,35] have discovered a macroscopic pilot wave effect in the form of “walking droplets”. These droplets mimic the behavior of particles that are being guided by their associated pilot wave and thereby give a macroscopic simulation of de Broglie’s ideas. We also mention that on a higher mathematical level Seip [39] quantizes the harmonic oscillator using families of curves of maximum modulus; it would be interesting to explore the links between Seip’s approach and ours (we are indebted to Franz Luef for having drawn our attention to this work). We end by noting that in his paper [38] on Einstein’s theory Holland deals with stationary state wavefunctions for which the spatial factor is real. Acknowledgements Maurice de Gosson has been supported by the Austrian Research Agency FWF (Projektnummer P20442-N13). The authors wish to thank the Editor for very useful remarks, opening new perspectives for future work. References [1] L. de Broglie, An Introduction to the Study of Wave Mechanics, Methuen, London, 1930. [2] E. Fermi, L’interpretazione del principio di causalità nella meccanica quantistica, Rend. Lincei 11 (1930) 980; reprinted in Nuovo Cimento 7 (1930) 361, English translation by G. Strini. [3] G. Benenti, G. Strini, Gaussian wave packets in phase space: the Fermi g F function, Am. J. Phys. 77 (6) (2009) 546–551. [4] G. Benenti, G. Strini, Quantum mechanics in phase space: first order comparison between the Wigner and the Fermi function, Eur. Phys. J. D 57 (2010) 117–121. [5] D. Bohm, Wholeness and the Implicate Order, Routledge, London, 1980. [6] D. Bohm, Hidden variables and the implicate order, in: B.J. Hiley, Peat F. David (Eds.), Quantum Implications: Essays in Honour of David Bohm, Routledge, London, 1987. [7] P.R. Holland, The Quantum Theory of Motion. An Account of the de Broglie– Bohm Causal Interpretation of Quantum Mechanics, Cambridge University Press, Cambridge, 1995.

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