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A Flea on Schrödinger’s Cat

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‘Another secondary readership is made up of those philosophers and physicists who—again like myself—are puzzled by so-called foundational issues: what the strange quantum formalism implies about the nature of the world it so accurately describes. (…) My presentation is suffused with a perspective on the quantum theory that is very close to the venerable but recently much reviled Copenhagen interpretation. Those with a taste for such things may be startled to see how well quantum computation resonates with the Copenhagen point of view. Indeed, it had been my plan to call this book Copenhagen Computation until the excellent people at Cambridge University Press and my computer-scientist friends persuaded me that virtually no members of my primary readership would then have any idea what it was about.’

David Mermin, Quantum Computer Science: An Introduction [87] (Preface)

Abstract

We propose a technical reformulation of the measurement problem of quantum mechanics, which is based on the postulate that the final state of a measurement is classical; this accords with experimental practice as well as with Bohr’s views. Unlike the usual formulation (in which the post-measurement state is a unit vector in Hilbert space), our version actually opens the possibility of admitting a purely technical solution within the confines of conventional quantum theory (as opposed to solutions that either modify this theory, or introduce unusual and controversial interpretative rules and/or ontologies).

To that effect, we recall a remarkable phenomenon in the theory of Schrödinger operators (discovered in 1981 by Jona-Lasinio, Martinelli, and Scoppola), according to which the ground state of a symmetric double-well Hamiltonian (which is paradigmatically of Schrödinger’s Cat type) becomes exponentially sensitive to tiny perturbations of the potential as ħ→0. We show that this instability emerges also from the textbook wkb approximation, extend it to time-dependent perturbations, and study the dynamical transition from the ground state of the double well to the perturbed ground state (in which the cat is typically either dead or alive, depending on the details of the perturbation).

Numerical simulations show that adiabatically arising perturbations may (quite literally) cause the collapse of the wave-function in the classical limit. Thus, at least in the context of a simple mathematical model, we combine the technical and conceptual virtues of decoherence (which fails to solve the measurement problem but launches the key idea that perturbations may come from the environment) with those of dynamical collapse models à la grw (which do solve the measurement problem but are ad hoc), without sharing their drawbacks: single measurement outcomes are obtained (instead of merely diagonal reduced density matrices), and no modification of quantum mechanics is needed.

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Notes

  1. We regard these as Wittgensteinian “hinge propositions” [114], on which modern physics is based.

  2. This is sometimes called Bohr’s doctrine of classical concepts [102], which accurately describes experimental practice! See also [75] for a detailed analysis.

  3. For our work it is a moot point whether Bohr endorsed this second point as well; it is hard to say.

  4. The original expositions [28, 51, 59, 65, 66, 111] might be hard to follow for non-mathematicians (see, however, [22]). Applications in chemistry and solid-state physics may be found in [23, 67, 84].

  5. This notion needs to be quantified, of course [2, 19, 78]. See also [75] and the present paper.

  6. This is true in particular for the Many-Worlds (aka “Everett”) Interpretation [101], or the Modal Interpretation of quantum mechanics [36], in which radical changes are proposed in the ontology and/or the usual interpretative rules of the theory, without clarifying in any way what is really going on during measurements (a question one indeed is not supposed to ask, according to received wisdom). Bohmian mechanics (as a modern incarnation of de Broglie’s pilot-wave theory) does a better job here [33, 38, 39], but its narrow applicability (at least in its current form), focusing as it does on position as the only physical observable, makes it unattractive to many (including the authors). See some kinship in Sect. 5.4, however.

  7. This applies, for example, to the famous paper by Danieri, Loinger, and Prosperi [34], to early papers on decoherence [117], and to much of the mathematical physics literature on the measurement problem, including the work of the senior author [40, 60, 72, 73, 109]. We now regard such papers as mathematically interesting but conceptually misguided, at least on this point. On the other hand, it is to the credit of especially the Swiss school that it drew attention to the idea that measurement involves limiting procedures, so that solutions of the measurement problem should at least incorporate the appropriate limits.

  8. In this sense even von Neumann’s book [89] is misleading, since he suggested that the act of observation may be identified with a cut in the chain now named after him. What is right about this idea is that observation is linked to a voluntary change of information, but it would have been preferable to point out that such a change, in so far as it defines measurement, should be a loss of quantum information.

  9. Furthermore, despite its outspoken ambition to derive classical physics from quantum theory [68, 103, 117], decoherence hardly (if at all) invokes limits like Planck’s constant going to zero, which are needed, for one thing, to derive the correct classical equations of motion (cf. [75, 76]).

  10. An introduction for philosophers to the material in this section may be found in [75, Chaps. 4 and 5].

  11. A C∗-algebra is a complex algebra A that is complete in a norm ∥⋅∥ satisfying ∥ab∥≤∥a∥∥b∥ for all a,b∈A, and has an involution a→a ∗ such that ∥a ∗ a∥=∥a∥2.

  12. Positivity of ω means that ω(a ∗ a)≥0 for all a∈A. If A has a unit 1, then a state may equivalently be defined as a positive linear functional ω:A→ℂ that satisfies ω(1)=1.

  13. A self-adjoint operator a on a Hilbert space is compact just in case it has a spectral decomposition a=∑ i λ i |e i 〉〈e i | (where |e i 〉〈e i | is the orthogonal projection onto the ray ℂe i ), where the eigenvectors (e i ) form an orthonormal basis of H, each nonzero eigenvalue λ i has finite multiplicity, and if the eigenvalues are listed in decreasing order of their absolute value (i.e., ∥a∥=|λ 1|≥|λ 2|≥⋯), then lim α→∞|λ α |=0.

  14. Although unbounded operators play a major practical role in physics (think of position and momentum), they may be awkward to deal with due to domain issues and in any case they can theoretically be avoided without any loss of generality. Indeed, unbounded self-adjoint operators a bijectively correspond to bounded operators (or constructions involving those) in at least four different ways [97]. First, one may pass to the unitary Cayley transform (a−i)(a+i)−1. Second, one may construct the associated one-parameter unitary group t↦exp(ita) by Stone’s Theorem. Third, one may work with the bounded spectral projections, from which the operator may be reconstructed by the spectral theorem. Fourth, one could take the resolvents (a−λ)−1, λ∉ℝ (whose typical integral kernels are Green’s functions).

  15. Classically, one could work with the (essentially) bounded integrable functions L ∞(ℝ2), whilst in quantum theory one could take the algebra B(L 2(ℝ)) of all bounded operators on L 2(ℝ) (as opposed to merely the compact ones). These broader classes can be constructed from the C 0-functions or compact operators by taking pointwise and weak (or strong) limits, respectively. Enlarging the class of observables like that also brings in new continuity conditions on the states (namely σ-additivity and σ-weak continuity, respectively). Imposing these leads to the same states as discussed above; without them, one obtains more.

  16. Often Weyl quantization \(Q^{W}_{\hbar}\) is used instead of Berezin quantization Q ħ , as in [9], but for Schwartz functions f on phase space these have the same asymptotic properties as ħ→0 [74]. The advantage of Berezin quantization is that it is well defined also for continuous functions vanishing at infinity, in that for any unit vector Ψ∈L 2(ℝ) the map f↦〈Ψ|Q ħ (f)|Ψ〉 defines a probability measure on phase space. In contrast, the Wigner function defined by \(f\mapsto\langle\varPsi| Q^{W}_{\hbar}(f)|\varPsi\rangle\) may fail to be positive, as is well known.

  17. In (2.18) we regard classical states as probability measures on phase space; hence the addition on the right-hand side is a convex sum of measures, which has nothing to do with addition in the particular phase space ℝ2 (whose linear structure is accidental and irrelevant).

  18. This appeal to “practice” does not mean that we are resigned to fapp (i.e., “for all practical purposes”) solutions to the measurement problem. As in [73], we remain convinced that the classical description of a measurement apparatus is a purely epistemic move, relative to which outcomes are defined. So even if it were possible to study a cat as a quantum system, there would be no measurement problem, since in that case there would be innumerable superpositions but not a single (undesirable) mixture of classical states.

  19. We share this rejection with the Bohmians [32]. The folk wisdom (shared by the Founding Fathers) that the Copenhagen Interpretation has no measurement problem relies on these secondary Copenhagenian claims, which indeed sweep the problem under the rug. Incidentally, these claims seem much more popular than Bohr’s doctrine of classical concepts, which is generally not well understood, and/or mistaken for the idea that the goal of physics is to explain experiments, or that reality does not exist, et cetera.

  20. The analogy with the thermodynamic limit will be discussed in Sect. 5.2. As to the limit ħ→0, we repeat [75, pp. 471–472] that although ħ is a dimensionful constant, in practice one studies the (semi)classical regime of a given quantum theory by forming a dimensionless combination of ħ and other parameters; this combination then re-enters the theory as if it were a dimensionless version of ħ that can indeed be varied. The oldest example is Planck’s radiation formula, with the associated limit ħν/kT→0, and another example is the Schrödinger operator (2.10), with mass reinserted, where one may pass to a dimensionless parameter \(\hbar /\lambda \sqrt{2m\epsilon}\), where λ and ϵ are typical length and energy scales, respectively.

  21. Paraphrasing Bell [6]: the difference between \(\rho_{0}^{\pm}\) and \(\psi_{\hbar}^{(0)}\) can be made ‘as big as you do not like.’.

  22. Families of unit vectors like \(\varPsi^{(i)}_{\hbar}\), where i=0,1,+,−, typically do not have a limit as unit vectors (or even as density matrices, including one-dimensional projections).

  23. As explained above, the nonlinearity inherent in the limit ħ→0 makes it impossible to find the limit of this ground state \(\varPsi^{(0)}_{\hbar}\) by just adding the results for two localized wave-functions like \(\varPsi^{+}_{\hbar}\) and \(\varPsi^{-}_{\hbar}\).

  24. The “Flea on the Elephant” terminology used in [111] for the phenomenon in question evidently motivated the title of the present paper, which has identified the proper host animal at last!

  25. Some of the details in this section depend on the latter assumption, but our overall scenario in Sect. 4 does not. For example, if the value and/or the curvature of one of the minima is decreased, then the ground state wave-function will localize above that minimum, as follows from standard minimax techniques taking single harmonic eigenfunctions as trial states [51, 98]. So collapse is actually easier in that case.

  26. Symmetric perturbations are excluded by 3, as these would satisfy \(d_{V}'=d_{V}''\).

  27. If δV has support on both sides of the real axis (which is possible in the case (3.6)), a more detailed analysis of its shape is necessary in order to predict the direction of collapse [21].

  28. Compare [98, p. 35], [111] for such arguments. Nonetheless, the effect of the flea is counterintuitive even from the point of view of quantum-mechanical tunneling: for example, with a perturbation of the kind displayed in Figs. 4, 5 and 6, which falls under case (3.6), one would expect tunneling from the right into the left-handed well to be discouraged, even increasingly so as ħ→0, because the potential barrier through which to tunnel has been heightened, but in fact the right-handed peak of the unperturbed ground state tunnels to the left so as to localize the ground state wave-function. See Sect. 5.3 for further discussion.

  29. Cf. [7] (for mathematicians) or [55] (for physicists), and references therein.

  30. This approximation is extremely well known also in physics [79], but has hardly been studied in the present context. It is too simple to display the behaviour (2.22)–(2.24), though. See also Sect. 5.1.

  31. A corresponding movie may be found on www.math.ru.nl/~landsman/flea.avi.

  32. This states that under an adiabatic perturbation δV=δV(t/T) the unperturbed ground state moving under the perturbed Hamiltonian converges to the ground state of the latter as t→T→∞ [52, 54].

  33. For potentials with n identical wells, this basis may be chosen to consist of the eigenfunctions of the n individual wells with Dirichlet boundary conditions [58, 59]. For the quantum Ising model (cf. [70]), where n=2, one may take the state with all spins up and the state with all spins down, etc.

  34. In this respect our approach has an edge over Bohmian mechanics and the grw theory, although it remains to be shown that typical environments induce the right perturbations for the mechanism to apply.

  35. Like the measurement problem, this seemingly paradoxical situation does not seem to bother physicists very much, although their Higgs mechanism relies on a resolution of it: apparently, in any finite volume the system refuses to choose a ground state (or vacuum), although all perturbative calculations underlying the successful Standard Model of elementary particle physics rely on such a choice. But it has been the subject of recent discussions in the philosophy of science [4, 20, 80, 86, 92], in which some claim that this “discontinuity” in passing from N<∞ to N=∞ is crucial for the possibility of emergence (‘More is Different’), whilst others try to find arguments for continuity and hence defend some form of reductionism.

  36. If it happens to be true that measurement outcomes emerge adiabatically, it would be a marked break with tradition, starting with von Neumann’s model, in which both the measurement interaction and the alleged collapse take place instantly [89]. However, the recent notion of a “weak measurement” seems to support our approach on this point [69, 82, 100]. We are indebted to Jos Groot for these references.

  37. See [26] for a recent discussion of randomness amplification, which focuses on the way experiments may be construed to amplify the randomness inherent in the (alleged) “free” choice of an experimentalist.

  38. What follows will hardly be new to specialists in these matters, but it needs to be stated clearly.

  39. Using the traditional scene where (Alice, Bob) are spacelike separated and perform experiments with settings (a,b) and outcomes (x,y), respectively, a stochastic hidden variable λ (or rather the corresponding theory) satisfies oi if Alice’s conditional probability P(x∣a,b,y,λ) of finding x given (a,b,y,λ) is independent of Bob’s outcome y, whereas the theory satisfies pi if her conditional probability P(x∣a,b,λ) of finding x given (a,b,λ) is independent of Bob’s setting b, and vice versa. These notions were originally introduced, in slightly different form, by Jarrett [64]. Controversies around this terminology and its use of the sort discussed in e.g. [18, 48, 85, 93, 106] seem irrelevant to our purposes.

  40. Muller also notes that this paper reveals the ‘empiricist presuppositions’ of the authors. For example, we are not worried about superpositions like Ψ=∑ i c i Ψ i in Sect. 2.1 as such, despite the fact that most quantum-mechanical observables, including the one (O) that is being measured, have no value in such a state. Compared to classical physics this is admittedly a curious feature of quantum theory, but it causes neither inconsistencies within the formalism nor disagreements with observation. Furthermore, it has nothing to do with Ψ being a superposition (which is a basis-dependent statement), and in our view it only leads to a problem (namely the measurement problem) if the device whose state Ψ represents is classical (e.g., in being macroscopic). For in that case quantum mechanics naively (i.e. without the flea mechanism) appears to assign numerous values to O, which seems unacceptable (rather than assigning no value, which for the stated reasons we can live with both as theorists and as alleged empiricists).

  41. As opposed to the extremely sophisticated and mathematically rigorous methods of Helffer and Sjöstrand [37, 58, 59], who somewhat confusingly suggest they use the ordinary wkb method.

  42. This result can also be found by applying the method of comparison equations, which is explained in [116]. Further references are [81] and [88].

  43. This can be explained by level crossing, i.e. certain energy levels of the two individual wells coincide.

  44. To keep the discussion straightforward, we ignored the special case δ∈{kπ|k∈ℤ∖{0}} here.

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Acknowledgements

The authors are indebted to Koen Reijnders for help with the appendix. They also wish to thank Jeremy Butterfield, Fred Muller, and an anonymous referee for their penetrating comments on the first draft of this paper, which have improved the current (final) version.Footnote 40

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  1. Robin Reuvers

    Present address: Pembroke College, Cambridge, CB2 1RF, UK

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    N. P. (Klaas) Landsman

  2. Faculty of Science, Radboud Honours Academy, Radboud University Nijmegen, Nijmegen, The Netherlands

    Robin Reuvers

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Appendix A: The Flea from wkb

Appendix A: The Flea from wkb

In this appendix, we study the “flea” type perturbation from the point of view of the wkb method of the physics textbooks (like [52, 71]).Footnote 41 As explained in [44, 45], the connection formulae stated in such books are actually correct only for simple potentials like a single well, but with due modifications (see below), the formalism will reproduce both the rigorous and the numerical results described in the main body of this paper.

1.1 A.1 Quantization Condition for an Asymmetric Double Well

We start by recalling some standard wkb formulas. The wkb wave-function in the classically allowed region without turning points (E>V(x)) can be written as

$$ \varPsi(x)\cong\frac{1}{\sqrt{p(x)}} \bigl[Ae^{\frac{i}{\hbar}\int ^{x}{p(y)dy}}+Be^{-\frac{i}{\hbar}\int^{x}{p(y)dy}} \bigr], $$
(A.1)

where

$$ p(x)=\left \{ \begin{array}{l@{\quad}l} \sqrt{ [E-V(x) ]} & \mbox{if } E\geq V(x)\\[4pt] \pm i\sqrt{ [V(x)-E ]} & \mbox{if } E< V(x) . \end{array} \right . $$
(A.2)

A similar formula holds for the classically forbidden region (E<V(x)), namely

$$ \varPsi(x)\cong\frac{1}{\sqrt{|p(x)|}} \bigl[Ce^{-\frac{1}{\hbar }\int ^{x}{|p(y)|dy}}+De^{\frac{1}{\hbar}\int^{x}{|p(y)|dy}} \bigr]. $$
(A.3)

These wave-functions can be connected across turning points via so-called connection formulas, stated in books like [52]. First, we need to distinguish between two kinds of turning points in the usual way: we use the coefficients A l , B l , C l and D l for a left-hand turning point and A r , B r , C r and D r for a right-hand one. The lower limit of the integrals in the above equations is always the coordinate of the turning point. The connection formulas for a left-hand turning point are given by

(A.4)

whilst those for a right-hand turning point are given by

(A.5)

Now consider a general asymmetric double well, as shown in Fig. 9. This figure also introduces part of the notation used.

Fig. 9
figure 9

An asymmetric double-well potential V. The minima are a and b. We assume that the particle has energy E. This provides us with turning points x 1, x 2, x 3 and x 4 and hence with five distinct regions. Four of these regions are named with Roman numerals

Full size image

We need some more notation for the wkb coefficients used in our calculation. As in (A.1) and (A.3), A,B and C,D denote the coefficients of the wkb wave-function in the classically allowed region and the classically forbidden region, respectively. The number attached to a letter shows to which turning point it belongs, e.g. A 1 and B 1 are the coefficients of the wkb wave-function in region II with respect to x 1 (i.e. x 1 is the lower boundary of the integral in (A.1)). We also need the following three quantities:

$$ \theta_1=\frac{1}{\hbar}\int^{x_2}_{x_1}p(x)dx, \quad\quad \theta_2=\frac{1}{\hbar}\int^{x_4}_{x_3}p(x)dx, \quad\quad K=\frac{1}{\hbar}\int^{x_3}_{x_2}\big|p(x)\big|dx. $$
(A.6)

A final quantity we need is

$$ \tilde{\phi}=\arg{ \biggl[\varGamma \biggl(\frac{1}{2}+i\frac{K}{\pi } \biggr) \biggr]}+\frac{K}{\pi}-\frac{K}{\pi}\ln \biggl( \frac{K}{\pi } \biggr). $$
(A.7)

We are interested in the limit K→∞, since this implies that the barrier is very high and broad, which corresponds to the classical limit ħ→0. Note that \(\tilde{\phi}\rightarrow0\) as K→∞. Our goal is the following quantization condition for the general double well in Fig. 9:

$$ \bigl(1+e^{-2K} \bigr)^{1/2}=\frac{\cos(\theta_1-\theta_2)}{\cos (\theta_1+\theta_2-\pi+\tilde{\phi})} . $$
(A.8)

This condition can be derived in the following way:

  1. 1.

    We start out in region I (coefficients C 1 and D 1). The wave-function needs to be square integrable, so we immediately see that C 1=0.

  2. 2.

    Using the left connection matrix from (A.4), we move to region II (coefficients A 1 and B 1). We can then write the wkb wave-function with respect to x 2 by using

    $$ \left ( \begin{array}{c} A_2\\ B_2\\ \end{array} \right )= \left ( \begin{array}{c@{\quad}c} e^{i\theta_1} & 0\\ 0 & e^{-i\theta_1}\\ \end{array} \right ) \left ( \begin{array}{c} A_1\\ B_1\\ \end{array} \right ), $$
    (A.9)

    which can be proved by changing the lower boundary of the integrals in the wkb wave-function (A.1). The result is

    $$ \left ( \begin{array}{c} A_2\\ B_2\\ \end{array} \right )=e^{i\pi/4} \left ( \begin{array}{c} -i e^{i\theta_1}\\ e^{-i\theta_1}\\ \end{array} \right )D_1. $$
    (A.10)
  3. 3.

    In a similar way, we start in region IV (coefficients C 4 and D 4), and see that D 4=0. After moving to region III with a connection matrix and rewriting the wave-function with respect to x 3, we find

    $$ \left ( \begin{array}{c} A_3\\ B_3\\ \end{array} \right )=e^{i\pi/4} \left ( \begin{array}{c} e^{-i\theta_2}\\ -i e^{i\theta_2}\\ \end{array} \right )C_4 . $$
    (A.11)
  4. 4.

    We now use a result derived in [44] to jump over the barrier and connect the wkb wave-functions in region II and III, viz.Footnote 42

    $$ \left ( \begin{array}{c} A_2\\ B_2\\ \end{array} \right )= \left ( \begin{array}{c@{\quad}c} (1+e^{2K} )^{1/2}e^{-i\tilde{\phi}} & ie^K\\ -ie^K & (1+e^{2K} )^{1/2}e^{i\tilde{\phi}}\\ \end{array} \right ) \left ( \begin{array}{c} A_3\\ B_3\\ \end{array} \right ). $$
    (A.12)
  5. 5.

    Combining the above results (i.e. inserting (A.10) and (A.11) in (A.12)), we find

    (A.13)
    (A.14)
  6. 6.

    The equality of the above two equations leads to the quantization condition (A.8).

As will be discussed in the next two subsections, Eqs. (A.8) and (A.13) have implications for the energy levels and the wave-functions in an asymmetric double well.

1.2 A.2 Energy Splitting in an Asymmetric Double-Well Potential

Assume that for a certain (unperturbed) symmetric double well and given energy E, the constants θ 1 and θ 2 equal some value θ. As in Fig. 9, we introduce a perturbation in the right-hand well. For example, by (A.7), this means that θ=θ 1>θ 2 for a positive perturbation. We therefore write θ 1=θ, θ 2=θ−δ with δ∈ℝ (e.g. δ>0 in Fig. 9). The quantization condition (A.8) then becomes

$$ \bigl(1+e^{-2K} \bigr)^{1/2}=\frac{\cos(\delta)}{\cos(2\theta -\delta-\pi+\tilde{\phi})}. $$
(A.15)

We can solve for θ, yielding two solutions

$$ \theta_{\pm}=\biggl(n+\frac{1}{2}\biggr)\pi+\frac{1}{2} \delta-\frac {1}{2}\tilde{\phi} \pm \frac{1}{2}\arccos \biggl[ \frac{\cos(\delta)}{ (1+e^{-2K} )^{1/2}} \biggr]. $$
(A.16)

This resembles the original quantization condition \(\theta=(n+\frac{1}{2} )\pi\) for a single well, which is derived using connection formulas in [52]. Here, the energy levels have split up in pairs around the original ones (where the minus sign in (A.16) corresponds to the lower energy by (A.7)). To see what this means, we will examine this equation for two special cases. We first set δ=0 and check if this reproduces known results for a symmetric double well:

$$ \theta_{\pm}=\biggl(n+\frac{1}{2}\biggr)\pi-\frac{1}{2} \tilde{\phi} \pm \frac{1}{2}\arccos \biggl[\frac{1}{ (1+e^{-2K} )^{1/2}} \biggr]. $$
(A.17)

Supposing that K is large, this means that

$$ \theta_{\pm}\approx\biggl(n+\frac{1}{2}\biggr)\pi\pm \frac{1}{2}e^{-K}, $$
(A.18)

since for K large, \(\tilde{\phi}\approx0\) and \(\arccos (\frac{1}{\sqrt{1+x^{2}}} )=\arctan{x}\approx x\) for small x. We find that the energy levels of the single well have split into two. As discussed in [45], this leads exactly to the familiar energy splitting for a symmetric double-well potential stated in texts like [71]. That means that our method for general double wells reproduces known results for a symmetric one. Now that this has been confirmed, let us look at (A.16) in the classical limit K→∞. Solving (A.16) for K→∞ (and so \(\tilde{\phi}\rightarrow0\)) gives

(A.19)
(A.20)

This differs from the symmetric well, which for K→∞ gives a twofold degeneracy for each energy level labeled by n. Equation (A.20) can be understood in the following way: in the classical limit, tunneling is suppressed. Therefore, the particle is localized in one of the wells, where it obeys the familiar quantization condition for a single well. If it is in the left well, then \(\theta_{1}=(n+\frac{1}{2})\pi=\theta_{-}\), but if it is in the right well, we have \(\theta_{2}=(n+\frac{1}{2})\pi =\theta_{+} -\delta\).

1.3 A.3 Localization in an Asymmetric Double-Well Potential

Now that we have analyzed the behaviour of the energy splitting, we turn to the wkb wave-function. With the notation used in the previous section, (A.13) leads to

$$ \frac{D_1}{C_4} =i \bigl[ \bigl(1+e^{2K} \bigr)^{1/2}e^{-i(2\theta _{\pm}-\delta+\tilde{\phi})}+e^Ke^{-i\delta} \bigr]. $$
(A.21)

Inserting (A.16), the reader can check that for δ∈[−π,π] one has

$$ \frac{D_1}{C_4} =\sin(\delta)e^{K} \mp\sqrt{\sin^2( \delta )e^{2K}+1}. $$
(A.22)

This allows us to derive localization of the wkb wave-function in the classical limit K→∞. As can be seen from (A.10), D 1 is a measure of the amplitude of the wkb wave-function in regions I and II in Fig. 9. In a similar way, (A.11) shows that C 4 is a measure of the amplitude of the wkb wave-function in regions III and IV. Therefore, the fraction D 1/C 4 indicates whether the wave-function is localized, and if so, where. Doing the same calculation again for δ∈[π,3π] gives the above result multiplied by −1. Of course, this can be generalized: for n∈ℤ and δ∈[(2n−1)π,(2n+1)π], the result (A.22) is correct for n even and should be multiplied by −1 for n odd. This will not affect our conclusions, as we will see. We consider some cases and check what (A.22) tells us:

  • For δ=0 (no perturbation), we find that \(\frac {D_{1}}{C_{4}}=\mp1\). The general double well has pairs of energy levels (labeled by n). Such a pair consists of a lower and higher lying level, corresponding to θ − and θ + in (A.16), respectively. Here, we see that for the lower level D 1=C 4, i.e. the wkb wave-function is even. However, for the higher level we find D 1=−C 4, which means the wkb wave-function is odd. This is a well-known fact and it is nice to see our method reproducing it. Note that this conclusion is not only independent of n, but also of K, as expected.

  • For δ>0,δ∉{kπ|k∈ℤ} (which corresponds to a positive perturbation in the right well, e.g. the potential in Fig. 9), we find, in the limit K→∞, that:

    $$ \frac{D_1}{C_4}\longrightarrow \left \{ \begin{array}{l@{\quad}l} \infty& \mbox{for $\theta_{-}$ in (A.16)}\quad \mbox{(lower energy)}\\[2pt] 0 & \mbox{for $\theta_{+}$ in (A.16)}\quad \mbox{(higher energy)} . \end{array} \right . $$

    Hence for low (high) energy, the wkb wave-function is localized on the left (right).

  • For δ<0, δ∉{kπ|k∈ℤ}, i.e., a negative perturbation in the right well, we find

    $$ \frac{D_1}{C_4}\longrightarrow \left \{ \begin{array}{l@{\quad}l} 0 & \mbox{for $\theta_{-}$ in (A.16)}\quad \mbox{(lower energy)}\\ [2pt] \infty& \mbox{for $\theta_{+}$ in (A.16)}\quad \mbox{(higher energy)}. \end{array} \right . $$

    For the lower (higher) energy, the wkb wave-function is localized on the right (left).

  • For δ∈{kπ|k∈ℤ∖{0}}, something peculiar happens, in that either \(\frac {D_{1}}{C_{4}}=\pm1\) or \(\frac{D_{1}}{C_{4}}=\mp1\). This implies that no localization takes place.Footnote 43

  • So far, we have interpreted δ as the result of a perturbation in the right well. However, our approach allows us to interpret a positive perturbation in the right-hand well as a negative one in the left-hand well, and vice versa. Therefore, the above results change places if we put the perturbation in the left-hand well.

Our method produces the results we would expect. However, to be precise, the above reasoning needs to be amended as follows. We have treated δ as a constant, but in reality it depends on K. The reason for this is that K affects θ 1 and θ 2, and therefore δ=θ 1−θ 2, via the quantization condition. Now consider a fixed energy level (i.e. fixed n and fixed sign ± in (A.16)) in a given double-well potential that has a perturbation in one of the wells. In the limit of completely decoupled wells (K→∞), we know this energy level has some fixed limit higher than the minimum of the potential. As long as the perturbation is below this energy level, we know that θ 1−θ 2≠0 by (A.7). This means that there exists some K 0 such that |θ 1−θ 2|≠0 for any K>K 0. We may then apply the above reasoning to verify that our conclusions about localization are still correct.Footnote 44

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(Klaas) Landsman, N.P., Reuvers, R. A Flea on Schrödinger’s Cat. Found Phys 43, 373–407 (2013). https://doi.org/10.1007/s10701-013-9700-1

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