© European Physical Society, EDP Sciences, 2025

In 1924 Louis de Broglie published the article “Tentative Theory of Light Quanta” in Philosophical Magazine, 47, 446-458 (1924). This paper introduced the relationship between the wavelength λ of waves and the momentum p of particles

$\lambda =h/p$(1)

connected for any physical system, not only for light quanta -photons -, also for matter, through the Planck’s action constant h. Although the nature of these “waves” was unknown, the de Broglie equation allowed a simple rationale for the ad-hoc quantization rules of the old quantum theory and it represented the first influential idea that triggered the avalanche of the construction of quantum mechanics. The Schrodinger “wave mechanics”, with the Schrodinger equation for these “matter waves”, and the Heisenberg matrix mechanics converged in a synthesis formulated in terms of states and observables. The interpretation of the “wave function” came from the Max Born “probability density amplitude” and the language was modelled by Niels Bohr in the so-called Copenhagen interpretation. The Bohr complementarity principle for the conjugate - incompatible - observables, appearing in the Heisenberg uncertainty relations, was even extended to explain the universal “wave-particle duality”.

The scientific revolution that quantum mechanics has represented has no precedent in the history of physics. The last 100 years have seen the appearance of novel phenomena in multiple fields of physics and the development of quantum technologies. These achievements do not hide, however, the pending issues about the foundations of the quantum world and the clarification of its conceptual basis.

In this article we discuss whether the concept of a “wave-particle duality”, which persists in the language of many scientists, has any actual basis in the behaviour of nature or it is simply a remnant of the historical origins of the theory. One still reads in standard textbooks and research articles sentences like: “The photon is a wave in its propagation and a particle in its detection” or “The existence of atomic interference is due to the matter waves of quantum physics”. What is behind these expressions is that waves interfere, particles do not.

The essential question can be stated as

Is the wave function ψ(x,t) a field in space-time?

A priori, one would say that the formulation of the theory in the image of time evolution of the states - instant realism - and the position representation as observable variable for a single particle, the description is in terms of a “true” wave function satisfying a wave equation with all desired wave properties. This formulation explains in particular the spatial interference in canonical devices at the quantitative level and it would give a credit to the language of “probability waves” in space-time. Is this “real” or an artifact of the interpretation of the representation used for the single particle behaviuor?

In order to answer this question, the key point is

What is “x”?, either a space point or the position observable.

The own formulation of the theory provides the response in favour of the second meaning, because

1. The same predictivity can be obtained by using the momentum representation, with the basis states chosen by the conjugate momentum observable, leading to a wave function ψ(p,t) in “momentum space”. What is “real” is the state Ψ(t) - the time is not a quantum observable -, independent of the representation used.

2. For a system of two particles (1, 2), even in the position representation, the wave function ψ(x₁, x₂; t) is defined in the configuration space, not in ordinary space. The two variables refer to the position observable for each of the two particles.

Then, why the spatial interference in a single particle double-slit experiment appears exactly like in the language of the wave-particle duality? In fact, the experimental result for “single photon interference”, where photons in the same state are prepared unconnected one-by-one, is sometimes advertised as: “Wave-particle duality observed in Young’s double slit experiment with sensitivity to individual photons”. Since a single particle cannot be shared between the two slits, one invokes its dual wave as the explanation. However, one realizes that the only ingredient which is needed is the interference of the probability amplitudes for the two alternatives, as a consequence of the linear superposition principle of the photon states:

• 3. If Ψ₁ is the state for definite propagation through the slit 1 and Ψ₂ is through slit 2, the state of the particle is given by the linear superposition of the two alternatives, leading to its propagation with the coherent sum of the two probability amplitudes for its arrival to the screen.

The components that interfere in quantum mechanics are not particles - neither dual waves -, they are probability amplitudes for different accessible untagged alternatives in the experiment. The fact that probability amplitudes add up like complex numbers is responsible for all quantum interferences. A photon, electron or atom is not shared by the two slits, as one would describe it by invoking its dual wave. This (coherent) sum of amplitudes, with a phase shift in its propagation, leads to the interference pattern for the probability density at each position in the screen. The question asking which is the slit followed by the photon is “unspeakable”, in John Bell’s terms. What the single particle interference experiment demonstrates is that a single particle contains all the information, although the experimental evidence of the interference pattern needs of course the statistical accumulation of individual arrivals of particles to the screen.

If the two alternatives in the experimental device are distinguishable, even if not actually distinguished, the linear superposition is not operating and the behaviour of the single particle is given in terms of classical probabilities by the incoherent sum of the two probabilities. To conclude:

• 4. The sentence “Single particle interference”, without prepositions in English, should be better understood as the more precise “Interference of probability amplitudes for a single particle”.

This argumentation proclaims that the double-slit interference experiment for a single particle has the same conceptual basis as any of the plethora of phenomena for “two-state systems” in quantum mechanics, with the superposition principle as the guiding explanation. For bound systems, the paradigmatic example of the covalent bonding of the molecular H₂⁺ ion follows the same reasoning: the electron state in the molecule is a linear superposition of the two atomic states of the electron with each proton. The symmetry between the two alternatives ensures that the ground state of the molecular ion (in the Born-Oppenheimer approximation) is a symmetric superposition of the two atomic states. This linear superposition implies a quantum logics in which the sentence saying that the electron is “either in one proton or the other” is wrong. What is correct is that these two alternatives have equal probability amplitudes in the ground state of H₂⁺.

• 5. In the ground state of the H₂⁺ molecular ion, the electron state is delocalized with equal probability amplitudes for the two localized states. The mysterious language of “the electron is shared by the two protons” in the covalent bonding is merely a remnant of the dual wave language for the electron, not its actual physics behaviour.

Historically, perhaps the most clarifying example of the concepts involved in the two-state systems is that of the ammonia molecule NH₃. The two geometrical states, with the nitrogen atom to the right or left of the plane of the three hydrogens, are two symmetrical alternatives to describe the molecule with definite energy E. The geometrical states would have been - by symmetry under parity - degenerate with energy E₀. However, quantum tunneling between these two states induces a mixing “A” between them, so that the Hamiltonian H of the system, in the geometrical basis, is given by the 2×2 matrix

$\text{H=}\left(\begin{array}{cc}{\begin{array}{c}E\end{array}}_0& A\\ A& E_0\end{array}\right)$(2)

The stationary states with definite time evolution that diagonalize H are then the two states of definite left-right symmetry, either symmetric or antisymmetric, with either equal or opposite probability amplitudes of the two geometrical states. The ground state in quantum mechanics has always the symmetry of H, so that this symmetric state and the antisymmetric excited state break the degeneracy E₀ of the geometrical states, as the reader may check with elementary algebra from H. The two states of ammonia with definite energy E are then not degenerate, leading to a phase shift in its coherent propagation. As a corollary, the production of the molecule in one of the two geometrical states leads to the beautiful phenomenon of “ammonia oscillations” in time between the two geometrical states. The precise quantum language for the left-right interference responsible of ammonia oscillations should be:

• 6. What is the transition probability as function of time that, having prepared the ammonia molecule in the left state, it appears as the right state ?

This is an appearance experiment. An alternative question would be to ask for the survival probability of being in the same initial state. These phenomena have the same conceptual basis as the spatial interference experiments, associated with the linear superposition property. If we have a localized nitrogen, the energy is not defined. If we have a definite energy, the relative position of the nitrogen atom is not defined. This proper language has no trace of the need of invoking any dual wave of the molecule. These quantum properties are essential in the behaviour of ammonia in presence of a time-dependent electric field, using the opposite electric dipole moments of the geometrical states. The inverted population preparation in the excited energy state leads to the stimulated emission of microwaves in the ammonia maser, the first one created by the Nobel Prize-winning Charles H. Townes in 1953.

Since the middle of the XX century, the field of particle physics has seen the appearance of particles with internal degrees of freedom, identified as charges beyond the electric charge, called “flavours”. In Fig. 1 we show the flavour of the elementary constituents of matter: quarks and leptons.

Fig. 1 Elementary constituents of matter with the flavour of quarks and leptons.

The first case was “strangeness”, needed for distinguishing the neutral K⁰ -meson from its antiparticle K¯⁰, as well as for baryons. K⁰ (ds¯) and K⁰ (sd¯) have zero electric charge and opposite strangeness, whereas the non-strange π⁰- meson (uu¯, dd¯) is equal to its antiparticle. Strangeness is conserved in strong and electromagnetic interactions but it is violated by weak interactions, contrary to the rigorous conservation of electric charge. Weak interactions mix then particle K⁰ and its antiparticle K¯⁰. In the basis of definite strangeness, the dynamics of the neutral Kaon can be described as a two-state system with a Hamiltonian H like that of Eq. (2). This leads to the two stationary states of definite mass and energy with symmetric (antisymmetric) superposition of opposite strangeness in the ground (excited) state. As a consequence, the ground (excited) state of definite mass and energy has equal (opposite) probability amplitudes of K⁰ or K¯⁰. The reader may have noticed that, instead of talking of two different particles K⁰, K¯⁰, due to the mixing the proper language is that of two states of the neutral Kaon. The strangeness mixing of states was first introduced by Gell-Mann and Pais in 1955.

• 7. The strangeness mixing and the non-degeneracy of the two states with definite mass predicts “neutral Kaon oscillations” between the two states of definite strangeness. This is a consequence of the interference of the probability amplitudes that a K⁰ (or K¯⁰) state has for the two non-degenerate mass alternatives.

These oscillations are actually observed because a K⁰, for example, can be prepared initially in different facilities. The oscillations are induced by an interference involving internal properties (flavour charges) that have nothing to do with wave propagation in space. And still the conceptual basis for the neutral Kaon oscillation and the Young’s double-slit experiment for single particle is the same in quantum physics: the linear superposition of the states with the interference of the two probability amplitudes. In the case of the neutral Kaon oscillation, the amplitudes for having one or other mass; in the spatial interference, the amplitudes for passing through slit 1 or slit 2. This common concept is illustrated in Fig. 2 with the schematic layout of any interference experiment in two-state systems, no matter whether the observables refer to either spatial or internal properties.

Fig. 2 Schematic layout of any quantum interference experiment in two-state systems. The essential ingredients are: (i) the mixing in the prepared state α as a coherent superposition of 1 and 2 -the “slits”-; (ii) its propagation -the “arms”- with a phase-shift Δ. The detector D - the “screen”- projects –“filters”- either α or β.

For the benefit of the reader, one should add that the physics of the neutral Kaon contains other “essential complications” in nature: (i) it is unstable with the two states of definite mass and time evolution having very different lifetimes, so at long times there is de-coherence and the interference pattern disappears; (ii) regeneration of the short-lived state by interaction with matter; (iii) the mixing “A” between matter (K⁰) and antimatter (K¯⁰) is not symmetric, so that “A” has to be complex with H_αβ=A and H_βα=A* leading to a violation of the matter-antimatter symmetry, first observed in 1964 by the Nobel Prize-winning James Cronin and Val Fitch. All in all, Feynman, Lee and Okun have elevated this two-state system to be a laboratory jewel in nature to understand the intricacies of the quantum world.

In the elementary constituents of matter, the three families of leptons - electron, muon, tauon - contain the corresponding neutrino species (or “flavours”) with weak interactions only - see Fig. 1 -. The physics of neutrinos is both fascinating and challenging to be observed: ordinary matter is (almost) transparent to neutrino propagation. If we avoid the “essential complication” of having three neutrino flavours (essential for a possible asymmetry between neutrino and antineutrino propagation in vacuum, still unknown in 2025) instead of two, we would have here another “two-state system” between the flavour states of electron-neutrinos and muon-neutrinos. To a good approximation, this is an appropriate description for the solar neutrino problem, with neutrinos emitted with the e-flavour in the nuclear fusion reactions in the Sun. Due to the flavour mixing, telling us that the states of definite weak interaction are distinct to the states of definite mass, the appropriate quantum language is that of “neutrino flavour oscillations”:

• 8. What is the survival probability that an e-neutrino produced in the Sun behaves as an e-neutrino when detected at Earth?

Bruno Pontecorvo, who predicted the existence of the second neutrino before its discovery in 1962, anticipated a “solar neutrino problem” in the answer to the point (8). A survival probability below 100 % is predicted when the two ingredients, flavour mixing and non-degeneracy of the two states of definite mass and energy, are present. No known symmetry between these flavour states – “Flavour Physics” is one of the open issues in particle physics research - generalizes the Hamiltonian H of Eq. (2) to have unequal E_α, E_β for the diagonal terms, instead of a common E₀. Such a real symmetric matrix is diagonalized by an orthogonal matrix

$\text{U}=\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right)$(3)

with a non-maximal superposition cosθ ≠ sinθ in general. The observed deficit of e-neutrinos, compared to the total arrival of neutrino flux from the Sun, in 2002, gave a survival probability of 1/3 and it proved the “solar e-neutrino flavour oscillations” quantum phenomenon. In 1998, “atmospheric mu-neutrino flavour oscillations” had been observed in the arrival of mu-neutrinos produced in the Earth atmosphere by cosmic ray interactions. These results imply that neutrinos have non-vanishing masses, much smaller than all other elementary constituents of matter. The 2015 Nobel Prize in Physics was awarded to Takaaki Kajita and Arthur McDonald for these discoveries.

Neutrino flavour oscillations are also observed with terrestrial nuclear reactors and accelerators by means of baselines - the distance between production and detection – up to more than 1000 Km. These experiments are today the method to explore open questions on neutrino properties. The interference of probability amplitudes involved in neutrino oscillations refers to flavour states, not to spatial properties. In fact, all neutrino states propagate in these experiments in the same spatial direction and there is no room to speak the language of dual waves in space.

Conclusion

With this conceptual discussion of the interferences of probability amplitudes involved in the physics of a variety of two-state quantum systems, one discovers that there is a common fundamental origin: the linear superposition of states. The alternative untagged properties present in the superposition can be either spatial or internal, with no need of invoking a mysterious inconsistent duality for explaining the case of spatial interference. Waves satisfy the superposition principle in space-time, but this is not that required in quantum physics. With the comparison of the different cases summarized in the points from (1) to (8), the reader discovers that the historical dilemma in quantum physics for the interference is not a matter of particles versus waves. The interference is between probability amplitudes of having alternative untagged properties present in the superposition of quantum states. Probability amplitude is the fundamental concept in quantum physics. In quantum logics, thanks to the linear superposition of states, things are not necessarily either one or the other.

About the Author

Jose Bernabeu is Emeritus Professor of the University of Valencia. His research accomplishments cover the fields of neutrinos, electroweak interactions, symmetries, flavour and quantum physics. He was Member of LHC-Committee and Head of EPS HEP-PP Board.

Acknowledgements

The author would like to thank Francisco Botella, Catalina Espinoza and Arcadi Santamaria for multiple scientific discussions on the subject of this article. This research has been funded by the Grants CIPROM/2021/054 (Prometeo, Generalitat Valenciana), PID2023-151418NB-I00 (MCIU/AEI/10.13039/501100011033/FEDER, UE) and CEX2023-001292-S (SO, MCIU/AEI).

All Figures

	Fig. 1 Elementary constituents of matter with the flavour of quarks and leptons.
In the text

	Fig. 2 Schematic layout of any quantum interference experiment in two-state systems. The essential ingredients are: (i) the mixing in the prepared state α as a coherent superposition of 1 and 2 -the “slits”-; (ii) its propagation -the “arms”- with a phase-shift Δ. The detector D - the “screen”- projects –“filters”- either α or β.
In the text