Schrödinger hoped that his wave mechanics would help to re-establish some sense of ‘visualizability’ of the physics going on inside the atom. In searching for a suitable interpretation of the wavefunction, he focused on the density of electrical charge, which he associated with the wavefunction ψ multiplied by its complex conjugate. Hidden in his words is the interpretation that would eventually come to dominate our understanding of the wavefunction. Max Born had no hesitation in concluding that the only way to reconcile wave mechanics with the particle description is to interpret the modulus-square of the wavefunction as a probability density. It was Wolfgang Pauli who proposed to interpret this not only as a transition probability or as the probability for the system to be in a specific state, as Born had done, but as the probability of ‘finding’ the electron at a specific position in its orbit inside an atom.
Schrödinger hoped that his wave mechanics would help to re-establish some sense of ‘visualizability’ of the physics going on inside the atom. In searching for a suitable interpretation of the wavefunction, he focused on the density of electrical charge, which he associated with the wavefunction ψ multiplied by its complex conjugate. Hidden in his words is the interpretation that would eventually come to dominate our understanding of the wavefunction. Max Born had no hesitation in concluding that the only way to reconcile wave mechanics with the particle description is to interpret the modulus-square of the wavefunction as a probability density. It was Wolfgang Pauli who proposed to interpret this not only as a transition probability or as the probability for the system to be in a specific state, as Born had done, but as the probability of ‘finding’ the electron at a specific position in its orbit inside an atom.
This chapter examines the interpretation of the wavefunction, and specifically what it reveals about the location of a particle. In quantum mechanics, all the properties of a system are expressed in terms of a wavefunction, which is obtained by solving the equation proposed by Erwin Schrödinger. Indeed, wavefunctions provide the essential foundation for understanding the properties of electrons in atoms and molecules, and are central to explanations in chemistry. The chapter then considers how, according to the Born interpretation, the probability density at a point is proportional to the square of the wavefunction at that point. A wavefunction is normalized if the integral over all space of its square modulus is equal to 1. Ultimately, a wavefunction must be single-valued, continuous, not infinite over a finite region of space, and have a continuous slope. The quantization of energy stems from the constraints that an acceptable wavefunction must satisfy.
Position probabilities play a privileged role in the interpretation of quantum mechanics. The quantum wave function (usually designated “Ψ(r)’) cannot itself represent a physical quantity, because it is complex valued. Its square — |Ψ(r)|2 — however, is real; and it is |Ψ(r)|2 which is the fundamental interpreted quantity of the theory. Schrödinger, who originated wave mechanics, interpreted |Ψ(r)|2 as the density of a charged particle; but this suggestion was inadequate. A principal reason is that, for most physical systems, |Ψ(r)|2 spreads out in time, whereas particles themselves do not. Schrödinger's interpretation was replaced by Born who introduced probabilities into quantum mechanics. Born urged that |Ψ(r)|2 represents not the density of the system in space, but rather a probability density. This is almost universally how quantum mechanics is taught today: |Ψ(r)|2 gives the probability that a system in state Ψ(r) is located at r.
Abstract In June 1925 Heisenberg made the breakthrough that would lead to matrix mechanics. The theory was abstract, offered no means for visualizing the mechanics of the atom, and its application required a lot of heavy lifting. Erwin Schrödinger subsequently made the breakthrough that would lead to wave mechanics, building on Louis de Broglie’s association of a wave with an electron. In Schrödinger’s wave equation the quantum numbers of the hydrogen atom arise naturally. He argued that his ‘wavefunction’ offered a realistic description of the electron in an atom, and sought to eliminate Bohr’s discontinuous quantum jumps. But in the spring of 1926, Max Born offered an enduring alternative. The modulus-square of the wavefunction is a measure of probability in an unvisualizable microphysics. In an intense debate with Bohr, Schrödinger demanded a theory based on logical concepts free from contradictions. Bohr was left to ponder what this might mean.
In quantum mechanics, the physical state of an electron is described by a wave function. According to the standard probability interpretation, the wave function of an electron is probability amplitude, and its modulus square gives the probability density of finding the electron in a certain position in space. In this article, we show that this central assumption of quantum mechanics may have an ontological extension. It is argued that microscopic particles such as electrons are indeed particles, but their motion is not continuous but discontinuous and random. On this view, the modulus square of the wave function not only gives the probability density of the particles being found in certain locations, but also gives the probability density of the particles being there. In other words, the wave function in quantum mechanics can be regarded as a representation of the state of random discontinuous motion of particles, and at a deeper level, it may represent the dispositional property of the particles that determines their random discontinuous motion.
When Max Born introduced his statistical interpretation of wave mechanics in mid-1926, the reactions were mixed. His most recent collaborator, Pascual Jordan, was delighted; his former assistant, Werner Heisenberg, was horrified. The interpretation served as grist for Neils Bohr's philosophical mill. For Bohr's former collaborator, John Slater, it inspired a strange hybrid interpretation, a sort of statistical version of Schrödinger's electrodynamic model. What was the interpretation that engendered these varied reactions? Neither Born nor most of his contemporaries saw in it anything that even suggests the interpretation that we now associate with his name. Rather, it was another former assistant of Born's, Wolfgang Pauli, with eyes as sharp as his tongue, who saw in Born's work the idea that |Ψ(q)| represents the probability distribution for position.
In this paper, we propose an ontological interpretation of the wave function in terms of random discontinuous motion of particles. Ac-cording to this interpretation, the wave function of an N-body quan-tum system describes the state of random discontinuous motion of N particles, and in particular, the modulus squared of the wave function gives the probability density that the particles appear in every possible group of positions in space. We present three arguments supporting this new interpretation of the wave function. These arguments are mainly based on an analysis of the mass and charge properties of a quantum system. It is realized that the Schrödinger equation, which governs the evolution of a quantum system, contains more information about the system than the wave function of the system, such as the mass and charge properties of the system, which might help understand the ontological meaning of the wave function. Finally, we briefly an-alyze possible implications of the suggested ontological interpretation of the wave function for the solutions to the measurement problem. The wavefunction gives not the density of stuff, but gives rather (on squaring its modulus) the density of probability. Probability of what exactly? Not of the electron being there, but of the electron being found there, if its position is ‘measured’. Why this aversion to ‘being ’ and insistence on ‘finding’? The founding fathers were unable to form a clear picture of things on the remote atomic scale. (Bell 1990)