It is often stated that complex numbers are essential in quantum theory. In this article, the need for complex numbers in quantum theory is motivated using the results of tandem Stern-Gerlach experiments
The Stern-Gerlach experiment holds key role for students, particularly those studying quantum mechanics, modern physics, atomic physics, and physics education.It is one of the simplest experiments that demonstrates that the microscopic world does not behave accordingly withdynamics of classical physics, thusthe Stern-Gerlach experiment allows students to follow a logical progression: Classical expectation → Experimental observation → Failure of classical explanation → Quantum hypothesis → Spin states → Mathematical formulation → Quantum measurement.
The Stern–Gerlach (SG) experiment (1922) was one of the earliest experiments for which the result could not be explained using classical physics by any means. Hence it demonstrated unambiguously the importance of quantum effects. The SG experiment can be explained using the quantum theory for the spin-½ particle, which is a quantum system that can be represented simply by 2 × 2 matrices. We will use the spin-½ particle to introduce some of the basic concepts of quantum mechanics, such as state space, operators, measurement, the uncertainty principle, and the evolution equation. Furthermore, the theory for the spin-½ particle can be readily generalized to finite dimensional quantum systems, i.e., systems that can be represented by finite dimensional matrices.
The Stern–Gerlach experiment can play an important role in teaching the formalism of quantum mechanics. In the context of a finite-dimensional Hilbert space students can learn how to prepare a specific quantum state starting from an arbitrary state, issues related to the time evolution of the wave function and quantum measurement. The Stern–Gerlach experiment can also be used to teach the distinction between the physical space where the experiment is performed and the Hilbert space where the state of the system lies, and how information about the state of the system in the Hilbert space can be exploited to interpret the possible outcomes of the experiment in physical space. Students can learn the advantages of choosing an appropriate basis to make predictions about the outcomes of experiments with different arrangements of Stern–Gerlach devices. The latter can also help students understand that an ensemble of identically prepared systems is not the same as a mixture. We discuss student difficulties with the Stern–Gerlach experiment based on written tests and interviews with advanced undergraduate and graduate students in quantum mechanics courses. We also discuss preliminary data which suggest that the Quantum Interactive Learning Tutorial on the Stern–Gerlach experiment is helpful in improving student understanding of these concepts.
ABSTRACT The bulk of experiments and observations that lined the path to quantum mechanics amounted to a backlog of data in need of explanation. A notable exception was the Stern–Gerlach experiment (SGE) designed to test the predictions of the old quantum theory as embodied by the 1913–1916 Bohr–Sommerfeld–Debye atom model. After its initial triumphs, such as the explanation of the astrophysically observed spectrum of , this model was frowned upon in some quarters, mainly because of its ad hoc nature. Otto Stern (1888–1969)—and his close colleague and friend, Max von Laue (1879–1960)—even took a vow: “If this nonsense of Bohr should, in the end, prove to be right, we will quit physics.” Stern would go further yet and, in 1920, devise an experiment—the SGE—to disprove Bohr's model. However, contrary to Stern's expectations, the SGE ruled unequivocally in favor of the model and its amendments. Wolfgang Pauli (1900–1958) quipped: “That should convert even the non‐believer Stern.” And it did: Confronted with the outcome of the SGE, Stern noted in 1961: “I still have objections to the idea of beauty of quantum mechanics. But she is correct.” Ironically, the SGE would epitomize quantum mechanics as we know it: whether quantum measurement, state preparation, coherence, or entanglement—in addition to the quantization of angular momentum and its spatial projections—the SGE has it all. It now seems that the SGE may even impact future studies of the relationship of quantum theory to other pillars of physics, most notably, general relativity. The descendants of the SGE include transformative techniques for the study of the quantum properties of nuclear, atomic, molecular, and condensed‐matter systems, with magnetic resonance imaging being among the most widely used. Key Points Edward Pickering; Williamina Fleming; Pickering series; Bohr‐Sommerfeld‐Debye model of the atom; Otto Stern; Stern‐Gerlach experiment; Bohr magneton; Old Quantum Theory; Spin; nuclear magneton; magnetic dipole moment of the proton; matter‐wave duality; Quantum Mechanics.
Complex numbers, i.e., numbers with a and an imaginary part, are essential for mathematical analysis, while their role in other subjects, such as electromagnetism or special relativity, is far less fundamental. Quantum is the only physical theory where these numbers seem to play an indispensible role, as the theory is explicitly formulated in terms of operators acting on complex Hilbert spaces. The occurrence of complex numbers within the quantum formalism has nonetheless puzzled countless physicists, including the fathers of the theory, for whom a version of quantum physics, where states and observables are represented by operators, seemed much more natural. In fact, previous works showed that such real quantum physics can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary quantum states. Thus, are complex numbers really needed for a quantum description of nature? Here, we show this to be case by proving that and complex quantum make different predictions in network scenarios comprising independent quantum state sources. This allows us to devise a Bell-type quantum experiment whose input-output correlations cannot be approximated by any quantum model. The successful realization of such an experiment would disprove quantum physics, in the same way as standard Bell experiments disproved local physics.