Quantum Elegance: Decoding Hydrogen’s Zeeman Effect
Quantum Mechanics & Magnetism | Ayush Varma
In the captivating realm of hydrogen stationary states, foundational quantum concepts such as expectation values, position operators, and angular momentum operators are indispensable as they serve as guiding lights for physicists navigating the intricacies of electrons within atoms. These quantum tools provide a nuanced understanding of quantum mechanics, particularly when applied to hydrogen, a fundamental building block. This article will attempt to interpret the hydrogen emission spectrum and the selection rules associated with quantum numbers, linking the concepts of magnetic moment and Larmor precession. The mechanics of these, in turn, describe the Zeeman effect. Lastly, we look at the various industrial and research applications of spectral shifting, both past and forthcoming.
Expectation values in quantum mechanics offer a statistical perspective to gauge average outcomes in quantum regimes. When looking at hydrogen stationary states, these values become useful tools for predicting the most probable results of measurements relating to observables like position and angular momentum [1]. They act as guiding beacons, steering physicists toward a comprehensive understanding of the behaviour of electrons within their orbitals, and portraying the inherently probabilistic nature of quantum systems [1-2].
The position operator, a fundamental quantum mechanical tool, represents the observable tied to a particle’s position (x). For hydrogen’s stationary states, this operator allows scientists to compute the average position of an electron within its orbital [2-3]. By acting on the wavefunction, the position operator unveils essential details about the probable location of the electron, offering insights into the spatial distribution of the electron cloud. The wavefunction, denoted by Ψ, encapsulates the quantum state of a system. Its square, |Ψ|2, provides the probability density of finding a particle in a specific position. Understanding the wavefunction is crucial as it forms the foundation for fundamentalising quantum states [3].
Simultaneously, the angular momentum operator plays a vital role, addressing the observable linked to a particle’s angular momentum (L). In the hydrogen atom, this operator is instrumental in quantifying the anticipated angular momentum of an electron in a specific orbital [4-6]. Quantum numbers associated with the angular momentum operator impart crucial information about the shape and orientation of the orbital, providing a comprehensive understanding of the electron’s rotational motion [5-6].
Distinguishing between sharp and fuzzy quantities is integral to quantum mechanics. Sharp quantities, like energy levels in stationary states, have precisely defined values. In contrast, fuzzy quantities, exemplified by position and momentum, exist in a state of uncertainty as per Heisenberg’s uncertainty principle [7]. This principle asserts that the more precisely one quantity is known, the more uncertain the other becomes, challenging classical notions of determinism [8-9].
With the basics covered, let’s dive into how this quantum toolkit aids in analysing the hydrogen emission spectrum. The emission spectrum, consisting of distinct lines, results from electrons transitioning between stationary states [9]. As an electron moves from a higher energy level to a lower one, it emits a photon with energy equal to the energy difference between the two levels [10]. This emitted energy corresponds to a specific wavelength or frequency, creating spectral lines [10].
The information derived from the wavefunction is crucial to understanding these transitions. The hydrogen wavefunction, shown in Figure 1, is the current orbital model internationally equipped by science as a whole. Analysing the wavefunction helps physicists predict and interpret the spectral lines in the hydrogen emission spectrum. The quantum numbers associated with the stationary states obtained from wavefunction solutions provide a roadmap for understanding the allowed energy levels and transitions [11-13].
Figure 1: A visualisation of the hydrogen wavefunction. The current electron probability distribution model reveals quantum behaviour. It showcases a H atom’s orbital structure and the distinct energy levels with spatial arrangement [13].
Optical transitions play a pivotal role in the hydrogen emission spectrum. When an electron transitions between two stationary states with different energy levels, it absorbs or emits a photon with energy corresponding to the energy difference between these states [14]. This process involves changes in the electron’s orbital configuration and is fundamental to the creation of spectral lines. Optical transitions are at the heart of spectroscopy, allowing scientists to probe the energy levels and quantum states of atoms. [14-15]
The selection rule is a critical concept in understanding these transitions. It stipulates which transitions are allowed and which are forbidden based on the conservation of angular momentum. The selection rule for hydrogen spectral lines involves a change in the principal quantum number (n). Specifically, the change in n must be ±1 for a transition to be allowed. This rule arises from the law of conservation of energy and angular momentum in quantum systems [16].
Figure 2: The hydrogen spectrum reveals the fingerprints of hydrogen’s energy transitions in the form of an array of spectral lines.
In the intricate dynamics of hydrogen stationary states and optical transitions, the conservation of momentum emerges as a pivotal factor. The selection rule for hydrogen spectral lines, governed by the conservation of angular momentum, dictates the allowed transitions based on changes in the principal quantum number [16].
Spectral shifting, a phenomenon where spectral lines change position, can be justified through these quantum concepts as energy levels and their transitions affect the wavelength or frequency, respectively, of emitted or absorbed photons. As we see in Figure 2, this provides a profound correlation between quantum principles and the observed shifts in the spectral lines of hydrogen [17]. We now explore the concept of magnetic moment, a property associated with the movement of particles, which is especially relevant when considering hydrogen. The magnetic moment (μ) of an electron is a measure of its intrinsic magnetic properties, arising from its spin and orbital motion [17]. The quantisation of the magnetic moment in quantum mechanics means that only discrete values can be taken along a particular direction [18].
This quantisation of the magnetic moment has significant implications in the realm of spectroscopy, particularly in the phenomenon known as Larmor precession [18]. Larmor precession refers to the motion of the magnetic moment of a charged particle, such as an electron, in response to an external magnetic field. The quantisation of the magnetic moment introduces discrete precession frequencies associated with specific transitions between energy levels.
The torque experienced by the magnetic moment in an external magnetic field is a crucial element in Larmor precession. The torque induces the rotational motion which enables precession [18]. The quantisation of the magnetic moment leads to discrete torque values, corresponding to specific transitions between energy levels. In the context of hydrogen, with an external magnetic field, the quantised magnetic moments of electrons lead to discrete changes in precession frequencies, contributing to the fine structure observed in the hydrogen spectrum. This phenomenon provides valuable information about the magnetic properties of electrons and the effects of external magnetic fields on atomic behaviour [18].
Integrating the influence of these physical quantities on the Zeeman effect within the hydrogen atom tells us a lot. The Zeeman effect is observed when atoms are subjected to an external magnetic field, causing a socalled spectral line splitting. The gyromagnetic ratio and Bohr magneton collectively govern the behaviour of electrons in response to the external magnetic field, influencing the extent and pattern of line splitting [18].
The gyromagnetic ratio determines the strength of the interaction between the magnetic moment of the electron and the external magnetic field. Larger gyromagnetic ratios result in more pronounced Zeeman splitting [18]. The Bohr magneton (μB), representing the magnetic moment associated with the orbital motion of an electron, plays a pivotal role in determining the magnitude of the Zeeman splitting. These are both usually derived from theoretical constants.
In summary, the interplay of physical quantities, such as the gyromagnetic ratio, Bohr magneton, quantum numbers, and Larmor frequencies, profoundly influences the behaviour of electrons within the hydrogen atom in the Zeeman effect. The quantisation of the magnetic moment and the discrete nature of transitions between energy levels give rise to the characteristic splitting patterns observed in the spectral lines. Understanding these quantum principles provides a deeper insight into the magnetic properties of atoms and the intricate dynamics of electrons in the presence of external magnetic fields. The Zeeman effect, with its spectral signatures, stands as a testament to the profound connections between quantum mechanics and experimental observations in the realm of atomic physics. The extent of industrial applications and impacts of spectral shifting are limited but handy. In material science and nanotechnology, the analysis of shifted spectral lines aids in characterising electronic and magnetic properties and steering advancements in material design. Shifted spectral lines help with detecting and quantifying pollutants, contributing to efforts to understand and mitigate environmental impacts.
In medical diagnostics, particularly in MRI scanning, the Zeeman effect’s principles play a crucial role. When a patient is placed in a strong external magnetic field (such as that of an MRI scanner), the Zeeman effect induces the splitting of nuclear magnetic resonance signals emitted by hydrogen nuclei in water molecules within the body. By precisely detecting these shifted signals, MRI scanners create detailed images of internal structures, offering unparalleled insights into tissues and organs. The role of the Zeeman effect in NMR spectroscopy and MRI scanning exemplifies its practical applications, showcasing how fundamental quantum principles contribute significantly to scientific research and healthcare diagnostics [19]. The precision of spectral analysis contributes to the accuracy of medical diagnoses and the development of advanced imaging techniques with detailed views of internal structures.
Beyond scientific research, spectral shifting studies on the electronic and magnetic properties of materials influences the development of advanced materials with tailored properties. The industry of nanomaterials hugely appreciates spectral splitting as precise control over electronics manufacture is needed. Spectral analysis is used to characterise and manipulate the electronic structure of nanomaterials, influencing their conductive properties. This has implications for the development of novel electronic devices with enhanced performance [19-20].
The use of spectral shifting in astronomics branches out into more specific domains. In solar physics, the analysis of shifted spectral lines allows scientists to unravel the delicate details of solar sunspots, providing insights into localised magnetic activity on the sun’s surface. In molecular and atomic rotational dynamics, spectral shifting assists in understanding orbital rotations accurately. Beyond this, celestial objects at any reachable point in our cosmos can be rigorously assessed based on the magnetic field knowledge gathered from the Zeeman effect. In turn, this information contributes to a broader understanding of the evolution of our universe [20].
Satellite technologies also benefit from precise navigation and communication offered by the leveraging of magnetic sensors, thanks to the Zeeman effect. Satellites are therefore able to accurately orientate and align with respect to earth’s magnetic field, creating stabilisation. As quantum technologies continue to advance, the applications of spectral analysis in quantum computing and communication hold promise for transformative breakthroughs. Spectral shifting, with its roots in quantum principles, becomes a key tool in manipulating and analysing quantum states, contributing to the development of more powerful and efficient devices [21-22].
Conclusion
Hydrogen stationary states, optical transitions, and the Zeeman effect unveils not only the fundamental principles of quantum mechanics but also their profound applications across scientific disciplines and industries. From unravelling the mysteries of celestial bodies to steering advancements in material science, environmental monitoring, and medical diagnostics, the impact of spectral shifting resonates across the spectrum of human knowledge and technological innovation. The Zeeman effect’s role in NMR spectroscopy and MRI scanning further emphasises its versatile applications, bridging the gap between fundamental quantum concepts and practical advancements in healthcare diagnostics.
[1] R. Nave, “Zeeman Effect,” hyperphysics.phy-astr.gsu.edu. http:// hyperphysics.phy-astr.gsu.edu/hbase/quantum/zeeman.html
[2] R. Doron et al., “Determination of magnetic fields based on the Zeeman effect in regimes inaccessible by Zeeman-splitting spectroscopy,” High Energy Density Physics, vol. 10, pp. 56–60, Mar. 2014, doi: https://doi. org/10.1016/j.hedp.2013.10.004.
[3] J. C. del Toro Iniesta, “On the discovery of the zeeman effect on the sun and in the laboratory,” Vistas in Astronomy, vol. 40, no. 2, pp. 241–256, Jan. 1996, doi: https://doi.org/10.1016/0083-6656(96)00005-0.
[4] P. G. Carolan, M. Forrest, N. J. Peacock, and D. L. Trotman, “Observation of Zeeman splitting of spectral lines from the JET plasma,” Plasma Physics and Controlled Fusion, vol. 27, no. 10, pp. 1101–1124, Oct. 1985, doi: https://doi.org/10.1088/0741-3335/27/10/003.
[5] W. Fang, J. Chen, Y. Feng, X.-Z. Li, and A. Michaelides, “The quantum nature of hydrogen,” International Reviews in Physical Chemistry, vol. 38, no. 1, pp. 35–61, Jan. 2019, doi: https://doi.org/10.1080/014423 5x.2019.1558623.
[6] J. Simola and J. Virtamo, “Energy levels of hydrogen atoms in a strong magnetic field,” Journal of physics, vol. 11, no. 19, pp. 3309–3322, Oct. 1978, doi: https://doi.org/10.1088/0022-3700/11/19/008.
[7] A. R. P. Rau and L. Spruch, “Energy Levels of Hydrogen in Magnetic Fields of Arbitrary Strength,” The Astrophysical Journal, vol.207, pp.671-679., Jul, 1976. https://adsabs.harvard.edu/full/1976ApJ...207..671R
[8] W. C. Myrvold, “What is a wavefunction?,” Synthese, vol. 192, no. 10, pp. 3247–3274, Jan. 2015, doi: https://doi.org/10.1007/s11229-014-0635-7.
[9] Y. Aharonov, J. Anandan, and L. Vaidman, “Meaning of the wave function,” Physical Review A, vol. 47, no. 6, pp. 4616–4626, Jun. 1993, doi: https://doi. org/10.1103/physreva.47.4616.
[10] W. Rosner, G. Wunner, H. Herold, and H. Ruder, “Hydrogen atoms in arbitrary magnetic fields. I. Energy levels and wavefunctions,” vol. 17, no. 1, pp. 29–52, Jan. 1984, doi: https://doi.org/10.1088/0022-3700/17/1/010.
[11] E. Leader and C. Lorcé, “The angular momentum controversy: What’s it all about and does it matter?,” Physics Reports, vol. 541, no. 3, pp. 163– 248, Aug. 2014, doi: https://doi.org/10.1016/j.physrep.2014.02.010.
[12] M. Mazilu, “Spin and angular momentum operators and their conservation,” Journal of optics, vol. 11, no. 9, pp. 094005–094005, Aug. 2009, doi: https://doi.org/10.1088/1464-4258/11/9/094005.
[13] M. A. Gorlach, A. N. Poddubny, and P. A. Belov, “Splitting of emissionspectrum lines in an anisotropic medium due to self-induced torque,” Physical Review A, vol. 89, no. 3, Mar. 2014, doi: https://doi.org/10.1103/ physreva.89.032508.
[14] V. S. Lisitsa, “New results on the Stark and Zeeman effects in the hydrogen atom,” Soviet Physics Uspekhi, vol. 30, no. 11, pp. 927–951,Nov. 1987, doi: https://doi.org/10.1070/ pu1987v030n11abeh002977.
[15] O. Stern, “A New Method for the Measurement of the Bohr Magneton,” Physical Review, vol. 51, no. 10, pp. 852–854, May 1937, doi: https://doi.org/10.1103/physrev.51.852.
[16] L. Deych, “Fine Structure of the Hydrogen Spectra and Zeeman Effect,” Springer eBooks, pp. 465–496, Jan. 2018, doi: https://doi. org/10.1007/978-3-319-71550-6_14.
[17] P. F. Winkler, D. Kleppner, T. Myint, and F. G. Walther, “Magnetic Moment of the Proton in Bohr Magnetons,” vol. 5, no. 1, pp. 83–114, Jan. 1972, doi: https://doi.org/10.1103/ physreva.5.83.
[18] J. S. Rigden, “Quantum states and precession: The two discoveries of NMR,” Reviews of Modern Physics, vol. 58, no. 2, pp. 433–448, Apr. 1986, doi: https://doi. org/10.1103/revmodphys.58.433.
[19] L. C. Balling and F. M. Pipkin, “Gyromagnetic Ratios of Hydrogen, Tritium, Free Electrons, andRb85,” Physical Review, vol. 139, no. 1A, pp. A19–A26, Jul. 1965, doi: https://doi. org/10.1103/physrev.139.a19.
[20] P. L. Bender and R. L. Driscoll, “A Free Precession Determination of the Proton Gyromagnetic Ratio,” IRE transactions on instrumentation, vol. I–7, no. 3/4, pp. 176– 180, Dec. 1958, doi: https://doi.org/10.1109/ ire-i.1958.5006784.
[21] A. Mojiri, R. Taylor, E. Thomsen, and G. Rosengarten, “Spectral beam splitting for efficient conversion of solar energy—A review,” Renewable and Sustainable Energy Reviews, vol. 28, pp. 654–663, Dec. 2013, doi: https://doi. org/10.1016/j.rser.2013.08.026.
[22] E. Landi Degl’Innocenti, “The Zeeman effect: applications to solar physics,” Astronomische Nachrichten, vol. 324, no. 4, pp. 393–396, May 2003, doi: https://doi.org/10.1002/ asna.200310143.
[23] G. Radda, “The use of NMR spectroscopy for the understanding of disease,” Science, vol. 233, no. 4764, pp. 640–645, Aug. 1986, doi: https://doi.org/10.1126/science.3726553.
Ayush is an astrophysics student who has particular interests in cosmic inflation and the Higgs field, with a dream of visiting CERN to witness the LHC in action. He plays competitive badminton and cricket, and enjoys watching astronomy documentaries. You can almost always find his talkative self outdoors. He claims to be a buff for Indian movies, and reads fictional thrillers.