The Relevance of Magnetism for the Development of Quantum Mechanics

Transcription

1 The Relevance of Magnetism for the Development of Quantum Mechanics Pascal Mauron Contents 1 Introduction 1 2 Crookes tube and the discovery of the electron Crookes tube Thomson s experiment Towards the Schrödinger equation Quantization, spin, and the Stern-Gerlach Experiment Black body radiation The Bohr atom model The Stern-Gerlach experiment Degeneracy and the Zeeman effect A brief refresher on the Hamiltonian for a central potential The normal Zeeman effect The g-factor The anomalous Zeeman effect Spin-orbit interaction and the hydrogen fine and hyperfine structures Introduction Towards the end of the nineteenth century, the achievements of what we today call classical physics were such that students were discouraged from this academic path on the reasoning that all the important laws of physics had been discovered, meaning that research would only be concerned with clearing minor problems or improving measurement methods [1]. The specificities of those minor problems can be found in any reference on the history of physics. This small paper focuses instead on the impact of phenomena of magnetic nature on the development of quantum mechanics: The discovery of the electron. The experimental proof of the quantization of angular momentum. The discovery of the electron spin. The experimental proof of quantum degeneracy. For the sake of completeness, phenomena that are not directly related to magnetism but where essential in the discovery of quantum mechanics are also briefly mentioned. 1

2 2 Crookes tube and the discovery of the electron Wave particle duality postulates that all particles exhibit both wave and particle properties. The idea of duality originated in a debate that dates back to the 17 th century. Competing theories of light were proposed by Christiaan Huygens and Isaac Newton: light was thought either to consist of waves (Huygens) or of particles (Newton) [2]. Assuming light to travel at different speeds in different media, refraction could be easily explained as the medium-dependent propagation of light waves. The resulting Huygens Fresnel principle was extremely successful at reproducing light s behaviour and, subsequently supported by Thomas Young s discovery of double-slit interference, was the beginning of the end for the particle light camp. After James Clerk Maxwell discovered that only four equations are required to describe self-propagating waves of oscillating electric and magnetic field, it quickly became apparent that visible light, ultraviolet light, and infrared light (phenomena thought previously to be unrelated) were all electromagnetic waves of differing frequency. The wave theory had prevailed or at least it seemed to. 2.1 Crookes tube Around 1870, cathode rays (revealed later to be streams of electrons) produced in a Crookes tube 1 were found to be deflected by both electric and magnetic fields, obeying Faraday-Henry s law of induction for currents in electric conductors [3]. E = B t (1) Figure 1: Electrons casting the shadow of a Maltese cross on the face of a Crookes tube (from [3]). As was the case two centuries before, there were two opposed theories to explain what was happening [4]: cathode rays were either corpuscles (i.e. electrically charged atoms) or aether vibrations, some new form of electromagnetic waves that were separate from what carried the current through the partial vacuum of the tube. The debate continued until comparisons 1 From the English physicist Sir William Crookes, also known as the inventor of the radiometer. 2

3 between experiments on cathode rays deflected by electric and magnetic fields allowed Joseph John Thomson ( Nobel Prize in Physics 1906) to measure the mass-to-charge ratio of a very light particle that was later named the electron. 2.2 Thomson s experiment This clever experiment (whose description here is freely adapted from an undergraduate laboratory course from Dartmouth University [5]) exploits the Lorentz interaction between electromagnetic fields and charged particles. F = q [E + v B] (2) The apparatus consists in a Crookes tube where deflecting electric and magnetic fields can be applied perpendicularly to each other (see Fig. 2), so that the electric and magnetic components of the Lorentz force are collinear: The electric field is provided by two electrodes placed inside the tube. The magnetic field (symbolised with a horseshoe magnet in the figure) is usually produced with a Helmholtz coil. Figure 2: Schematic view of the apparatus used in an experiment to determine the massto-charge ratio of electrons in a Crookes tube (from [5]) Electron velocity The first step is to determine the velocity of the particle. For this, both the magnetic and the electric deflections are switched on, and the goal is to adjust them until the Lorentz force is as close to zero as possible. At this point the cathode rays follow straight lines again, as they would without the external fields. F = 0 v = E B (3) 3

4 The amplitudes of both the electric and the magnetic fields can be computed from the construction details of the apparatus: The amplitude of the electric field E depends on the voltage V applied between the deflector plates separated by a distance d. The amplitude of the magnetic field B is directly proportional to the current I circulating in the Helmholtz coils. E = V d e z B = k H I e y v = Mass-to-charge ratio of the electron V dk H I Between the electrodes is the electric field constant, meaning that if we now switch off the magnetic field, the cathode rays follow a parabolic curve. (4) ΣF = qe = ma (5) Figure 3: Sketch of the particle deviation with only the electric field on. Measuring the coordinates of the impact point P x,y and combining all these equations with the geometry given in Fig. 3 gives the sought for mass-to-charge ratio: The deflection y P is proportional to the acceleration due to the electric field. The particles impact when the deflection equals half the distance d between the electrodes. The distance x P travelled by the particle between the electrode plates is proportional to its speed, which is a priori the same as before if we did not change the applied voltage used to generate the cathode rays. m e q = E 2y P ( xp v ) 2 = (x P ki) 2 V (6) Thomson s experiment showed that the corpuscles in cathode rays had less than one thousandth of the mass of the hydrogen ion, the least massive ion known. The last piece of this puzzle was brought in 1908 by Robert Andrews Millikan (Nobel Prize in Physics 1923), who determined the magnitude of the elementary electrical charge with his oil drop experiment. 4

5 2.3 Towards the Schrödinger equation The planetary model of the atom Thomson s view of the atom was that of negatively charged plums (the electrons), surrounded by a positively charged pudding. To account for the electrical neutrality of atoms, the sum of charges was of course zero. This was challenged by an experiment performed in 1908 by Hans Geiger and Ernest Marsden, under the guidance of Ernest Rutherford. When alpha particles were shot at a very thin gold foil, some of the exiting particles were deflected at high angles, which led Rutherford to formulate his planetary model of the atom (1911), based on Coulomb scattering between particles of opposed charges. Less than a decade later, Rutherford was the first to deliberately transmute one element into another: bombarding pure nitrogen with alpha particles produced oxygen and something new that he identified as hydrogen nuclei. Hydrogen being known as the lightest element, Rutherford postulated this nucleus to be a new particle, which he dubbed the proton (1920) [6] Compton scattering and de Broglie s suggestion Although classical electromagnetism predicted that the wavelength of scattered rays should be equal to the initial wavelength, multiple experiments had found that it was in fact longer (i.e. corresponding to lower energy). In 1923, Arthur Holly Compton (Nobel Prize in Physics 1927) explained this shift by attributing particle-like momentum to Einstein s photons, by assuming that each photon interacted with only one electron. Conservation of mass-energy and momentum leads to a wavelength shift that depends on the mass of the impacted particle and on the scattering angle. The same year, Louis de Broglie (Nobel Prize in Physics 1929), guided by the analogy of Fermat s principle in optics, was led to suggest that the dual wave-particle nature of radiation should have its counterpart in a dual particle-wave nature of matter. His expression relating the wave vector k and the momentum p of a particle was verified through the observation of electron diffraction by a crystal lattice [7]. p = k (7) Figure 4: Sketch (left, from [8]) and example (right, from [9]) of electron diffraction. 5

6 2.3.3 The Schrödinger equation In view of the consequences and achievements of quantum mechanics, the next steps are surprisingly straightforward. Considering the kinetic and radiative energy of a wave-like particle to be identical E = ω = p2 (8) 2m and using these definitions for a wave function ψ ψ (x, t) = ψ 0 e j(kx ωt) = ψ 0 e j (px Et) (9) directly 2 leads to the well-known Schrödinger equation. E ψ (x, t) = j 2 ψ (x, t) = t 2m 2 ψ (x, t) (10) x2 3 Quantization, spin, and the Stern-Gerlach Experiment According to the Bohr van Leeuwen theorem of statistical mechanics, the thermal average of the magnetization is always zero 3. This makes magnetism in solids solely a quantum mechanical effect that cannot be accounted for by classical physics [10]. Only quantization allows to solve the discrepancy between theory and experimentation. 3.1 Black body radiation In 1900, John William Strutt (3 r d Baron Rayleigh, Nobel Prize in Physics 1904) and James Jeans presented a derivation of the energy density u of a black body, based on the equipartition of energy over all possible oscillators of angular frequency ω at a temperature T. Alas, the Rayleigh-Jeans law is known for predicting the so-called ultraviolet catastrophe: the energy output should diverge towards infinity as wavelength approaches zero. This dilemma was solved the same year by Max Planck (Nobel Prize in Physics 1918), by assuming that the resonators considered using Boltzmann s statistics could only emit energy in quantized form, as a multiple of the elementary unit energy E 0 = ω 0. E n = ne 0 = n ω 0, n N (11) Due to this quantization, calculation of the partition function of the system is done with a summation instead of an integral, which removes the divergence at high frequencies. 3.2 The Bohr atom model Rutherford s planetary model of the atom that we mentioned before faced insurmountable difficulties. Following the Larmor equation of classical electrodynamic, an electron in orbit should continuously radiate energy, and one can estimate that it would take it approximately s to spiral into the nucleus from an orbit of the size of m [7]. Furthermore, an electron in periodic motion around an atom nucleus should radiate with the same frequency as that motion. 2 Of course, we neglect here the fact that the spatial derivative is actually an operator. 3 The intuitive proof is that the rate of work Ẇ done by the Lorentz force F on a particle with charge q and velocity v does not depend on the magnetic field B. Ẇ = F v = q (E + v B) v = qe v 6

7 This was in clear disagreement with experimental results that a) some atoms are stable (!) and b) they emit light only in very narrow bands. Along the same lines as Planck s hypothesis of quantized energy levels of a resonator, Niels Bohr (Nobel Prize in Physics 1922) postulated in 1913 that an atom may only exist in states of discrete energy levels, the difference of which should correspond to the emission spectrum of the atom, thus avoiding the radiation allowed by a continuous energy spectrum [11]. ω = E Applying classical mechanics to this problem leads to a quantization of the angular momentum L, in units of, where n is called the primary quantum number. (12) L = n, n N (13) From this condition we can calculate two widely used constants pertaining to the hydrogen atom: the Bohr-radius a 0, which corresponds to the radius of the lowest orbit of a planetary electron of mass m e (ground state n = 1), and the Rydberg constant Ry, which gives the energy level for the state n. a 0 = 2 m e e 2 = 0, m (14) E n = Ry 1 n 2 (15) Ry = m ee = 13, 606 ev (16) The Bohr atom model was very successful for hydrogen or helium, being able to explain the Rydberg formula for the spectral emission lines of atomic hydrogen. But it was unable to explain more complex effects such the Zeeman effect or the fine structure, which we will discuss later in more detail. 3.3 The Stern-Gerlach experiment In 1922, Otto Stern and Walter Gerlach designed an experiment to test the quantization of angular momentum (also known as Bohr-Sommerfeld quantization) [7] [12]. A beam of electrically neutral silver atoms was directed through an inhomogeneous magnetic field B, with the apparatus described in Fig. 5. Figure 5: Schematic view of the Stern-Gerlach experiment (from [12] and [7]). 7

8 The magnetic moment µ produced by an electric charge q moving along a circular path of radius r depends on its instantaneous velocity v. Consequently, the magnetic moment is a function of the angular momentum L. µ = 1 2 q r v = q 2m L (17) Figure 6: Magnetic moment of a charged particle in orbit, from [8]. Eq. 17 is the base of the definition of the Bohr magneton µ B, which is the magnetic moment of an electron of mass m e at the lowest possible angular momentum L =. µ B = e (18) The interaction of a magnetic moment µ with a magnetic field B gives rise to a change in energy V and thus a forcef on the atom. V = µ B (19) F = V = {µ x B x + µ y B y + µ z B z } (20) With the magnetic field primarily pointing in the z-direction, precession of the magnetic moment about this same axis 4 is sufficiently fast for the average values of µ x and µ y to vanish. Thus the (average) net force points only in the z direction, leading to a deviation of the particle proportional to the angular momentum and the gradient of the magnetic field. F z = µ z B z = q B 2m z L z (21) 4 Known as the Larmor precession, which for a (classical) electron is characterised by a frequency of over 10 GHz/T esla. L = q 2m L B = q q BL ez ωlarmor = 2m 2m B 8

9 The experiment was carried out with a beam of silver atoms from a hot oven impacting on a target. Silver atoms were chosen because all shells are complete up to a single outer electron 5. Thus the sum of the angular momentums of all the inner electrons is zero (Hund s rule for spatial symmetry), meaning that this large atom acts magnetically as a single electron. Furthermore, this single electron has zero orbital angular momentum (5s 1 with orbital quantum number l = 0): one would expect there to be no interaction with an external magnetic field 6 Figure 7: Gerlach s postcard to Niels Bohr, dated February 8 th, Stern and Gerlach s experimental results rose two questions [13] that could not be answered classically: Although this electron has actually zero angular momentum, it behaves as if it possessed a magnetic moment, which could only be the result of some current loop. Instead of the continuous smear that would be produced by all possible orientations of the dipoles, the target revealed that the field separated the beam into two distinct parts, indicating a quantization of this newly discovered magnetic moment of the electron. In 1924, Wolfgang Pauli (Nobel Prize in Physics 1945) proposed a new quantum degree of freedom (or quantum number) with two possible values, in order to resolve inconsistencies between observed molecular spectra and the developing theory of quantum mechanics. He formulated the Pauli exclusion principle, perhaps his most important work, which stated that no two electrons could exist in the same quantum state, identified by four quantum numbers including his new two-valued degree of freedom. The idea of spin originated with Ralph Kronig. In 1925, Samuel Abraham Goudsmit and George Eugene Uhlenbeck postulated that the electron had an intrinsic angular momentum, independent of its orbital characteristics. In classical terms, a ball of charge could have a magnetic moment if it were spinning such that the charge at the edges produced an effective current loop. This kind of reasoning led to the use of the name electron spin to describe the intrinsic angular momentum, as if the electron were a spinning ball of charge. 5 The use of silver also proved to be fortuitous, as the high sulfur content of Stern s bad cigar turned the silver atoms deposited on the target to silver sulfide, which was similar to developing a photographic film [14]. 6 The set of orthogonal systems known as spherical harmonics had been studied by Pierre-Simon de Laplace at the end of the 18 th century, and the existence of the quantum numbers l emerged from a discussion of elliptical orbits based on the Bohr atom theory. Finally, the shell terminology comes from Arnold Sommerfeld (of the Bohr-Sommerfeld quantization), and their existence was confirmed by X-ray diffraction experiments performed in 1913 [15]. 9

10 This picture of the rotation of a particle around some axis is correct so far as spins obey the same mathematical laws as quantized angular momenta do 7. However, this spinning ball picture is not realistic: the electron should be rotating so fast that its radius would move faster than light. The spin is simply another degree of freedom available to this particle. Spins also possess some peculiar properties that distinguish them from orbital angular momenta [16]: The spin quantum numbers may take on half-odd-integer values. Although the direction of its spin can be changed, an elementary particle cannot be made to spin faster or slower. The spin of a charged particle is associated with a magnetic dipole moment µ s with a g-factor differing from 1 (which we will discuss later). This could only occur classically if the internal charge of the particle were distributed differently from its mass. 4 Degeneracy and the Zeeman effect In 1896, Dutch physicist Pieter Zeeman (Nobel Prize in Physics 1902) discovered what we call now the Zeeman effect: the splitting of spectral lines by a strong magnetic field 8. Historically, one distinguishes between the normal and one anomalous Zeeman effects. Both have actually the same cause, but there was no good explanation for the anomalous one until the discovery of the electron spin. Figure 8: One of the applications of the ( normal ) Zeeman effect: the magnetic field of a sunspot is proportional to the splitting of the hydrogen line (from [18]). 7 The total angular momentum J is the vector sum of the orbital angular momentum L and of the intrinsic spin momentum S, using the Clebsch-Gordan coefficients. 8 The explanation came from his compatriot Hendrik Lorentz (with whom he shared the Nobel Prize), who described the polarization of the light emitted by a excited charged particle in the presence of a magnetic field. Still a few years before Thomson s discovery of the electron, Zeeman s work showed that the particles had to be negatively charged and a thousandfold lighter than the hydrogen atom [17]. 10

11 4.1 A brief refresher on the Hamiltonian for a central potential In the three-dimensional case, the time-independent Schrödinger equation for a central potential (from a nucleus with charge Ze and a single electron) in spherical coordinates takes the form 9 2 2m ( 2 r r ) ψ (r) + L2 ψ (r) + V (r) ψ (r) = E ψ (r) (22) r 2mr2 This allows to separate the variables of the corresponding wave function ψ (r) into a radial part R nl (r) and an angular part Y lml (θ, φ) (spherical harmonics). ψ (r) = R nl (r)y lml (θ, φ) (23) Y lml (θ, φ) are eigenfunctions of the square of the operator of angular momentum L 2, and as this operator commutes with the Hamiltonian and with one of its components (usually chosen to be L z ) 10, all three operators H, L 2 and L z have a (complete) set of common eigenstates. These eigenstates are characterised by a set of parameters [19]: The primary quantum number n describes the electron shell of the atom (energy level). The azimuthal quantum number l describes the subshell (named s for l = 0 then p, d, f, g...), giving the magnitude of the angular momentum. The magnetic quantum number m l yields the projection of the angular momentum along a specified axis. The introduction of spin adds a fourth parameter: the spin projection quantum number m s. E n = m ( ) Ze 2 2 2n 2 L 2 = 2 l(l + 1) L z = m l (24) S z = m s The allowed energy states E n are the same as with the Bohr model, but because of the wave function we can map out the probability distribution for the electron without having to rely on classical orbits. Furthermore, while the energy levels are characterised by the primary quantum number n, the mathematical derivation shows that there are really two quantum numbers. n = n r + l + 1 { n N l N (25) Finally, for each eigenvalue l there are actually (2l + 1) different eigenfunctions of quantum number m l, for a total of n 2 levels of degeneracy. l m l l, m l N (26) n 1 (2l + 1) = n 2 (27) l=0 9 Where m = m 1m 2 m 1 +m 2 is actually the reduced mass. 10 Conservation of energy and momentum! 11

12 The underlying mathematics for spin are the same as for the projection of the angular momentum, meaning that an electron with spin s = 1 2 actually doubles the level of degeneracy to 2n 2. s m s s, m s = ± 1 (28) The normal Zeeman effect The Hamiltonian H of an atom in a magnetic field can be given as the sum of the unperturbed Hamiltonian of the atom H 0 and of the perturbation H B due to the magnetic field. H = H 0 + H B (29) As already exposed while discussing the Stern-Gerlach experiment, the perturbation is proportional to the projection of the magnetic moment µ on the magnetic field B. H B = µ B = e L B (30) Due to the separation of variables, only the m l eigenvalues to the z-component of the angular momentum are of interest here. E nlml = nlm l H B nlm l = eb nlm l L z nlm l = e B m l (31) The magnetic field breaks the degeneracy, giving rise to (2l + 1) energy shifts of magnitude E nlml, known as the Zeeman effect. For hydrogen and a magnetic field of 1 Tesla, the shift corresponds to 58 µev. Figure 9: Zeeman splitting of the 6-fold degeneracy of the p-state of the hydrogen atom (from [8]). The angular momentum produces a triplet (m l ±1) with twofold degeneracies due to spin. The value of g S 2 compensates the half-integral spin angular momentum, and the result is a quintuplet with twofold degeneracy of the central state. 12

13 4.3 The g-factor When the Zeeman effect was observed for hydrogen, the observed splitting was consistent with an electron orbit magnetic moment given by [13] µ orbital = e L = µ B L (32) But when the effects of electron spin were discovered by Goudsmit and Uhlenbeck, they found that the observed spectral features were matched by assigning to the electron spin a magnetic moment µ spin = g S e S (33) where g S is the electron spin g-factor. A g-factor is a dimensionless quantity which characterizes the magnetic moment of a particle or nucleus. It is a proportionality constant that relates the observed magnetic moment µ to the appropriate angular momentum, usually in units of the Bohr magneton [20]. Any particle having a magnetic moment has an associated g-factor: protons, neutrons, muons, nuclei. The g-factor has a direct influence on the Larmor precession, and the associated differences between nuclei are the basis for magnetic resonance. More precise experiments showed that the value was slightly greater than 2, and this fact took on added importance when that departure from 2 was predicted by quantum electrodynamics. In fact, the value of g S has been determined to be [8] and is one of the most precisely known values in all physics. 4.4 The anomalous Zeeman effect g S = (15) (34) The normal Zeeman effect only accounts for an odd number of lines. The presence of even number of lines that could be irregularly separated as well was a mystery until the discovery of spin [7]. Because of spin, the level of degeneracy doubles and the new perturbation H B leads to a slightly different expression for the energy shift. H B = e (L + g S S) B (35) The key to this new problem is the rule of vector addition of two quantum-mechanical angular momenta, established by Alfred Landé in Accordingly, we calculate the eigenstates of J 2 and J z (with quantum numbers j and m j, respectively 11 ), where J = L + S is the total angular momentum. For an electron with g S 2 and a magnetic magnetic field B defined along the z-axis, we need to calculate eb jm j l L z + 2S z jm j l eb jm j l J z + S z jm j l (36) 11 These quantum numbers define the same kind of eigenstates as given in Eq. 24 for the angular momentum L. 13

14 Eq. 36 is linear, and the first term is easy to calculate. eb jm j l J z jm j l = e B m j (37) The problem with the second term is that J is a constant of the motion, while S and L are not. As a consequence, both S and L precess around J (see Fig. 10). As with the Stern-Gerlach experiment, this precession is fast enough for us to have to consider only their average values, which are their respective projections onto J. Figure 10: Precession of the angular momentum L and magnetic momentum S around the total angular momentum J (from [7]). S = S J J 2 J = S J 2 j(j + 1) J (38) Using L = J S and squaring on both sides, S J = 1 [ J 2 + S 2 L 2] = 2 [j(j + 1) + s(s + 1) l(l + 1)] (39) 2 2 we can now define S z as a function of J z. S z = J z j(j + 1) + s(s + 1) l(l + 1) 2j(j + 1) (40) Adding Eq. 37 and 40 with ( s = 2) 1 ( and j = l ± s = l ± 1 2) gives the sought for energy shifts. The term in parentheses is known as the Landé g-factor g j. Fig. 11 gives a general representation of the anomalous Zeeman effect. E Z = m j e B ( 1 ± l g j = 2l + 1 2l + 2 2l + 1 ) = m j e B g j (41) 2l 2l + 1 If the external field is strong enough so that the spin-orbit coupling can be neglected, Eq. 41 can be simplified for hydrogen to Eq. 43, which gives the Zeeman splitting shown in Fig. 9. E Z = (42) e B (m l + 2m s ) (43) 14

15 Figure 11: General representation of the anomalous Zeeman effect (from [7]). 4.5 Spin-orbit interaction and the hydrogen fine and hyperfine structures Further interactions are all actually variants of the Zeeman effect: some magnetic moment and charges in relative movement to each other. The mathematical treatments are in essence the same as presented for the anomalous Zeeman effect. In order to keep this paper small, these effects are only discussed in broad terms Spin-orbit interaction The Stern-Gerlach experiment was not the only experimental evidence that suggested an additional property of the electron. The other was the closely spaced splitting of the hydrogen spectral lines, called fine structure, which is caused by the phenomenon called spin-orbit coupling. We have already seen how an external magnetic field affects the Hamiltonian of an electron in a central potential. But because of the intrinsic magnetic momentum known as spin, this spin-orbit coupling acts like a permanent Zeeman effect due to the magnetic field generated by the moving electrons. Ignoring relativistic effects, the magnetic field generated by the electron depends on the electric field E and on the velocity of the particle, i.e. its momentum L, according to the Maxwell equations. Assuming a central potential V (r), this can be expressed as B = v E c 2 = L m e c 2 E r = 1 c 2 v r r dv (r) dr (44) µ = e g S S (45) The interaction with the intrinsic magnetic moment corresponds to a spin-orbit perturbation H SO, given here for a point of charge Ze interacting with a single electron. The reduction by a prefactor of 2 comes from relativistic effects known as Thomas precession [7]. It corresponds to an internal magnetic field on the electron of about 0.4 Tesla [13]. H SO = µ B = 1 [ ] e 2 m 2 ec 2 S L1 dv = Ze2 1 S L r dr 4πɛ 0 2m 2 ec 2 r 3 (46) 15

16 Figure 12: Schematic representation of the hydrogen fine structure due to spin-orbit interaction (from [13]) The hyperfine structure The last effect we will discuss (even if very briefly) is the very tiny hyperfine splitting, which is a permanent Zeeman effect due to the magnetic field generated by the magnetic dipole moment of the nucleus. If the spin operator of the nucleus is denoted by I, then the magnetic dipole operator is M = Z e g N 2M N I (47) where Z is the nuclear charge, M N the nuclear mass, and g N its gyromagnetic ratio (g-factor). The hyperfine structure is a factor m e /M N smaller than the typical spin-orbit coupling. Figure 13: Schematic representation of the hydrogen hyperfine structure due to interaction between the total angular momentum of the electron and the magnetic dipole moment of the nucleus (from [13]). 16

17 References [1] Wikipedia, Status December 2012 [2] Wikipedia, Status December 2012 [3] Wikipedia, Status December 2012 [4] Wikipedia, Status December 2012 [5] Undergraduate physics laboratory of Dartmouth University, ~phys1/labs/lab3.pdf, Status December 2012 [6] Wikipedia, Status December 2012 [7] Stefan Gasiorowicz, Quantum Physics, 3 rd edition, John Wiley & Sons, 2003 [8] Matthias R. Gaberdiel, Quantenmechanik I, Skript zur Vorlesung der ETH Zürich, Wintersemester [9] Wikipedia, Status December 2012 [10] Wikipedia, Status December 2012 [11] Wikipedia, Status December 2012 [12] Wikipedia, Status December 2012 [13] HyperPhysics at the Georgia State University, hbase/hframe.html, Status December 2012 [14] Bretislav Friedrich and Dudley Herschbach, Stern and Gerlach: how a bad cigar helped reorient atomic physics, Physics Today 56, 53, December 2003 [15] Wikipedia, Status December 2012 [16] Wikipedia, Status December 2012 [17] Wikipedia, Status December 2012 [18] Sami K. Solanki, Handout_L1.pdf, Max Planck Institute for Solar System Research, Saas-Fee Advanced Course 39, 2009 [19] Wikipedia, Status December 2012 [20] Wikipedia, Status December