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T.C UNIVERSITY of GAZIANTEP DEPARTMENT OF ENGINEERING OF PHYSICS SIMULATION OF THE STERN GERLACH EXPERIMENT WITH MATHEMATICA (An Interactive Application) GRADUATION PROJECT IN DEPARTMENT OF ENGINEERING PHYSICS ESRA DE By RMENC and ARDA KANDEM R JUNE 2012 SUPERVISOR Assoc. Prof. Dr Okan ÖZER ACKNOWLEDGEMENTS We would like to thank to Assoc. Prof. Dr Okan ÖZER for helping and supporting us throughout the project. It is thanks to his meticulous care and painstaking exactness that the project has taken its present form. Furthermore, he has checked the whole project with remarkable patience, pointing out errors which might have otherwise gone unnoticed. We also thank to Assist. Prof. Dr Ahmet B NGÜL. He has shared with us his previous studies on the problem. We must say a special thank to Deniz BERKYÜREK who is our friend helped us about internet preferences of our program. ABSTRACT Understanding of discrete values of angular momentum, or spin as it is called on the atomic level, are one of the hallmarks of quantum mechanics. And the Stern-Gerlach experiment, used to measure spin, is one of the most-used examples for illustrating ideas in quantum mechanics. ÖZET Elektronun spin hareketi yaptı ının kanıtı olan Stern-Gerlach Deneyinin teorisi üzerine çalı ıldı, Mathematica programı ile simülasyonu yapıldı. Programın internet ortamında çalısması sa landı. TABLE OF CONTENTS CONTENTS Page TABLE OF CONTENTS .......................................................................................... 1 LIST OF FIGURES .................................................................................................. 3 CHAPTER 1: INTRODUCTION .............................................................................. 4 CHAPTER 2: SPIN .................................................................................................... 4 2.1. Description of Spin ............................................................................................ 6 2.2. Spin Quantum Number ...................................................................................... 6 2.3. Classical Spin Angular Momentum .................................................................... 7 2.4. Quantum Spin Angular Momentum ................................................................... 9 2.5. Electron Spin ................................................................................................... 11 2.6. Electron Intrinsic Angular Momentum ............................................................. 12 2.7. Magnetic Moment ........................................................................................... 13 2.8. Electron Spin Magnetic Moment ...................................................................... 14 2.9.Spin Direction ................................................................................................... 15 2.9.1. Spin Projection Quantum Number and Spin Multiplicity ............................... 15 2.9.2. Spin Vector ................................................................................................... 16 2.10. History ........................................................................................................... 17 CHAPTER 3: EXPERIMENTAL PROPERTIES ................................................. 18 3.1. The Stern Gerlach Experiment ............................................................................ 18 3.2. Why Neutral Silver Atom .................................................................................... 23 3.3. Why Inhomogeneous Magnetic Field .................................................................. 24 CHAPTER 4: PHYSICAL PARAMETERS FOR THE SIMULATION ……….. 25 4.1. The Slit ............................................................................................................ 25 4.2. The Magnetic Field .......................................................................................... 26 4.3. Equations of Motion ........................................................................................ 27 4.4. Ag Atoms and Their Velocities ........................................................................ 28 4.5. Quantum Effect ............................................................................................... 30 4.6. Assumptions .................................................................................................... 31 4.7. The Schema of the Experiment ....................................................................... 32 4.8. Results ............................................................................................................. 33 CONCLUSIONS ...................................................................................................... 45 APPENDIX A. CODE OF MATHEMATICA AND PHP CODES OF THE SIMULATION ......................................................................................................... 46 REFERENCES ........................................................................................................ 47 LIST OF FIGURES LIST OF FIGURES Page Figure 1. Spin Orientations....................................................................................... 12 Figure 2. Experimental Set Up ................................................................................. 20 Figure 3. Motion Of Silver Atom ............................................................................. 22 Figure 4. Expected And Observed Distribution ........................................................ 24 Figure 5. Experimental Set Up In Detail ................................................................... 25 Figure 6. Slit ............................................................................................................ 25 Figure 7. Electromagnets .......................................................................................... 26 Figure 8. Maxwell-Boltzman Distribution Function ................................................. 29 Figure 9. Spin Vector ............................................................................................... 30 Figure 10. Experiment ............................................................................................... 32 CHAPTER 1 INTRODUCTION The Stern–Gerlach experiment involves sending a beam of particles through an inhomogeneous magnetic field and observing their deflection. The results show that particles possess an intrinsic angular momentum that is most closely analogous to the angular momentum of a classically spinning object, but that takes only certain quantized values. The experiment is normally conducted using electrically neutral particles or atoms. This avoids the large deflection to the orbit of a charged particle moving through a magnetic field and allows spin-dependent effects to dominate. If the particle is treated as a classical spinning dipole, it will precess in a magnetic field because of the torque that the magnetic field exerts on the dipole. If it moves through a homogeneous magnetic field, the forces exerted on opposite ends of the dipole cancel each other out and the trajectory of the particle is unaffected. However, if the magnetic field is inhomogeneous then the force on one end of the dipole will be slightly greater than the opposing force on the other end, so that there is a net force which deflects the particle's trajectory. If the particles were classical spinning objects, one would expect the distribution of their spin angular momentum vectors to be random and continuous. Each particle would be deflected by a different amount, producing some density distribution on the detector screen. Instead, the particles passing through the Stern–Gerlach apparatus are deflected either up or down by a specific amount. This was a measurement of the quantum observable now known as spin which demonstrated possible outcomes of a measurement where the observable has discrete spectrum. Although some discrete quantum phenomena, such as atomic spectra, were observed much earlier, the Stern–Gerlach experiment allowed scientists to conduct measurements of deliberately superposed quantum states for the first time in the history of science. By now it is known theoretically that angular momentum of any kind has a discrete spectrum, which is sometimes imprecisely expressed as "angular momentum is quantized”. If the experiment is conducted using charged particles like electrons, there will be a Lorentz force that tends to bend the trajectory in a circle. This force can be cancelled by an electric field of appropriate magnitude oriented transverse to the charged particle's path. In summary: The Stern–Gerlach experiment had one of the biggest impacts on modern physics: In the decade that followed, scientists showed using similar techniques, that the nuclei of some atoms also have quantized angular momentum. It is the interaction of this nuclear angular momentum with the spin of the electron that is responsible for the hyperfine structure of the spectroscopic lines. The Stern–Gerlach experiment has become a paradigm of quantum measurement. In particular, it has been assumed to satisfy von Neumann projection. According to more recent insights, based on a quantum mechanical description of the influence of the inhomogeneous magnetic field, this can be true only in an approximate sense. Von Neumann projection can be rigorously satisfied only if the magnetic field is homogeneous. Hence, von Neumann projection is even incompatible with a proper functioning of the Stern–Gerlach device as an instrument for measuring spin. CHAPTER 2 DESCRIPTION OF SPIN In quantum mechanics and particle physics , spin is a fundamental characteristic property of elementary particles, composite particles (hadrons), and atomic nuclei. All elementary particles of a given kind have the same spin quantum number, an important part of the quantum state of a particle. When combined with the spin-statistics theorem, the spin of electrons results in the Pauli exclusion principle, which in turn underlies the periodic table of chemical elements . The spin direction (also called spin for short) of a particle is an important intrinsic degree of freedom. Wolfgang Pauli was the first to propose the concept of spin, but he did not name it. In 1925, Ralph Kronig, George Uhlenbeck, and Samuel Goudsmit suggested a physical interpretation of particles spinning around their own axis. The mathematical theory was worked out in depth by Pauli in 1927. When Paul Dirac derived his relativistic quantum mechanics in 1928, electron spin was an essential part of it. Spin quantum number As the name suggests, spin was originally conceived as the rotation of a particle around some axis. This picture is correct so far as spins obey the same mathematical laws as quantized angular momenta do. On the other hand, spins have some peculiar properties that distinguish them from orbital angular momenta: 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 with a g-factor differing from 1. This could only occur classically if the internal charge of the particle were distributed differently from its mass. Hence the allowed values of are , , , , , etc. The value of for an elementary particle depends only on the type of particle, and cannot be altered in any known way . The spin angular momentum S of any physical system is quantized. The allowed values of S are: (1) where is the Planck constant. In contrast, orbital angular momentum can only take on integer values of , even values of . That is why rather than was defined as the quantum mechanical unit of angular momentum. When spin was discovered it was too late to change. All known matter is ultimately composed of elementary particles called fermions, and all elementary fermions have Classical Spin Angular Momentum A particle moving through space possesses angular momentum, a vector, defined by (2) where r and p are the position vector and momentum respectively of the particle. This is some - times referred to as orbital angular momentum since, in particular, it is an important consideration in describing the properties of a particle orbiting around some centre of attraction such as, in the classical picture of an atom, electrons orbiting around an atomic nucleus. Classically there is no restriction on the magnitude or direction of orbital angular momentum. From a classical perspective, as an electron carries a charge, its orbital motion will result in a tiny current loop which will produce a dipolar magnetic field. The strength of this dipole field is measured by the magnetic moment which is related to the orbital angular momentum by (3) Thus, the expectation on the basis of this classical picture is that atoms can behave as tiny little magnets. The classical idea of spin follows directly from the above considerations. Spin is the angular momentum we associate with a rotating object such as a spinning golf ball, or the spinning Earth. The angular momentum of such a body can be calculated by integrating over the contributions to the angular momentum due to the motion of each of the infinitesimal masses making up the body. The well known result is that the total angular momentum or spin is given by (4) where is the moment of inertia of the body, and is its angular velocity. Spin is a vector which points along the axis of rotation in a direction determined by the right hand rule: curl the fingers of the right hand in the direction of rotation and the thumb points in the direction of . The moment of inertia is determined by the distribution of mass in the rotating body relative to the axis of rotation. If the object were a solid uniform sphere of mass and radius , and rotation were about a diameter of the sphere, then the moment of inertia can be shown to be (5) ! If the sphere possesses an electric charge, then the circulation of the charge around the axis of rotation will constitute a current and hence will give rise to a magnetic moment which can be shown, for a uniformly charged sphere of total charge , to be given by " # (6) The point to be made here is that the spinning object is extended in space, i.e. the spinning sphere example has a non-zero radius. If we try to extend the idea to a point particle by taking the limit of momentum must vanish unless 0 we immediately see that the spin angular is allowed to be infinitely large. If we exclude this last possibility, then classically a point particle can only have a spin angular momentum of zero and so it cannot have a magnetic moment. Thus, from the point-of-view of classical physics, elementary particles such as an electron, which are known to possess spin angular momentum, cannot be viewed as point objects – they must be considered as tiny spinning spheres. But as far as it has been possible to determine by high energy scattering experiments, elementary particles such as the electron behave very much as point particles. Whatever radius they might have, it is certainly very tiny: experiment suggests it is $ % & . Yet they are found to possess spin angular momentum of a magnitude equal (for the electron) to ' ( which requires the surface of the particle to be moving at a speed greater than that of light. This conflict with special relativity makes this classical picture of an elementary particle as a tiny, rapidly rotating sphere obviously untenable. The resolution of this problem can be found within quantum mechanics, though this requires considering the relativistic version of quantum mechanics: the spin of a point particle is identified as a relativistic effect. We shall be making use of what quantum mechanics tells us about particle spin, though we will not be looking at its relativistic underpinnings. On thing we do learn, however, is that spin is not describable in terms of the wave function idea that we have been working with up till now. Quantum Spin Angular Momentum Wave mechanics and the wave function describe the properties of a particle moving through space, giving, as we have seen, information on its position, momentum, energy. In addition it also provides, via the quantum mechanical version of (7) a quantum description of the orbital angular momentum of a particle, such as that associated with an electron moving in an orbit around an atomic nucleus. The general results found are that the magnitude of the angular momentum is limited to the values )*+ + ,( + - - -.- / (8) which can be looked on as an ‘improved’ version of the result used by Bohr, the one subsequently ‘justified’ by the de Broglie hypothesis, that is (. The quantum theory of orbital angular momentum also tells us that any one vector component of , 0 1( 0 say, is restricted to the values 1 %+- %+ - %+ -/ -+ % -+ % -+ (9) 0 This restriction on the possible values of mean that the angular momentum vector can have only certain orientations in space – a result known as ‘space quantization’. All this is built around the quantum mechanical version of , and so implicitly is concerned with the angular momentum of a particle moving through space. But a more general perspective yields some surprises. If special relativity and quantum mechanics are combined, it is found that even if a particle, a point object, has zero momentum, so that the orbital angular momentum is zero, its total angular momentum is, in general, not zero. The only interpretation that can be offered is that this angular momentum is due to the intrinsic spin of the particle. The possible values for the magnitude S of the spin angular momentum turn out to be )* . - - - - -/ ,( (10) and any one vector component of S, S z say, is restricted to the values 1( 0 2 % -% -% -/ - % - % - (11) i.e. similar to orbital angular momentum, but with the significant difference of the appearance of half integer values for the spin quantum number s in addition to the integer values. This theoretical result is confirmed by experiment. In nature there exist elementary particles for which . ! - - -/ (12) such as the electron, proton, neutron, quark (all of which have spin ), and more exotic particles of higher half-integer spin, while there exist many particles with integer spin, the photon, for which , being the most well known example, though because it is a zero rest mass particle, it turns out that particular interest here is the case of 0, that is 0 3 (. 0 can only have the values 3 . Of for which there are two possible values for Particle spin is what is left after the contribution to the angular momentum due to motion through space has been removed. It is angular momentum associated with the internal degrees of freedom of a point particle, whatever they may be, and cannot be described mathematically in terms of a wave function. It also has no classical analogue: we have already seen that a point particle cannot have spin angular momentum. Thus, particle spin is a truly quantum property that cannot be described in the language of wave functions – a more general mathematical language is required. It was in fact the discovery of particle spin, in particular the spin of the electron, that lead to the development of a more general version of quantum mechanics than that implied by wave mechanics. There is one classical property of angular momentum that does carry over to quantum mechanics. If the particle is charged, and if it possesses either orbital or spin angular momentum, then there arises a dipole magnetic field. In the case of the electron, the dipole moment is found to be given by 5 %4 78 2 where 6 (13) 6 and %5 are the mass and charge of the electron, is the spin angular momentum of the electron, and g is the so-called gyromagnetic ratio, which classically is exactly equal to one, but is known (both from measurement and as derived from relativistic quantum mechanics) to be approximately equal to two for an electron. It is the fact that electrons possess a magnetic moment that has made it possible to perform experiments involving the spin of electrons, in a way that reveals the intrinsically quantum properties of spin. Electron Spin Two types of experimental evidence which arose in the 1920s suggested an additional property of the electron. One was the closely spaced splitting of the hydrogen spectral lines, called fine structure. The other was the Stern-Gerlach experiment which showed in 1922 that a beam of silver atoms directed through an inhomogeneous magnetic field would be forced into two beams. Both of these experimental situations were consistent with the possession of an intrinsic angular momentum and a magnetic moment by individual electrons. Classically this could occur if the electron were a spinning ball of charge, and this property was called electron spin. Quantization of angular momentum had already arisen for orbital angular momentum, and if this electron spin behaved the same way, an angular momentum quantum number was required to give just two states. This intrinsic electron property gives: 9 % :; ; 5 < ;= 8>+ ; 5 <> A 8 5<B: ; 5 <C 2 ? % 2( 0 5 8 - 2 @ (14) (15) Figure1. Spin Orientations Electron Intrinsic Angular Momentum Experimental evidence like the hydrogen fine structure and the Stern Stern-Gerlach experiment suggest that an electron has an intrinsic angular momentum, independent of its orbital angular momentum. These experiments suggest just two possible states for thiss angular momentum, and following the pattern of quantized angular momentum, this requires an angular momentum quantum number of . With this evidence, we say that the electron has spin . An angular momentum and a magnetic moment could indeed arise from a spinning sphere of charge, but this classical picture cannot fit the size or quantized nature of the electron spin. The property called electron spin must be considered to be a quantum concept without detailed classical analogy. The quantum numbers associated with electron spin follow the characteristic pattern: ( - - 2 @ (16) The properties of electron spin were first explained by Dirac (1928), by combining quantum mechanics with theory of relativity. An electron spin is an intrinsic property of electrons. Electrons have intrinsic angular momentum characterized by quantum number . In the pattern of other quantized angular momenta, this gives total angular momentum (D 4 '. 7 ( (17) The resulting fine structure which is observed corresponds to two possibilities for the zcomponent of the angular momentum. 0 @ ( (18) This causes an energy splitting because of the magnetic moment of the electron 2 % 5 8 (19) Magnetic Moment Particles with spin can possess a magnetic dipole moment, just like a rotating electrically charged body in classical electrodynamics. These magnetic moments can be experimentally observed in several ways, e.g. by the deflection of particles by inhomogeneous magnetic fields in a Stern–Gerlach experiment or by measuring the magnetic fields generated by the particles themselves. The intrinsic magnetic moment angular momentum , is of a Spin % 82 particle with charge q, mass m, and spin (20) where the dimensionless quantity 82 is called the spin g-factor. For exclusively orbital rotations it would be (assuming that the mass and the charge occupy spheres of equal radius). The electron, being a charged elementary particle, possesses a nonzero magnetic moment. One of the triumphs of the theory of quantum electrodynamics is its accurate prediction of the electron g-factor, which has been experimentally determined to have the value % E . F. .G , with the digits in parentheses denoting measurement uncertainty in the last two digits at one standard deviation. The value of arises from the Dirac equation, a fundamental equation connecting the electron's spin with its electromagnetic properties, and the correction of E . F. / arises from the electron's interaction with the surrounding electromagnetic field, including its own field. Composite particles also possess magnetic moments associated with their spin. In particular, the neutron possesses a non-zero magnetic moment despite being electrically neutral. This fact was an early indication that the neutron is not an elementary particle. In fact, it is made up of quarks, which are electrically charged particles. The magnetic moment of the neutron comes from the spins of the individual quarks and their orbital motions. Electron Spin Magnetic Moment Since the electron displays an intrinsic angular momentum, one might expect a magnetic moment which follows the form of that for an electron orbit. The H %component of magnetic moment associated with the electron spin would then be expected to be 3 0 (21) I but the measured value turns out to be about twice that. The measured value is written 3 8 0 (22) I where 8 is called the gyromagnetic ratio and the electron spin 8 % factor has the value 8 E . and 8 for orbital angular momentum. The precise value of 8 was predicted by relativistic quantum mechanics in the Dirac equation and was measured in the Lamb shift experiment. A natural constant which arises in the treatment of magnetic effects is called the Bohr magneton. The magnetic moment is usually expressed as a multiple of the Bohr magneton. I 5( 6 FE & ! J K ML !E&NN.N G JO 5PML (23) The electron spin magnetic moment is important in the spin-orbit interaction which splits atomic energy levels and gives rise to fine structure in the spectra of atoms. The electron spin magnetic moment is also a factor in the interaction of atoms with external magnetic fields (Zeeman effect). The term "electron spin" is not to be taken literally in the classical sense as a description of the origin of the magnetic moment described above. To be sure, a spinning sphere of charge can produce a magnetic moment, but the magnitude of the magnetic moment obtained above cannot be reasonably modeled by considering the electron as a spinning sphere. High energy scattering from electrons shows no "size" of the electron down to a J resolution of about VB QRST U, and at that size a preposterously high spin rate of some M would be required to match the observed angular momentum. Spin Direction Spin projection quantum number and spin multiplicity In classical mechanics, the angular momentum of a particle possesses not only a magnitude (how fast the body is rotating), but also a direction (either up or down on the axis of rotation of the particle). Quantum mechanical spin also contains information about direction, but in a more subtle form. Quantum mechanics states that the component of angular momentum measured along any direction can only take on the values W where W ( W W X Y% - % % -/- % - Z is the spin component along the B-axis (either , [, or H), (24) W is the spin projection quantum number along the B-axis, and is the principal spin quantum number (discussed in the previous section). Conventionally the direction chosen is the z-axis: 0 where 0 ( 0 0 X Y% - % % -/- % - Z is the spin component along the H-axis, number along the H-axis. 0 (25) is the spin projection quantum One can see that there are 0 possible values of . The number " " is the multiplicity of the spin system. For example, there are only two possible values for a spin- particle: 0 and which the spin is pointing in the 0 % H or %H directions respectively, and are often referred to as "spin up" and "spin down". For a spinvalues are - -% -% . These correspond to quantum states in E particle, like a delta baryon, the possible Spin vector For a given quantum state, one could think of a spin vector \ ] whose components are the expectation values of the spin components along each axis, i.e., \ ] ^\ _ ]- \ ` ]- \ 0 ]a. This vector then would describe the "direction" in which the spin is pointing, corresponding to the classical concept of the axis of rotation. It turns out that the spin vector is not very useful in actual quantum mechanical calculations, because it cannot be measured directly — _, `, and 0 cannot possess simultaneous definite values, because of a quantum uncertainty relation between them. However, for statistically large collections of particles that have been placed in the same pure quantum state, such as through the use of a Stern-Gerlach apparatus, the spin vector does have a well-defined experimental meaning: It specifies the direction in ordinary space in which a subsequent detector must be oriented in order to achieve the maximum possible probability (100%) of detecting every particle in the collection. For spinparticles, this maximum probability drops off smoothly as the angle between the spin vector and the detector increases, until at an angle of 180 degrees—that is, for detectors oriented in the opposite direction to the spin vector—the expectation of detecting particles from the collection reaches a minimum of 0%. As a qualitative concept, the spin vector is often handy because it is easy to picture classically. For instance, quantum mechanical spin can exhibit phenomena analogous to classical gyroscopic effects. For example, one can exert a kind of "torque" on an electron by putting it in a magnetic field (the field acts upon the electron's intrinsic magnetic dipole moment—see the following section). The result is that the spin vector undergoes precession, just like a classical gyroscope. This phenomenon is used in nuclear magnetic resonance sensing. Mathematically, quantum mechanical spin is not described by vectors as in classical angular momentum, but by objects known as spinors. There are subtle differences between the behavior of spinors and vectors under coordinate rotations. For example, rotating a spin- particle by .G degrees does not bring it back to the same quantum state, but to the state with the opposite quantum phase; this is detectable, in principle, with interference experiments. To return the particle to its exact original state, one needs a& degree rotation. A spin-zero particle can only have a single quantum state, even after torque is applied. Rotating a spin- same quantum state and a spin- particle N degrees can bring it back to the particle should be rotated F degrees to bring it back to the same quantum state. The spin particle can be analogous to a straight stick that looks the same even after it is rotated N degrees and a spin particle can be imagined as sphere which looks the same after whatever angle it is turned through. History Spin was first discovered in the context of the emission spectrum of alkali metals. In 1924 Wolfgang Pauli introduced what he called a "two-valued quantum degree of freedom" associated with the electron in the outermost shell. This allowed him to formulate the Pauli exclusion principle, stating that no two electrons can share the same quantum state at the same time. Pauli's theory of spin was non-relativistic. However, in 1928, Paul Dirac published the Dirac equation, which described the relativistic electron. In the Dirac equation, a four-component spinor (known as a "Dirac spinor") was used for the electron wavefunction. In 1940, Pauli proved the spin-statistics theorem, which states that fermions have half-integer spin and bosons integer spin. In retrospect, the first direct experimental evidence of the electron spin was the SternGerlach experiment of 1922. However, the correct explanation of this experiment was only given in 1922. CHAPTER 3 THE STERN-GERLACH EXPERIMENT This experiment, first performed in 1922, has long been considered as the quintessential experiment that illustrates the fact that the electron possesses intrinsic angular momentum, i.e. spin. It is actually the case that the original experiment had nothing to do with the discovery that the electron possessed spin: the first proposal concerning the spin of the electron, made in 1925 by Uhlenbach and Goudsmit, was based on the analysis of atomic spectra. What the experiment was intended to test was ‘space-quantization’ associated with the orbital angular momentum of atomic electrons. The prediction, already made by the ‘old’ quantum theory that developed out of Bohr’s work, was that the spatial components of angular momentum could only take discrete values, so that the direction of the angular momentum vector was restricted to only a limited number of possibilities, and this could be tested by making use of the fact that an orbiting electron will give rise to a magnetic moment proportional to the orbital angular momentum of the electron. So, by measuring the magnetic moment of an atom, it should be possible to determine whether or not space quantization existed. In fact, the results of the experiment were in agreement with the then existing (incorrect) quantum theory – the existence of electron spin was not at that time suspected. Later, it was realized that the interpretation of the results of the experiment were incorrect, and that what was seen in the experiment was direct evidence that electrons possess spin. It is in this way that the Stern-Gerlach experiment has subsequently been used, i.e. to illustrate the fact that electrons have spin. But it is also valuable in another way. The simplicity of the results of the experiment (only two possible outcomes), and the fact that the experiment produces results that are directly evidence of the laws of quantum mechanics in action makes it an ideal means by which the essential features of quantum mechanics can be seen and, perhaps, ‘understood’. Otto Stern and Walter Gerlach performed an experiment which showed the quantization of electron spin into two orientations. This made a major contribution to the development of the quantum theory of the atom. The actual experiment was carried out with a beam of silver atoms from a hot oven because they could be readily detected using a photographic emulsion. The silver atoms allowed Stern and Gerlach to study the magnetic properties of a single electron because these atoms have a single outer electron which moves in the Coulomb potential caused by the 47 protons of the nucleus shielded by the 46 inner electrons. Since this electron has zero orbital angular momentum (orbital quantum number + ), one would expect there to be no interaction with an external magnetic field. Stern and Gerlach directed the beam of silver atoms into a region of non-uniform magnetic field (see experiment sketch). A magnetic dipole moment will experience a force proportional to the field gradient since the two "poles" will be subject to different fields. Classically one would expect all possible orientations of the dipoles so that a continuous smear would be produced on the photographic plate, but they found that the field separated the beam into two distinct parts, indicating just two possible orientations of the magnetic moment of the electron. But how does the electron obtain a magnetic moment if it has zero angular momentum and therefore produces no "current loop" to produce a magnetic moment? In 1925, Samuel A. Goudsmit and George E. 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 "electron spin" to describe the intrinsic angular momentum. Figure 2. Experimental Set Up This experiment confirmed the quantization of electron ron spin into two orientations. This made a major contribution to the development of the quantum theory of the atom. The potential energy of the electron spin magnetic moment in a magnetic field applied in the z direction is given by b % Ed I 8 d0 3 where 8 is the electron spin 8 - factor and relationship of force to potential energy gives c0 % eb eH 3 I ed0 eH I d0 I (26) is the Bohr magneton. Using the (27) The deflection can be shown to be proportional to the spin and to the magnitude of the magnetic field gradient. The original experimental arrangement took the form of a collimated beam of silver atoms heading in, say, the [ direction and passing through a non-uniform magnetic field directed (mostly)in the H direction. Assuming the silver atoms posses a non-zero magnetic moment µ,the magnetic field will have two effect. First the magnetic field will exert a torque on the magnetic dipole so that the magnetic dipole vector will precess about the direction of the magnetic field. This will not affect the z component of µ,but the x and y components will change with time. Secondly, and more importantly here, the non-uniformity of the field means that the atoms experience a sideways force given as given before; c0 where b field. % Ed % eb eH 3 I ed0 eH (28) % 0 d is the potential energy of the silver atom in the magnetic Different orientations of the magnetic moment vector values of 0 will lead to different , which in turn will mean that there will be forces acting on the atoms which will differ depending on the value of 0 . The expectation based on classical physics is that due to random thermal effects in the oven, the magnetic dipole moment vectors of the atoms will be randomly oriented in space, so there should be a continuous spread in the H component of the magnetic moments of the silver atoms as they emerge from the oven, ranging from %f 0 f to f 0 f. A line should then appear on the observation screen along the H direction. Instead, what was found was that the silver atoms arrived on the screen at only two points that corresponded to magnetic moments of 0 where I 3 I I 5( 6 (29) is known as the Bohr magneton. Space quantization was clearly confirmed by this experiment, but the full significance of their results was not realized until some time later, after the proposal by Uhlenbach and Goudsmit that the electron possessed intrinsic spin, and a magnetic moment. The full explanation based on what is now known about the structure of the silver atom is as follows. There are & electrons surrounding the silver atom nucleus, of which 47 form a closed inner core of total angular momentum zero – there is no orbital angular momentum, and the electrons with opposite spins pair off, so the total angular momentum is zero, and hence there is no magnetic moment due to the core. The one remaining electron also has zero orbital angular momentum, so the sole source of any magnetic moment is that due to the intrinsic spin of the electron. Thus, the experiment represents a direct measurement of one component of the spin of the electron, this component being determined by the direction of the magnetic field, here taken to be in the H direction. There are two possible values for 0, corresponding to the two spots on the observation screen, as required by the fact that for electrons, i.e. they are spincomponent of spin are 0 the two values for . particles. The allowed values for the H 3 ( which, with the gyromagnetic value of two, yields 0 Of course there is nothing special about the direction H, i.e. there is nothing to distinguish the H direction from any other direction in space. What this means is that any component of the spin of an electron will have only two values, i.e. 3 ( _ ` 3 ( (30) Figure 3. Motion Of Silver Atom H < c g i h 3 ed0 jk eH I (31) The actual experiment was carried out with a beam of silver atoms from a hot oven because they could be readily detected using a photographic emulsion. The silver atoms allowed Stern and Gerlach to study the magnetic properties of a single electron because these atoms have a single outer electron which moves in the Coulomb potential caused by the & protons of the nucleus shielded by the & inner electrons. Since this electron has zero orbital angular momentum (orbital quantum number + ), one would expect there to be no interaction with an external magnetic field as mentioned before. So we can conclude that the answer of the why neutral silver atom is choosen for this experiment like that; Why Neutral Silver atom? • No Lorentz force (l of the atom is zero. • mn o p) acts on a neutral atom, since the total charge (q) Only the magnetic moment of the atom interacts with the external magnetic field. • Electronic configuration: q . . q .V r q V r! So, a neutral Ag atom has zero total orbital momentum. • Therefore, if the electron at 5s orbital has a magnetic moment, one can measure it. Stern and Gerlach directed the beam of silver atoms into a region of non-uniform magnetic field. A magnetic dipole moment will experience a force proportional to the field gradient since the two "poles" will be subject to different fields. Classically one would expect all possible orientations of the dipoles so that a continuous smear would be produced on the photographic plate, but they found that the field separated the beam into two distinct parts, indicating just two possible orientations of the magnetic moment of the electron. But how does the electron obtain a magnetic moment if it has zero angular momentum and therefore produces no "current loop" to produce a magnetic moment? In 1925, Samuel A. Goudsmit and George E. 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 "electron spin" to describe the intrinsic angular momentum. Here is the shortly answer of the why inhomogeneous magnetic field is used for this experiment. Why inhomogeneous magnetic Field? • In a homogeneous field, each magnetic moment experience only a torque and no deflecting force. • An inhomogeneous field produces a deflecting force on any magnetic moments that are present in the beam. In the experiment, they saw a deflection on the photographic plate. Since atom has zero total magnetic moment, the magnetic interaction producing the deflection should come from another type of magnetic field. That is to say: electron’s (at ! orbital) acted like a bar magnet. If the electrons were like ordinary magnets with random orientations, they would show a continues distribution of pats. The photographic plate in the SGE would have shown a continues distribution of impact positions. However, in the experiment, it was found that the beam pattern on the photographic plate had split into two distinct parts. Atoms were deflected either up or down by a constant amount, in roughly equal numbers. Figure 4. Expected And Observed Distribution Apparently, H component of the electron’s spin is quantized. CHAPTER 4 PHYSICAL PARAMETERS FOR THE SIMULATION Figure 5. Experimental Set Up In Detail The Slit The detector is a hot, straight platinum wire extending a short distance in the 3 direction about ,H . The beam, defined by a pair of parallel slits, also extends a few mm in the 3 direction. Figure 6. Slit Initial position ( r - - [r ), of each atom is selected randomly from a uniform distribution. That means: the values of x0 and z0 are populated randomly in the range of stu v_ - 9u v_ w, and at that point, each atom has the velocity The choosen tu v_ and 9u v_ values for our simulation are : tu v_ 9u v_ E - h- . E ! ! The Magnetic Field In the simulation, for the field gradient ed MeH along H axis, we assumed the following 3-case: • uniform magnetic field • constant gradient • : ed0 MeH : ed0 MeH LM field gradient is modulated by a Gaussian : ed0 MeH Figure 7. Electromagnets Rxy %z We also assumed that along beam axis: ed_ MeH (32) ed_ Me { (33) ed0 Me (34) d` (35) Equations of Motion Potential Energy of an electron: b % 2 Ed Components of the force: H % eb eH c` c_ 2_ % % eb e[ ed0 eH % 2_ d_ % 2` d` B :5 ed_ Me { eb e 2_ 20 ed_ e ed0 eH Consequently we have, 20 20 % c0 20 ed0 eH (36) V ed0 Me ed0 { e * B :5 d` ed0 eH (37) , B :5 ed0 MeH c_ { c` 20 d0 (38) (39) (40) 2 |}~• ed0 eH (41) (42) Differential equations and their solutions: _ since hr_ V V< c_ r €• { hr_ < (43) (44) r (45) and y component of the acceleration is; ` since hr` [ c` V [ V< €• [r h and [r [ (46) h` < (47) h< (48) and finally the H component of the acceleration is; 0 H since hr0 V H V< c0 20 €• Hr hr0 < H Hr ed0 MeH €• 0< (50) 0< (51) So the final positions on the photographic plate in terms of h, H Here r Hr r [ 0 4 7 h (49) ‚ and Hr are the initial positions at [ and ‚: (52) (53) ‚D 0 h (54) . Ag Atoms and Their Velocities We use Maxwell-Boltzman velocity distribution function to decide the probable velocity of silver atoms. Initial velocity v of each atom is selected randomly from the Maxwell-Boltzman distribution function: cu ƒ ' „ " zL # around peak value of the velocity: • • • Components of the velocity at h`r h , and h_r h 5 h † zL h0r . r- EN (55) (56) - Hr are assumed to be: Temperature of the oven is chosen as L Mass of an Ag atom is …% zL M h‡ Note that: M J O z8. j. Figure 8. Maxwell-Boltzman Distribution Function The Monte-Carlo Simulation is also used to get the most probable velocity values for silver atoms and we can see from the graph that the most probable velocity values are near the most probable velocity value, P‡ . Quantum Effect Spin vector components: ˆ _- `- 0 In spherical coordinates: _ f f ~U‰ • |}~ Š (57) ` f f ~U‰ • ~U‰ Š (58) f f |}~• (59) 0 where the magnitude of the spin vector is: f f '. ( (60) Figure 9. Spin Vector Angle Š can be selected as: Š where ‹ is random number in the range ‹ - E (61) However, angle • can be selected as follows: if 0 is not quantized, |}~• will have uniform random values: |}~• ‹% (62) 0 else if is quantized, :; • will have only two random values: |}~• 3 (M 0 3 '.(M (63) '. Assumptions We have to make assumptions to get good results for Stern-Gerlach Experiment. Here is the geometric assumptions used in experiment; • • : tu v_ and ‚ !: : and 9u v_ E! : There are some physical assumptions ; • • „ - or „ - Ag atoms are selected. Velocity (h) of the Ag atoms is selected from Maxwell–Boltzman distribution function around peak velocity. • The • Field gradient along H axis is assumed to be: of the Ag (For the silver atom: Melting point L ed0 MeH ed0 MeH ed0 MeH • temperature is takes as .! j ; Boiling point L .! j) j. for uniform magnetic field LM constant field gradient along z axis Rxy %z 9- component of spin is source field gradient is modulated by Gaussian function. 0: - either quantized according to quantum theory such that |}~• ' - or |}~• is not quantized and assumed that it has random orientation. The Scheme of the experiment Figure 10. Experiment Results „ Vd MVH Not Quantized „ Vd MVH Quantized „ Vd MVH Not Quantized „ Vd MVH Quantized „ Vd MVH :; < Not Quantized < Œ „ Vd MVH Quantized :; < < Œ „ Vd MVH :; < Not Quantized < Œ „ Vd MVH Quantized :; < < Œ „ Vd MVH :; < Not Quantized <E 5 %z „ Vd MVH Quantized :; < <E 5 %z „ Vd MVH :; < Not Quantized <E 5 %z „ Vd MVH Quantized :; < <E 5 %z CONCLUSION During EP 499 Graduation Project researches, we had a chance to learn about the meaning of spin, electron’s spin and effects of quantization to the motion of electron in magnetic field. We specially worked on the theory part of The Stern-Gerlach Experiment. We tried to understand the working principles of it and then we worked on simulation of the experiment using Mathematica Program (with my partner in this Project). We learned lots of codes to work it. Also we learned about simulation properties of Mathematica program. Then we thought that it could work on internet media. At the last step of our project, we focused on this idea and we successed it using “html”, “jscript” and “php” codes. APPENDIX A. CODE OF MATHEMATICA AND PHP CODES OF THE SIMULATION Please, get contact with Assoc. Prof. Dr Okan OZER: [email protected] PLEASE VISIT: http://www1.gantep.edu.tr/~ozer/projects/EsraAndArda/TheSternGerlachExperiment.php REFERENCES [1] R. A. Serway and J. W. Jewett, Physics for Scientists and Engineers with Modern Physics, International Edition (Brooks/Cole, 2007). [2] J. Basdevant and J. Dalibard, Quantum Mechanics, 1st Ed. (Springer, 2005). [3] S. Gasiorowicz, Quantum Physics, 3rd Ed. (Wiley, 2003). [4] A. Beiser, Concepts of Modern Physics, 6th Ed. (McGraw-Hill, 2002). [5] R. Shankar, Principles of Quantum Mechanics, 2nd Ed. (Springer 1994).