T.C UNIVERSITY of GAZIANTEP DEPARTMENT OF …€¦ · · 2012-06-22The Stern–Gerlach experiment involves sending a beam of particles through an ... This avoids the large deflection

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

By ESRA DE��RMENC� and ARDA KANDEM�R

JUNE 2012

SUPERVISOR

Assoc. Prof. Dr Okan ÖZER

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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.

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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.

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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.

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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

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CONCLUSIONS ...................................................................................................... 45

APPENDIX A. CODE OF MATHEMATICA AND PHP CODES OF THE

SIMULATION ......................................................................................................... 46

REFERENCES ........................................................................................................ 47

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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

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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.

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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.

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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.

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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:

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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

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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

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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 � 0 we immediately see that the spin angular

momentum must vanish unless � 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

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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 say, is restricted to the values

�0 �1(��1 %+- %+ � �-%+ � �-/�- + % �- + % �- + (9)

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This restriction on the possible values of �0 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

0 �1(��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 0 can only have the values 3�. Of

particular interest here is the case of � �� for which there are two possible values for

0, that is 0 3 ��( .

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

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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

�2 %4 5��67 8 (13)

where �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�<>� ? 0 �2(�-��2 @�� (14)

A 8�5<B:��;�5�<C��2 % 5��8 (15)

Electron Intrinsic Angular Momentum

Experimental evidence like the

experiment suggest that an electron has an intrinsic

its orbital angular momentum. These experiments suggest just two possible states for

this angular momentum, and following the pattern of

requires an angular momentum quantum number of

With this evidence, we say that the electron has

a magnetic moment could indeed arise

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

characteristic pattern:

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The properties of electron spin were first explained by Dirac (1928), by combining

quantum mechanics with theory of relativity.

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Figure1. Spin Orientations

Electron Intrinsic Angular Momentum

Experimental evidence like the hydrogen fine structure and the Stern

suggest that an electron has an intrinsic angular momentum, independent of

orbital angular momentum. These experiments suggest just two possible states for

s 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

could indeed arise from a spinning sphere of charge, but this



classical analogy. The quantum numbers associated with electron spin follow the

� ��(��-�� -��2 @��


quantum mechanics with theory of relativity.

Stern-Gerlach

angular momentum, independent of

orbital angular momentum. These experiments suggest just two possible states for

quantized angular momentum, this

An angular momentum and

from a spinning sphere of charge, but this



associated with electron spin follow the

(16)


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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 z-

component 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 � of a Spin % �� particle with charge q, mass m, and spin

angular momentum , is

� 82�� (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

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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

�0 3��I (21)

but the measured value turns out to be about twice that. The measured value is written

�0 3��8�I (22)

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 LM !E&NN.N�G��JO 5P LM (23)

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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

resolution of about ��J��QRSTU , and at that size a preposterously high spin rate of some

�� VB � �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 (�W ��W X Y%�- %�� % ��- / - � % �- �Z (24)

where W is the spin component along the B-axis (either �, [, or H), �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 (�0��0 X Y%�-%�� % ��- / - � % �- �Z (25)

where 0 is the spin component along the H-axis, �0 is the spin projection quantum

number along the H-axis.

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One can see that there are �� possible values of �0 . The number "�� " is

the multiplicity of the spin system. For example, there are only two possible values for

a spin- �� particle: �0 � �� and �0 % �� . These correspond to quantum states in

which the spin is pointing in the �H or %H directions respectively, and are often referred

to as "spin up" and "spin down". For a spin- �� particle, like a delta baryon, the possible

values are � �� - � �� - % �� - % ��E

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 spin- ��

particles, 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.

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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- � particle �N� degrees can bring it back to the

same quantum state and a spin- � 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 wave-

function. 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 Stern-

Gerlach experiment of 1922. However, the correct explanation of this experiment was

only given in 1922.

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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

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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.

This experiment confirmed the quantization of

orientations. This made a major

of the atom.

The potential energy of the

applied in the z direction is given by

b %��

where 8 is the electron spin

relationship of force to potential energy

c0

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

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Figure 2. Experimental Set Up

This experiment confirmed the quantization of electron spin

This made a major contribution to the development of the quantum theory

of the electron spin magnetic moment in a magnetic field

applied in the z direction is given by

�E d �I 8� d0 3�Id0

electron spin 8 - factor and �I is the Bohr magneton.

force to potential energy gives

%ebeH 3�I ed0eH


of the magnetic field gradient.

original experimental arrangement took the form of a collimated

ron spin into two

contribution to the development of the quantum theory

in a magnetic field

(26)

Using the

(27)


original experimental arrangement took the form of a collimated beam of

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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 %ebeH 3�I ed0eH (28)

where b %��E d %�0d is the potential energy of the silver atom in the magnetic

field.

Different orientations of the magnetic moment vector � will lead to different

values of �0 , 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�0f to f�0f. 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 3�I��I 5(��6 (29)

where �I 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

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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 spin- �� particles. The allowed values for the H

component of spin are 0 3 ��( which, with the gyromagnetic value of two, yields

the two values for �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�hi� 3�I��jk ed0eH

(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;

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Why Neutral Silver atom?

• No Lorentz force (l mn�o�p) acts on a neutral atom, since the total charge (q)

of the atom is zero.

• 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.

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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.

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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

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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 ��- h- ��. The choosen tu v_ and 9u v_ values for our simulation are :

tu v_ �E�!�

9u v_ �E��!�

The Magnetic Field

In the simulation, for the field gradient �ed eHM � along H axis, we assumed the

following 3-case:

• uniform magnetic field : ed0 eHM �

• constant gradient :�ed0 eHM �� L �M

• field gradient is modulated by a Gaussian : ed0 eHM ��Rxy��%z��

Figure 7. Electromagnets

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We also assumed that along beam axis:

ed_ eHM � (32)

ed_ e�M { � (33)

ed0 e�M � (34)

d` � (35)

Equations of Motion

Potential Energy of an electron:

b %�2�E d %�2_d_ % �2`d` % �20d0 (36)

Components of the force:

��B�:5� ed_ e�M { �� V� ed0 e�M ��

c_ %ebe� �2_ ed_e� � �20 ed0e� { � (37)

c̀ %ebe[ ��*�B�:5��d` �, (38)

H %ebeH �2_ ed0eH � �20 ed0eH �20 ed0eH ��B�:5� ed0 eHM �� (39)

Consequently we have,

c_ { � (40)

c̀ � (41)

c0 �20 ed0eH �2 |}~� ed0eH (42)

Differential equations and their solutions:

_ V��V<� c_�� { � (43)

� �r � hr_< (44)

since hr_ � ��

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� �r (45)

and y component of the acceleration is;

` V�[V<� c̀�� (46)

[ [r � h`< (47)

since hr` h and [r �

[ h< (48)

and finally the H component of the acceleration is;

0 V�HV<� c0�� 20 ed0 eHM�� (49)

H Hr � hr0�< � �� 0<� (50)

since hr0 �

H Hr � �� 0<� (51)

So the final positions on the photographic plate in terms of h, � and �:

� �r (52)

[ � � � (53)

H Hr � �� 0 4�h7� � �D� 0�h

(54)

Here �r and Hr are the initial positions at [� ��.

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

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Maxwell-Boltzman distribution function:

cu � '�� " ��zL#� �M h�5�� %�h��zL�

(55)

around peak value of the velocity:

h� �zL �M (56)

Note that:

• Components of the velocity at ��r- �- Hr� are assumed to be:

h`r � �h , and h_r � �h0r � ��.

• Temperature of the oven is chosen as L� ��j.

• Mass of an Ag atom is � �EN��J�O�z8.

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�.

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Quantum Effect

Spin vector components: �� _- ` - 0�

In spherical coordinates:

��_ � � ff�~U��|}~��

(57)

��` � � ff ~U��~U��

(58)

��0 � ff�|}~��

(59)

where the magnitude of the spin vector is:

ff '.� ( (60)

Figure 9. Spin Vector

Angle � can be selected as:

� �� (61)

where � is random number in the range ��-��E However, angle � can be selected as follows:

if 0 is not quantized, |}~� will have uniform random values:

|}~� �� % �� (62)

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else if 0 is quantized, :;�� will have only two random values:

|}~� 0 3( �M'.( �M 3 �'. (63)

Assumptions

We have to make assumptions to get good results for Stern-Gerlach Experiment.

Here is the geometric assumptions used in experiment;

• �� :� and �� :�

• tu v_ � �!�:� 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 temperature of the Ag source is takes as L� ��j.

(For the silver atom: Melting point L� ��.!�j ; Boiling point ��.!�j)

• Field gradient along H axis is assumed to be:

�ed0 eHM � for uniform magnetic field

ed0 eHM �� L �M constant field gradient along z axis

ed0 eHM ��Rxy��%z�� field gradient is modulated by Gaussian function.

• 9- component of spin is 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

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The Scheme of the experiment

Figure 10. Experiment

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Results

� ��

Vd VH�M �� Not Quantized

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� ��

Vd VH�M � �� Quantized

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� �� Vd VH�M ��

Not Quantized

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� �� Vd VH�M � �� Quantized

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� ��

Vd VH�M � �:;��< �<� � ��

Not Quantized

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� ��

Vd VH�M �:;��< �<� � �� Quantized

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� �� Vd VH�M � �:;��< �<� � ��

Not Quantized

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� �� Vd VH�M � �:;��< �<� � �� Quantized

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�� Vd VH�M :;��< �<�E��5��%z�� Not Quantized

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� ��

Vd VH�M :;��< �<�E��5��%z�� Quantized

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� �� Vd VH�M :;��< �<�E��5��%z�� Not Quantized

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� �� Vd VH�M :;��< �<�E��5��%z�� Quantized

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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.

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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

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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).

Documents

T.C UNIVERSITY of GAZIANTEP DEPARTMENT OF …€¦ · · 2012-06-22The Stern–Gerlach experiment involves sending a beam of particles through an ... This avoids the large deflection