Stern-Gerlach Experiment and Spin

Transcription

1 Stern-Gerlach Experiment and Spin 1 Abstract Vedat Tanrıverdi Physics Department, METU tvedat@metu.edu.tr The historical development of spin and Stern-Gerlach experiment are summarized. Then some questions on spin are stated. 2 Introduction Spin is a subtle topic in quantum mechanics. Without understanding its algebra we can not understand many things in quantum mechanics. It is one of the key topics to be able understand subatomic world. But the algebra is not the whole story, there are more things on this subject. However, in many quantum mechanics books, spin is considered just as an intrinsic property, and framework for the spin is just constructed by showing related algebra for the spin[1, 2, 3, 4, 5]. Some of them mention about it more[6, 7, 8, 9, 10]. Only a few quantum mechanics books mention about it in detail and explain problems of some classical approaches[11]. In this work, the historical development of spin is summarized and some problems about it are stated. 3 Stern-Gerlach Experiment 3.1 History of Stern-Gerlach experiment and spin First consideration of the electron spin was a speculation of Compton in 1921 that electrons can be tiny spinning gyroscopes and magnetic field can be originated from this motion[12]. This speculation was only a consideration 1

2 and did not find any response for the following years and forgotten. We know some sources magnetic field now, but still we have some further questions. Similarly for the spin, we know some properties of ferromagnetism. However, we still do not have a mechanism which explains underlying reasoning for ferromagnetism. In fact we should ask more questions about magnetism itself, we still need to understand more on it. Apart from above speculation, there were different ongoing experiments related with the atoms. In 1922, Stern and Gerlach wanted to measure possible magnetic moments of different atoms. Stern and Gerlach found different discrete traces in their experiments, not all of them was similar to the one used in these days to teach Stern-Gerlach experiment. They observed many discrete lines, some of them were originating from orbital angular momentum of the electron but not all of them. Stern and Gerlach was familiar with the dependence of magnetic moment on mass of the particle and was knowing that effect is originating from the electron due to the length of deflection, they also know that all of the effects are not because of orbital angular momentum. Among their experiments, there was a special one; experiment with silver atoms. In this one, there were two discrete traces, which can not be originated from orbital angular momentum or related quantity m l value. Because the possible number of m l values have to be always odd. Spin was an unknown property of the electron at that time, so their knowledge was not enough to understand observed pattern. Stern and Gerlach thought that this pattern is related with the space quantization, proposed by Sommerfeld[11]. Sommerfeld s proposal was successful in fitting experimental results in some cases but not at all. The experiments with alkalis were not agree with the Sommerfeld s space quantization, hence space quantization thought was eliminated. So there have to be another explanation. In 1924, Pauli was studying on spectral lines observed in Zeeman effect. Available quantum numbers was not enough to explain observed pattern and Pauli introduced another quantum number to explain observed spectral lines[13]. With this quantum number, total quantum numbers describing electron became four and in fact to define a point particle we can use three degrees of freedom each corresponding one spatial coordinate. He just introduced this quantum number without any explanation, on the other side this quantum number was successful to predict experimental result, which made it usable and part of the quantum mechanics. However, it was not containing any information for the underlying reason. Not explaining or not even questioning underlying reason is an odd act 2

3 in science. It has a contradiction with the scientific understanding. If we leave our understanding reasons of nature, we can not do scientific research with reasoning. We do only research depending on experimental relation. Though in some cases it is useful, in some cases it prevents our further developmental procedure. This situation can be a drawback for science in different ways. Sometimes our understanding reasons and relations can take decades, however we should still continue to ask questions. In 1925, Goudschmit and Uhlenbeck tried to explain results of Stern- Gerlach experiment with an internal magnetic moment of the electron. In fact, Goudschmit and Uhlenbeck inspired from Pauli s work describing electron with four quantum numbers. However a point particle should had three degrees of freedom. Despite this fact, Pauli did not make any explanation concerning underlying reason for using four quantum numbers. Goudschmit and Uhlenbeck took this as a mystery and followed this mystery, and it led them to consider a structure for the electron. They assumed that electron could be a rotating sphere and this rotation could be source of the fourth degree of freedom. Having a shape brings new degrees of freedom. With Ehrenfest s help they found Abraham s studies on rotating sphere with charge and his results was showing that necessary g-factor in the electron s magnetic moment can be explained classically. They also modelled this magnetic moment by considering electron as a rotating solid spherical ball with a radius and charge. Their adviser, Ehrenfest, considered them as young enough to make such considerations and send their work to a journal[14]. After Ehrenfest s sending their work, Goudschmit and Uhlenbeck decided not to publish their work due to inconsistency of their model with either electron s mass or speed of light. However it was too late, and their work was published. At that days the origin of the double traces in the Stern-Gerlach experiment with silver atoms was not clear enough, though it was known that it should be originated from electron. In 1927, Phipps and Taylor did Stern- Gerlach experiment with hydrogen atoms, and they also observed double traces pattern. This result was removing doubts about the source of these double traces. Hydrogen atom has only one electron in s shell with l = 0 and m l = 0. This result was another evidence for the electron s intrinsic magnetic dipole moment. Today we still use Pauli s description of the electron with four quantum number, which should mysteriously correspond to four degrees of freedom. This is a contradiction with the consideration of the electron as a point particle. 3

4 3.2 Stern-Gerlach Experiment The schematic represantation of Stern-Gerlach experiment is shown in Fig.1. In the Stern-Gerlach experiment, the oven evaporates atoms, in general silver atoms, and give them some kinetic energy. The temperature of the oven is arranged not to ionize atoms and to move freely with some kinetic energy. There is a hole in the oven and atoms can escape from this hole. The escaped atoms are firstly collimated and then they enter in an inhomogeneous magnetic field. This magnetic field,b, causes a potential energy U = µ B due to interaction with magnetic moment, µ. The nonuniform structure of B the magnetic field causes a force F = µ z B z = g z s µ b m z s in the z-direction, z where g s 2, m s = ±1/2, µ b is the Bohr magneton and h is taken as 1. This force results with a deflection of the atoms in the z-direction. Figure 1: A diagram showing Stern-Gerlach experiment setup. Silver atoms, evaporated in the oven, are first collimated and passed through a inhomogeneous magnetic field and hit a screen leaving two discrete traces. This is not consisted with the thought that electrons have magnetic moment any direction and interaction with external magnetic field leaves continuous trace. (Adapted from a wikipedia file[15].) In the Stern-Gerlach experiment with silver atoms, there are 2 discrete trace each corresponding magnetic moments in different directions. In the figure, we see also the classical expectation related with possible continuous magnetic moment of electron. The discrete traces in the experiment show that there are two possible force in opposite directions related with the interaction between magnetic moments and the magnetic field. These two opposite forces resulted with the thought that there are two discrete magnetic moments for the silver atoms. These two discrete magnetic moments 4

5 can not be explained by m l value. Firstly, m l can not have 2 values, because of 2l + 1 possible values. Also for the silver atom s, only uncoupled electron is in 5s state, l = 0, which leaves only one m l value, m l = 0. To be able to obtain compatible results with the experiment something else was needed. The quantum mechanical calculation technique for this discrete pattern made possible after Pauli s invention of forth quantum number m s. In quantum mechanics, the electron is a spin half particle and its possible m s values are 1/2 and 1/2. These possible m s values gives two different magnetic moments each corresponding one of the discrete traces in the Stern-Gerlach experiment. 3.3 Quantum mechanical interpretation of Stern-Gerlach experiment For an atom, there are three possible magnetic moment source: magnetic moment due to electrons orbital angular momenta, spin of electrons and nuclear magnetic moment. The magnitude of these depends inversely on the mass of the source of magnetic moment, µ e h. So electronic magnetic 2m moments and nuclear magnetic moments differ in order, and nuclear magnetic moment is much smaller than electronic magnetic moments due to nucleonelectron mass ratio m N /m e This ratio tells us that the deflection in the Stern-Gerlach experiment is originating from electron. There are two reason for magnetic moment arising from electron; orbital angular momenta and spin. Before looking through to these, we should look through electron s description. Firstly, the electron is considered as a point particle. In quantum mechanics without any causal reason, except consistency with the experiments, we use four quantum numbers to define an electron orbiting around the nucleus: principal quantum number n, orbital angular momentum quantum number l, magnetic quantum number m l and electron spin quantum number m s. One of the electronic magnetic moment is related with orbital motion of electrons and it can be calculated by using m l. It takes integer values between l and l, in total 2l + 1 values. Since l is always integer, there must be odd number of possible m l values. Other electronic source of atomic magnetic moment is electron spin. In quantum mechanics, electron spin is half integer and there are only two possible corresponding m s values, 1/2 and 1/2. If an atom has an uncoupled electron at the outer shell, this electron can 5

6 cause magnetic moment, other coupled ones in full shells do not contribute to atom s magnetic moment since they cancel each other s effect. If we want to calculate magnetic moment of an atom due to its electrons, we should only consider unfilled shells. The electrons contribution to the magnetic moment of the atom comes from electrons in the unfilled outer shells and can be calculated from their m l and m s values. This contribution is two fold; due to orbital motion and electron spin. This is the reason why in Stern- Gerlach experiments with different atoms one can obtain different patterns. We already explained silver atom case with m l = 0 and m s = ±1/2. Since m l is zero, it does not contribute to magnetic moment and there are two possible m s value each corresponding one of the traces. Half integer value of the electron spin is firstly invented to be able to obtain a framework which is consistent with the experiments. There were two different traces, and electron spin is considered as 1/2 to be able to get consistent structure with quantum mechanics, in which the increment of quantum numbers is always 1. The remaining multiplication factors, multiplication of g-factor for electron spin and µ b, are arranged to be able to obtain consistent results with experiment. 4 Discussion As we see in the historical part the invention of the spin is totally experimental. This experimental invention of the spin works very well. It explains Zeeman effect, Stern-Gerlach experiment and also some other stuffs. It also makes possible to work another active research area; spintronics. This all shows that spin is a useful tool for us and it enables us to calculate many things. On the other side in books, spin is left as an intrinsic property. However these successes do not mean that spin can not be modeled and worked on. It is a topic we can not leave it as an intrinsic property. There must be some reason for it and as scientist we can and should search for that reason. It is scientist duty to search mysterious things. Today most physicist are not considered it as a mysteries thing. We should also consider on why we have lost our inspiration on such mysteries. As it is asked in a paper How are we prisoners of conventional thinking? [16]. We should also consider this question. Does conventional thinking cause it? Or something else? Let us go over the problematic things related with spin. We still do not 6

7 know why electron has four quantum number and in many books spin is just defined as an intrinsic property of the electron even without mentioning degrees of freedom problem. In many books it is written that spin is an intrinsic property of the electron. What is that intrinsic property? Can t we understand that intrinsic property and work on it? Or at least can t we develop some models for electron spin? We can still ask What is the spin? and try to find answers. We don t need to repeat repeated sentence Spin is an intrinsic property. This sentence does not give any causal answer, it is an answer that says nothing in explicit. We do not know explicitly what the spin is. It is still a mystery for us and as a scientist we should ask How this intrinsic property, spin, explicitly occurs?. Not mentioning about even degrees of freedom problem is hiding problems. We need to face with the problems, not hide them. If we hide we can never solve problems. It is our duty to ask questions, find problems and try to solve them, sometimes we should try to approach from different angles, sometimes we should directly face with them. Without asking questions, without constructing relations and models we can not continue our scientific developmental procedure. We should ask always questions, it is the start of science; in all areas, spin is only one of them. References [1] A. Messiah, Quantum Mechanics. North-Holland, [2] A. Das and A. C. Melissinos, Quantum Mechanics, A Modern Introduction. Gordon and Breach, [3] S. Brandt and H. D. Dahmen, The Picture Book of Quantum Mechanics. Springer, Third edition, [4] L. E. Ballentine, Quantum Mechanics, A Modern Development. World Scientific, [5] G. Auletta, Foundations and Interpretation of Quantum Mechanics. World Scientific, [6] E. Mertzbaher, Quantum Mechanics. John Wiley & Sons, Third edition

8 [7] L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics. McGraw-Hill, [8] E. S. Abers, Quantum Mechanics. Pearson Education, [9] J. L. Powel and B. Crasemann, Quantum Mechanics. Addison Wesley, [10] H. C. Ohanian, Principles of Quantum Mechanics Printice-Hall, [11] R. Eisberg and R. Resnick, Quantum Mechanics. John Wiley & Sons, [12] A. H. Compton, Journal of the Franklin Institute 192, 145 (1921). [13] W. E. Pauli, Naturwissenschaften 12, 741 (1924). [14] G.E. Uhlenbeck and S. Goudsmit, Naturwissenschaften 47, 953 (1925)., Nature 117, 264 (1926). [15] experiment.png [16] C. Quigg, arxiv: