Aharonov-Bohm effect. Dan Solomon.

Transcription

1 Aharonov-Bohm effet. Dan Solomon. In the figure the magneti field is onfined to a solenoid of radius r 0 and is direted in the z- diretion, out of the paper. The solenoid is surrounded by a barrier that the eletron annot penetrate. We assume an infinite solenoid so there is no z-dependene. In ylindrial oordinates the magneti field is, B zˆ for r r Br, 0 for r r0 0 0 An interferene pattern appears on the sreen. A hange in the magneti field will ause a hange in the interferene pattern. This is the Aharonov-Bohm effet. This is an unexpeted result. Why? Call the region in the solenoid Region. Call the region outside of the solenoid Region. The eletron is onfined to Region. The magneti field is zero in Region. Yet the motion of the eletron is dependent on the field in Region even though the eletron never enters Region. How does the eletron know about the field in region. First onsider a Classial partile. The equation of motion is for a point partile in an eletromagneti field is m dv dt q E v B ; m is the mass, q is the harge, v is the veloity, E is the eletri field, B is the magneti field. Non-relativisti quantum mehanis. The Shrodinger Equation for the wave funtion xt, is, i t i qa qa0 where P x, t x, t is the probability density and A, 0 the eletromagneti potential and E A t A 0 ; B A. A is One key potential differene between the quantum and lassial equations of motion is that the eletromagneti potential appears expliitly in the quantum equation of motion whereas the lassial equations of motion are diretly dependent on the eletromagneti field. What effet does this have on the physis of the problem? Gauge Transformation. Note that for a given eletromagneti field E and B the potentials are not unique. A hange in the gauge is a hange in the eletromagneti potential that produes no hange in the eletromagneti field. Suh a hange is given by A A, A0 A0 t.

2 The Shrodinger equation is gauge invariant. The result of a gauge transformation is to produe hange in the phase of the wave funtion x, t exp iq x, t therefore is unhanged. Reall the eletron is onfined to Region where B 0 What is A in this region? Is A 0? No. It is given by Stokes theorem - AdR AdS B ds m. One possible solution is, C S, ˆ ; ; ˆ ˆ ˆ A x y r r x y yx xy r Cartesian oordinates are used and ˆx and ŷ are unit vetors in the x and y diretions, respetively. S This potential annot be gauged away even though the magneti field in the region outside of the solenoid is zero. Eah path the eletron takes through the slits sees a different potential ausing a different phase shift in the wave funtion traversing that path. This is why the interferene pattern will vary as the magneti field in the solenoid is varied. Cohmology. We an shrink Region down to a point so that Region I = M be the set of -forms where the manifold 0 potential A as element of M, i.e., A M where A x, y a x, ydx a x, ydy. The magneti field B M x y 0. Let M. Then we an express the and is given by Ker d : M M B da. The -th de Rahm ohomology group is H M 0 Im d : M M We have shown that A is losed over M ( B da 0 ). But A is not exat.

3 Fiber Bundles. We need to know the i e phase of the wave funtion at eah point in spae. Imagine a line assoiated with eah point and eah point on the line being a possible phase angle. The olletion of fibers is alled a fiber bundle. Now apply this to the A-B effet. In our ase the Cartesian oordinates x, y M. is the fiber oordinate and xy, is a setion. The basi elements of a fiber bundle is the struture E, M,, F, G where E is the total spae, M is the base spae, F is the fiber spae, is a projetion mapping : E M, and G is a struture group whih ats on the fiber. Let x M then, F, the fiber at x, is given by x Fx x. A setion on a fiber bundle is a ontinuous map : B E suh that x x. The setion xy, give us our phase angle. But where does the potential ome from? The Connetion -form. Define,,. The potential is given by, d a x y dx a x y dy x, y x, y A x, y x, yd a x, ydx ax, ydy. The magneti x y field is then a x a y B da dxdy. Questions Raised: Is the Potential a mathematial onstrut (this is the ase in lassial physis) or is it a real field. How an it be a real field if it is not unique and an t be measured at a point? 3

4 Now bak to our analysis of the Aharonov-Bohm effet. We will ontinue where we left off in Setion. For our purposes the base spae M R R is the entire xy plane. The fiber F xy, ontains information about the phase angle at the point xy,. If p F xy, then p is mapped to the interval 0,. It is important to understand that the fiber does tell us what the phase angle is at the point. It is a olletion of all possible phase angles. This is way all fibers are idential. Any arbitrary point has the same olletion of possible phase angle. The setion xy, xy, (or equivalently i e x, y. is how we assoiate a given phase angle ) with eah point in spae. Sine I have already that stated the magnitude of the wave funtion is known then when the setion is given the wave funtion is ompletely determined. However there is one thing that is missing from this piture. The eletromagneti potential is also required. In the following I shall explain how that is inluded. 6. The onnetion -form. The base spae M is a manifold and has a tangent vetor spae assoiated with it. Also eah fiber F is a manifold and the entire spae E is a manifold. In our example we an thing of the spae E xy, as being defined by 3-oordinates. There are the oordinates xy, orresponding to the base M. The oordinate orresponding to the fiber is the. Therefore E has a tangent spae and an assoiated dual spae of -forms. For the moment define the onnetion -form by d xy,. Reall that is a setion. Let the eletromagneti potential be equal to the pull-bak xy. This an be evaluated as, x, y x, y A x, y x, yd dx dy x y The magneti field is B da 0. Now suppose we have a s ituation where B is not zero. We an over this ase by modifying the -form,, d a x y dx a x y dy,. Let (0.) be given by, (0.) 4

5 Then taking the pullbak of the -form we obtain, x, y x, y Ax, y x, yd dx dy a x, ydx ax, ydy x y Rearrange terms to obtain, x, y x, y A x, y a x, y dx a x, y dy x y (0.3) (0.4) The magneti field is then, where we have used the fat that 7. Disussion and Conlusion. a x a y B da dxdy x, y x, y yx xy 0 to obtain the above expression. (0.5) The Aharonov-Bohm effet raises three separate, but related, issues. The first is the question of the whether or not we an desribe physis in terms of loal fields. In physis a field allows us to avoid the idea of ation at a distane. A field allows us to desribe physial proesses at a point in terms of numbers at that point. These numbers desribe the field its magnitude, diretion, et. A typial example is the eletromagneti field EB,. In this ase the potential AA, 0 is also a field sine if we know AA, 0 is a small region of a point then EB, an be determined using Eq. Error! Referene soure not found.. However there is key differene between the two. The eletromagneti field an be diretly measure at a point using instruments. Also, and most important, the eletromagneti field is a loal quantity in that the eletromagneti field in a region doesn t hange its value if the value of the eletromagneti field far away hanges value. This is not true of the potential. The potential at a point is effeted by field remote from that point (See disussion in Setion ). The other problem is that the potential is not unique. We an always add a gauge transformation to a potential without effeting the fields. Therefore in lassial physis the eletromagneti field EB, is onsidered real and the potential AA, 0 is onsidered to be a mathematial devie with underlying reality. The situation hanges if we go to quantum mehanis. In this ase the equation of motion is desribed in terms of the potential. The experimental fat of the Aharonov-Bohm effet seems to indiate that the potential is, indeed, a real field. The problem with this is that, unlike the eletromagneti field it annot be measured at a point. It an be derived from the eletromagneti fields but only if they are know every. Also, as previously mentioned, the potential is not unique but an be modified by a gauge transformation. This problem is unresolved but is debated in the literature. 5

6 The seond question may be expressed in the form - What information is neessary to solve for the equations of motion. Consider the problem in this paper. Reall that the magneti field in Region is zero and there is a magneti flux in Region and the partile is restrited to Region. In lassial physis you need the initial state of the partile (position and veloity). You also need to know the magneti field in Region. In quantum mehanis you to know the initial wave funtion inluding the phase. You also need to know the magneti field in Region. So far we are analogous to lassial physis. However for quantum mehanis you need an additional piee of information and that is the magneti field in Region. Given this you an solve the Shodinger equation. However a philosophial problem ours beause in order to solve the Shrodinger equation you must onvert the magneti flux through Region into a potential in Region. This an be done by finding an A that solves Eq. Error! Referene soure not found.. One solution is given by Error! Referene soure not found.. However this solution is not unique. There are an infinite number of others. So if we deide that the potential is a real field then whih of the possible potentials is the real one. The problem is that there are too many degrees of freedom. Does the fiber bundle approah help. We have shown that all the information in the magneti field is stored in the onnetion -form. However for a given magneti field the onnetion -form is not unique either beause you an always add the term, x, y x, y x dx to the onnetion -form without hanging the magneti field. This orresponds to a gauge transformation. One again we have too many degrees of freedom. In onlusion the Aharonov-Bohm effet hallenges our lassial oneptions of loality and the reality of the potential versus the eletromagneti field. y dy (0.6) (In the following I will address different ways this problem an be resolved) (Figure and Figure ). Mostafa E. El Demery. New Perspetives on the Aharonov-Bohm effet. Masters Thesis for St. Edmund s College. May 3, 03.. Greg Naber. Gauge Fields and Geometry: A piture Book. (Mathematis Student Organization: Drexel University) May 5, 0. GEOMETRY-A-PICTURE-BOOK.pdf 3. Alexandre Guay. Geometrial aspets of loal gauge symmetry Ambroz Kregar. Aharonov-Bohm effet. KREGAR, A. Aharonov-Bohm effet, e-print de seminário, Universidade de Ljubljana (0). 6

7 5. Yair Guttmann. Fiber Bundle Gauge Theories and Field s Dilemma A. Heil, A kirsh, B Reifenhauser, and H Vogel. Appliation of topologial ideas to the Aharonov-Bohm effet. Eur. J. Phys., Vol. 9 (988) Y. Aharonov and Bohm. Signifiane of Eletromagneti Potentials in the Quantum Theory. Phys. Rev. Vol. 5, No A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, J. Zwanziger. The Geometri Phase in Quantum Systemes. Springer-Velag, Berlin Heidlberg, (003). 9. Reinold A. Bertlmann. Anomalies in Quantum Field Theory. Oxford University Press. (000) 7