So now we're going to compare magnetic field B versus magnetic vector-potential A. This is like comparing electric field versus electrostatic potential phi. So Melody, do you remember why we use electrostatic potential instead of electric field? Yes. So I think the integral is easier because one of them is a scalar and other one is a vector. Exactly. So now we have to pose the same questions here. Is the vector potential merely a device which is useful in making calculations? Is the vector potential a real field? Those two. Now a supplementary question follows. Isn't the magnetic field a real field as it is responsible for the force on a moving particle? Let's go further. So, we will present some thoughts and the real field. The whole idea of a field is a rather abstract thing as we just mentioned before. It is about my cultural. When we think about pairwise interaction between human beings, we cannot do that because it's too heavy duty calculation. So we just summed them up to one parameter like culture. Is this culture right for me? Then it's like, can I be there and get well with the people there? Likewise, in the field concept, it was the interaction between the charge of interests and rest of the charges and put that as a mathematical function. So, it was an abstract thing. The value of the magnetic field is not very definite as you see. When you have charged and you are moving with the charge, it is stationary so there's no magnetic field. But if you stop moving along with the charge then you start to see the magnetic field. So, magnetic field is not very definite. By choosing a suitable moving coordinate system, you can make a magnetic field at a given point disappear as I just mentioned. Now, what we mean by here by a real field is a mathematical function we use for avoiding the idea of action at a distance. Maybe this wording is more complicated so let me repeat it. A mathematical function we use for avoiding the idea of action at a distance. So Melody, when you hear action at a distance, what do you think? Well, I think it's kind of like earlier, we discussed the idea of most of the forces we think about are physical. So one object touching another but for things like gravitation and magnetic force, you have a source and then you see some action over here and you want to connect the two. Exactly. Very good. We see that in animation and movies as well like in Avengers or other type of sci-fi. Now, continuing with that discussion. So, if we have a charged particle at the position of P, it is affected by other charges located at some distance from P, as Melody just mentioned. One way to describe the interaction is to say that the other charges makes some condition in the environment at P. Makes some condition, it influences. Invisible but it makes some change. If we know that condition, which we described by giving the electric and magnetic fields, then we can determine completely the behavior of the particle with no further reference to how those conditions came about. So now you following our logic. If those other charges were altered in some way but the conditions at P that are described by the electric and magnetic field at P remain the same, this is a big if, then the motion of the charge will also be the same. A real field is a set of numbers we specify in such a way that what happens at a point depends only on the numbers at that point. So that brings us back to the original definition of a field which is function of space, the physical properties function of as a function of space. Now homework. How does the fact that the vector potential is not unique affect the question of whether the vector potential is a proper real field for describing magnetic effects? Very difficult question but I hope you can find the answer. Now, let's discuss about the usefulness of vector potential A. Now it is easier to manipulate mathematically. It is a scalar integral plus three differential equations versus three integrals with higher complexity to get the force. Let's take a look at here. Phi of one is equal to one over four pi epsilon nought integral rows of two over r12 dv2 versus E of one is equal to one over four pi epsilon nought. Integral round two times the unit vector connecting one and two over r12 squared dv2. So we know the usefulness of less static potential right away as Melody just described. Mathematically this is the obvious choice. Now, let's take a look at the vector potential. Here the advantages are much less clear as you can see. A sub one is equal to one over four pi epsilon nought c squared times integral, j of two, this is a vector, dv2 over r12 versus p of one is equal to one over four pi epsilon nought c square, integral j of two cross E12 over r12 squared dv2. Of course, if you have a cross product, it adds complexity so you may say, "Look, this is simpler yes, but still you have to deal with three components of the vector." So it is true that in many complex problem is easier to work with A but it would be hard to argue that this ease of technique would justify making us learn about one more vector field. You agree? I agree. Good. So, let's think another way. We have introduced A because it does have an important physical significance. Not only is it related to energies of currents, but it is also real physical field. Oh! Now you have the answer to your question. But anyway, in classical mechanics, it is clear that we can write the force on a particle using Lorentz force equation which states f equals Q times parenthesis E plus V cross B. Given the forces, everything about the motion is determined. If the force is given by equation of motion, we can determine the motion. In any region where b is equal to zero, even if A is not 0 zero such as outside a solenoid, there is no discernible effect of A therefore, for a long time it was believed that A was not a real field. It turns out however that the field A is in fact a real field and we'll see the effect. A very important difference. So this is one of the very popular experiments in this field. Booms experiments. So, here we want to describe how the vector and scalar potentials enter into quantum mechanics and it is just because momentum and energy play a central role in quantum mechanics that A and phi provide the most direct way of introducing electromagnetic effects into quantum descriptions. So again, in quantum mechanics we like to play with momentum and energy instead of force and velocity. So, we are going to set up a thought experiment, imaginary experiment. Electrons with the same energy leaves the source and traveled towards, so this is the source, maybe thermionic emission like in the electric bulb or some source that we use for cathode ray tube. Now it's like a dinosaur, we don't see that cathode ray tube brown the television anymore and if we spit electrons out here with the same energy, you can imagine it will travel in the vacuum because there's nothing to be scattered. Beyond the wall is a backstop, here we have a movable detector like fluorescent screen, which measures the rate i, at which electrons arrive at a small region at the distance x from the axis of symmetry. So you can imagine if you have this fluorescent dye, the intensity of light will be proportional to the electrons arriving at that spot. You have a slit experiment. You have two slits here and you have learned, probably from high school physics or University physics, if you do this you will have interference patterns. So, depending on the path difference between two rays, you either have constructive interference or destructive interference at a given distance that's why you have oscillation of the intensity profile. Now we're going to do the same kind of calculation that we did for this slit experiments. If the distance between the screen and the slit is L, capital L, and if the difference in the path length of four electrons going through the two slits is A, the phase difference delta of the two waves is given by delta is equal to A over lambda bar, where lambda bar is lambda over two pi and lambda is the wavelength of the space variation of the probability amplitude. This is pretty complicated wording. But it just means the wave nature of your electron traveling from one place to the other. If the distance between the slit and the screen is large, very large, that is larger than the x, much larger than the x, then which is the distance here, then A is equal to x over L times d. Again, x over L, that will be this ratio times d this slip. That will be the difference here. So here we assume the lengths of these two edges are the same. Approximately and the difference is only here. Then delta is equal to x over capital L times d over lambda bar because we have put this here. So, we would like to state the law that determines the behavior quantum mechanical particles in electromagnetic field, which replaces the Lorentz force equation. So, we want to replace this. Now, since what happens is determined by amplitudes, the law must tell us how the magnetic influences affect the amplitude. We're no longer dealing with the acceleration of a particle. Now, the law: the phase of the amplitude to arrive via any trajectory is changed by the presence of magnetic field by an amount equal to the integral of the vector potential along the whole trajectory times the charge of the particle over Planck's constant. So, in a nutshell, magnetic change in phase is equal to q over Haba, the integral of trajectory A.ds, okay? So, if there is any vector potential, and if you have the line integral that will be the phase change, okay? Now, the effect of an electrostatic field is to produce a phase change given by the negative of the time integral of the scalar potential Pi. So, electric change in phase is minus q over Haba, integral Pi dt. Two expressions below together give the correct result for any electromagnetic field, static or dynamic, which replaces this Lorentz force. We will see how that happen. For now, it might be a distant truth. So, here in the box, you see the magnetic change in phase and electric change in phase. They are symmetric, only have sign change, and you place A by Pi, and s by t. A by Pi and s by t. So, the interference of the waves at the detector depends on the phase difference, as we mentioned. So, we just thought about the phase difference when there were no fields. That was really clear. The path difference was really clear. Now, if I turn on any magnetic field, we have to add this term here. That is the same for the path number one and path number two. So, you will have two additional terms for both of the phases here. Again, if it is constructive interference, then we have maximum amplitude. If it is destructive, we have minimum amplitude. That we know, right? So, if I add them up, if I ignore the path integral along the slit, this distance, it's like line integral. It's a closed loop integral. So, that's why the Delta, which is the difference between this and this becomes the circular integral around this loop, one and two. So, that's Delta B is equal two plus q over Haba round integral of A.ds. So, this equation tells us how the electron motion is changed by the magnetic field. With it, we can find the new positions of the intensity maxima and minima at the backstop. So, depending on how we turn on the magnetic field, the maximum peak will shift left and right. So, in this slide, for a moment, we will think about the arbitrariness in A. You may remember for the vector potential, we can arbitrarily set the reference. In the beginning, we set the reference to have a certain character, which was divergence of vector potential V zero. So, we put a condition like this. The divergence of A is zero. Now, we're going to change this. We're going to choose another type of reference, and there's a reason behind that, and see how that works for quantum mechanics. So, you already know from the boundary condition we just discussed, the two different vector potential functions A and A prime whose difference is the gradient of some scalar function Delphi, both represent the same magnetic field since the curl of a gradient is zero. So, it is the same classical force. It will give you the same classical force. Now, if in quantum mechanics, the effects depend on the vector potential, which of the many possible A functions is correct? So, let's take a look at here. You can have A prime.ds, and A.ds. But if you put this Delphi here, you see Delphi lie the circular integral is always zero. So, you can see those two will always be the same. So, the same arbitrariness in A continues to exist for quantum mechanics as well. So both A and A prime give the same phase differences and the same quantum mechanical interference effects. So, as long as you take Delphi difference between two different vector potential, you will have the same effect, so you cannot discern those vector potentials, okay? So, now it's a matter of choice. It's a matter of convenience. The line integral of A around a closed path is the flux of B through the area confined by the path, which here is the flux between paths one and two. So, you can see this equation with the help of B equals V equals del cross A, as Stoke's theorem, then this part becomes flux of B between one and two. Flux of B between one and two. This result depends only on B field and therefore only on the curl of A, thus both A and A prime give the same phase differences and the same quantum mechanical interference effect, okay? Now, continuing on our discussion of the real fieldness of B or A. Now, real field has no action on a particle from a distance. Real field is acting at that point. No matter what is happening around it. We can set up an example where B is zero at any place along the trajectory of the particle, so that is not possible to think of B acting directly on them. If we arrange a situation in which electrons are to be found only outside of the solenoid, no B field but only A, you know we have circulation of A outside the solenoid, but B field is non-existing. There still be an influence on the motion, which is classically impossible since the force depends only on B on electrons. So now, we are now jumping into quantum mechanics. Welcome to quantum mechanics. So, even though you have no B, you have an effect. In Lorentz force, if B is zero, you don't have any force. But without force, what happens? Nothing will happen. But here, you will see something happens. So, Delta, the phase difference is equal to Delta when magnetic field is zero plus q over haba, line integral of A around a close loop. Quantum mechanically, we can find out that there is a magnetic field inside a solenoid by going around it, okay? So, the interference in the presence of a solenoid. So, suppose that we put a very long solenoid of small diameter just behind a wall and between the two slits of which diameter is much smaller than d. So, you see here the solenoid that we put behind the wall. So, any of our line, the electrons will not see the inside of the solenoid. There will be no appreciable probability that the electrons will get near the solenoid. What will be the effect on our interference experiment? So, if we are from classical physics, then because we don't have any magnetic field that interferes with my electrons that are moving here, we wouldn't think there will be any effect. But there's a big but. We compare the situation with and without a current through the solenoid. If we have no current, we have neither B nor A, and we get the original pattern of electron intensity at the backstop. That's trivial. If we turn the current on in the solenoid and build up a magnetic field B inside, then there is an A outside, and there's a shift in the phase difference proportional to the circulation of A outside the solenoid, which is equal to the flux of B inside the two paths. This is hilarious. This is like the electric field in a spherical volume is depending on a very small volume of a charge. So, that means the flux inside the solenoid is influencing the path outside the solenoid. You can see there for any pairs of paths as the flux of B inside is constant, there is the same phase change. Because this doesn't change, the flux doesn't change, no matter where you go, the phase shift will be constant. Meaning it will be uniform phase shift. So, that means, you will have uniform motion of your interference pattern. So, this corresponds to shifting the entire pattern in x by a constant amount. So, if you see this change, that's quantum mechanics. So, the shift x sub zero of entire spectrum along x is del equals x over l times d over Lambda bar, Lambda is equal Lambda of b equals zero plus q over h bar, the circle integral of a.ds and if I merge them, x naught is equal to minus L over D Lambda bar times q over h bar, this integral. You can see that, right? This was x put in here, sending them to the left side, flipping around and using this. So, x naught is minus l over d Lambda bar q over h bar, times flux of b between one and two. Why? Because the circulation of one and two, connecting one and two is really the B field from the Stokes' theorem, the B flux, flux of b. So, this was derived by Dr Amaronov and and Dr Bomm in the Physical Review in 1959. So, if you're interested, you can take a look in the paper for more details. So we're going to think about connection between quantum, mechanical and classical formulas. We will show why it turns out that if we look at things on a large enough scale, it will look as though the particles are acted on by a force equal to QV cross the curl of A. The curl of A is magnetic field. So we will consider the slit experiment where we have a magnetic field which is uniform in a narrow strip of width W which is small as compared with L. So we'll think about uniform magnetic field zone here, with a width of W which is smaller than the capital L, and see what happens. Now, the phase shift by a universe strip of magnetic field can be understood, so the shift and face differ by the flux of B between the path which is approximately B times WD, because B is uniform WD is the area, and if you just multiply them, that's the flux. So delta is equal to delta when b is equal to zero plus q over harbor, times Bwd. Therefore, the effect will be to shift the whole pattern upward by an amount delta x, so delta x is equal to l lambda bar over d times delta delt, is equal to l lambda bar over d bracket del prime minus del p is equal zero, is equal to L times lambda bar cured Harbor bw. So you can see the effect here is to move all the patterns upward uniformly. Such as shift of delta x is equivalent to deflect and all the tragic tourists by the small angle alpha. So delta x is equal to l lambda bar cubed h bar PW is like translated to alpha is equal to delta x over l because l, the ratio becomes like angle if it is small enough. Then you can see if I put L from here, you will have lambda bar over h bar, q p w, right? That's easy manipulation. So classically, you would also expect a thin strip of magnetic field to deflect all trajectories through some small angle alpha prime as we just discussed. As the electrons go through the magnetic field, they fill a transverse force QV cross B which lasts for time w over B, so while they're here, like when they take off, like when the airplane is taking off, you feel the magneto moving in the X-Men right? So, only on this. So the change in their transfer momentum is just equal to this impulse,. So F times delta t is equal to the delta p, and delta p sub x is equal to qwB. You can see that right? So the angular deflection is equal to the ratio of this transverse momentum. So alpha prime will be delta px over p, which is equal to qwB over P. You can see how that connects in this diagram. So if you have momentum pushing upward is like adding two velocity vectors together because mass is constant. So that's how we can explain the deflection here. So then with that knowledge we can compare the results from the classical approach with the one from the quantum mechanical approach which has the, which have the same quantity, alpha prime is delta px over P is equal to qwB over p. Now for quantum mechanics, alpha is equal to delta x over L which is lambda over harbor qBw. So you see, qBw here and lambda bar over h bar, and the connection between classical mechanics and quantum mechanics in this case if they are the same, lambda bar is equal to harbor over p, and we know that. We know the momentum p is ha over lambda, or harbor over lambda bar, right? So a particle momentum p corresponds to quantum amplitude varying with the wavelength, with this equality alpha is alpha prime. So, either way, you can understand the connection between classical mechanics and quantum mechanics. So let's give- let's think about some thoughts from the analysis we see how it is that the vector potential A which appears in quantum mechanics in an explicit form produces a classical force which depends only on its derivatives curl of A. In quantum mechanics what matters is the interference between nearby paths, it always turns out that the effects depend only on how much they feel A change from point to point, and therefore only on the derivatives and not on the value itself. Nevertheless, the vector potential A together with the scalar potential phi, appears to give most direct description of physics, right? In the general theory of quantum electrodynamics, one takes a vector and scalar potential as the fundamental quantities in a set of equations that replaced the Maxwell equations, so we will then jump up from here to a new level of knowledge, okay. So what is true for Statics is false in dynamics. So this is a table that summarize those differences. So false in general, true only for Statics is including Coulomb's law. So Coulomb's law is only approximate. So we have to change to Lorentz force or Gauss's law, and you see here the curl of electric field is not always zero, it can change according to Faraday's law, if there isn't a change in magnetic field, and electric field is not a simple function of the gradient of the electrostatic potential. You have to take into account the time derivative of the vector potential as well. In that way, you now start to understand even if there is no gradient of electrostatic potential, you can see electric current, electric field, and the same goes with this equations we learned, and also Ampere's law is only true for magneto statics and the magnetic field coming from this is only limited to magneto static, so you have to think about the Maxwell's equations and then do this complicated math, the force equation. Continuing with our discussion, the Poisson's equations is only true for Statics, so the real Poisson's equations has more complicated terms as you can see here. The saying goes with the general equation for the electrostatic potential and magneto static vector potential which includes the past, what happens in the past. So now it becomes really really complicated. However, you can see the energy, the total energy of the field is written by this.product E and.product B and only for magnetostatic or electrostatics, you can use this equation that we just learned in the previous lecture. Okay, with that, we're going to wrap up our lecture here and hope to see you again. Thank you, bye bye. Bye.
