The only physical requirement we can make is that there can be only one value of the wave function for each point. One should be able to go even further. Now I turn to a second point. I would like to discuss next a very interesting situation that was noticed by Josephson 19 while analyzing what might happen at a junction between two superconductors. These are the standard equations for two quantum mechanical states coupled together. At low temperatures, when the energy of a system has been reduced very, very low, instead of a large number of states being involved, only a very, very small number of states near the ground state are involved.

But here to get one electron away from what all the others are doing is very hard because of the tendency of all Bose particles to go in the same state. But there are certain situations in which the peculiarities of quantum mechanics can come out in a special way on a large scale. A current once started, just keeps on going forever.

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The only physical requirement we can make is that there can be only one value of the wave function for each point. Suppose we have two superconductors which are connected by a thin layer of insulating material as in Fig. These equations are not the whole story.

Perhaps we can predict that the measurement of magnetic fields will—by using the effects of quantum-mechanical interference—eventually become almost as precise as the measurement of wavelength of light. The moment that you have billions in the same state that is, in the same electromagnetic wave , you can measure the wave function, which is the vector potential, directly. The only physical requirement we can make is that there can be only one value of the wave function for each point. You would think the current would be something like the density of particles times the velocity.

As you know, very many metals become superconducting below a certain temperature 9 —the temperature is different for different metals. It is recent and modern and would be a perfectly legitimate talk to give at a research seminar. The electric field outside a solenoid with an increasing current.

First, notice that the interference between two junctions can be used to make a sensitive magnetometer. For example, in one dimension Eq. The wave function changes with time according to the Schrödinger equation in Eq. Suppose we have two superconductors which are connected by a thin layer of insulating material as in Fig.

### Modern electrical batteries (first the boring historical stuff):

So we would expect all the pairs to be moving in the same state. Then the two amplitudes should be related in the following way: We can say that it only gets out by moving through the surface—and that is local conservation. The two possibilities differ by the vector potential. We are really getting control of nature on a very delicate and beautiful level.

First, there is no electrical resistance. Suppose that we take a piece of superconducting material which is in one lump. That is to say that the current takes on its maximum values where the flux linkage has just those quantized values we found in Eq. It was at this point that Born made an essential contribution to our ideas regarding quantum mechanics. I will begin by reminding you of some of the properties of the Schrödinger equation.

So we substitute the wave function of Eq. Now we can use Eq. My subject is the Schrödinger equation in a classical setting—the case of superconductivity. London 14 predicted that the flux trapped by a superconducting ring would be quantized and said that the possible values of the flux would be given by Eq. This is just another example of the quantum-mechanical penetration of a barrier.

But look what can be done. Now what are the consequences? I will, however, consider only the situation at essentially zero temperature—or, in any case, I will disregard the complications produced by those electrons which are not in pairs. When Schrödinger first discovered his equation he discovered the conservation law of Eq.

### Undergraduate

So everywhere in a lump of superconducting material the current is necessarily proportional to the vector potential: I am sorry to say, gentlemen, that to participate in this adventure it is absolutely imperative that you learn quantum mechanics as soon as possible. The sequence of events is sketched in Fig.

The electron is either here, or there, or somewhere else, but wherever it is, it is a point charge. The sequence of events is sketched in Fig. For example, in one dimension Eq.

- Even more interesting is a related phenomenon discovered experimentally by Meissner. In other words, it starts up its own current—and in just the right amount to push the field out. Everything now fits together, and the measurements show the existence of the predicted purely quantum-mechanical effect on a large scale. Finally, I would like to describe a very dramatic and interesting experiment which has recently been made on the interference of the currents from each of two junctions. Now think of what would happen in the following circumstance.
- That gives you the order of magnitude. I will begin by reminding you of some of the properties of the Schrödinger equation. These equations are not the whole story. This is just another example of the quantum-mechanical penetration of a barrier.

Taking the gradient of the whole of Eq. At first this was quite mysterious, 17 but we now understand why it should be so. Suppose this flux nearly instantaneously builds up from zero to something. The moment that you have billions in the same state that is, in the same electromagnetic wave , you can measure the wave function, which is the vector potential, directly.

I start with zero vector potential and then I turn on a vector potential. So we substitute the wave function of Eq. Since the electron is jumping backwards, I showed this phase shift with a minus sign. There are three pieces. Finally, I would like to describe a very dramatic and interesting experiment which has recently been made on the interference of the currents from each of two junctions. These equations are not the whole story.

### Undergraduate

To write them in a shorter form I will—following Eq. Any complex function can, of course, be written this way. If you would think of the ring as a classical object with an ideally perfect that is, infinite conductivity, you would think that whatever flux was initially found through it would just stay there—any amount of flux at all could be trapped. That means that the velocity can be expressed in terms of velocity potential.