L6.4 Three dimensional current and conservation.

published on July 17, 2020

Three-dimensional case Now, in the future homework, you will be doing the equivalent of this calculation here with the Laplacians– it's not complicated– so that you will derive with the current is

And the current must be a very similar formula as this one And indeed, I'll just write it here The current is h bar over m, the imaginary part of psi star And instead of ddx, you expect the gradient of psi That is the current for the probability in three dimensions

And the analog of this equation, d rho dt plus dj dx equals 0, is d rho dt plus divergence of j is equal to 0 That is current conservation Perhaps you do remember that from your study of electromagnetism That's how Maxwell discovered the displacement current

When he tried to figure out how everything was compatible with current conservation Anyway, that argument I'll do in a second so that it will become clearer

So one last thing here– it's something also– you can check the units here of j is 1 over l squared times 1 over t, so probability per unit area and unit time So what did we have?

We were doing the integral of the derivative of the integral given by n It was over here, dn dt We worked hard on it And dn dt was the integral of d rho dt So it was the integral of d rho dt dx

But we showed now that d rho dt is minus dj dx So here you have integral from minus infinity to infinity dx of dj dx And therefore, this is–

I should have a minus sign, because it was minus dj dx This is minus the current of x equals infinity times p minus the current at x equals minus infinity nt And as we more or less hinted before, since the current is equal to h over 2im psi star

Psi dx minus psi psi star dx, as you go to plus infinity or minus infinity, these things go to 0 given the boundary conditions that we put Because psi or psi star go to 0 to infinity, and the derivates are bound at the infinity

So this is 0, dn dt 0 All is good And two things happened In the way of doing this, we realized that the computation we have done pretty much

Established that this is equal to that, because dn dt is the difference of these two integrals, and we showed it's 0 So this is true And therefore, we suspect h is a Hermitian operator And the thing that we should do in order to make sure it is

Is put two different functions here, not two equal functions This worked for two equal function, but for two different functions, and check it as well And we'll leave it as an exercise It's a good exercise So this shows the consistency

But we discover two important ideas– one, the existence of a current for probability, and two, h is a Hermitian operator

So last thing is to explain the analogy with current conservation I think this should help as well So the interpretation that we'll have is the same as we have in electromagnetism

And there's a complete analogy for everything here So not for the wave function, but for all these charge densities and current densities So we have electromagnetism and quantum mechanics We have rho

Here is the charge density And here is the probability density If you have a total charge q in a volume, here is the probability to find the particle in a volume

There is a j in Maxwell's equations as well, and that's a current density Amber's law has that current It generates the curl of b

And here is a probability current density So that's the table So what I want to make sure is that you understand why these equations, like this or that,

Are more powerful than just showing that dn dt is 0 They imply a local conservation of probability You see, there has to be physics of this thing So the total probability must be 1 But suppose you have the probability distributed over space

There must be some relation between the way the probabilities are varying at one point and varying in other points so that everything is consistent And those are these differential relations that say that whenever you see a probability density change anywhere, it's because there is some current

And that makes sense If you see the charge density in some point in space changing, it's because there must be a current So thanks to the current, you can learn how to interpret the probability much more physically

Because if you ask what is the probability that the particle is from this distance to that distance, you can look at what the currents are at the edges and see whether that probability is increasing or decreasing So let's see that Suppose you have a volume, and define

The charge inside the volume Then you say OK, does this charge change in time? Sure, it could So dq dt is equal to integral d rho dt d cube x over the volume

But d rho dt, by the current conservation equation– that's the equation we're trying to make sure your intuition is clear about– this is equal to minus the integral of j– no, of divergence of j d cube x over the volume OK

But then Gauss's law Gauss's Lot tells you that you can relate this divergence to a surface integral dq dt is therefore minus the surface integral, the area of the current times that So I'll write it as minus jda, the flux of the current,

Over the surface that bounds– this is the volume, and there's the surface bounding it So by the divergence theorem, it becomes this And this is how you understand current conservation You say, look, charge is never created or destroyed So if you see the charge inside the volume changing,

It's because there's some current escaping through the surface So that's the physical interpretation of that differential equation, of that d rho dt plus divergence of j is equal to 0 This is current conservation

And many people look at this equation and say, what? Current conservation? I don't see anything But when you look at this equation, you say, oh, yes The charge changes only because it escapes the volume, not created nor destroyed

So the same thing happens for the probability Now, let me close up with this statement in one dimension, which is the one you care, at this moment And on the line, you would have points a and b And you would say the probability to be within a and b is the integral from a to b dx of rho

That's your probability That's the integral of psi squared from a to b Now, what is the time derivative of it? dp ab dt would be integral from a to b dx of d rho dt

But again, for that case, d rho dt is minus dj dx So this is minus dx dj dx between b and a And what is that? Well, you get the j at the boundary So this is minus j at x equals bt minus j at x equal a, t

So simplifying it, you get dp ab dt is equal to minus j at x equals bt plus j of a, t Let's see if that makes sense

You have been looking for the particle and decided to look at this range from a to b That's the probability to find it there We learned already that the total probability to find it anywhere is going to be 1, and that's going to be conserved,

And it's going to be no problems But now let's just ask given what happens to this probability in time Well, it could change, because the wave function could go up and down Maybe the wave function was big here

And a little later is small so there's less probability to find it here But now you have another physical variable to help you understand it, and that's the current That formula we found there for j of x and t in the upper blackboard box formula

Is a current that can be computed And here you see if the probability to find the particle in this region changes, it's because some current must be escaping from the edges And let's see if the formula gives it right Well, we're assuming quantities are

Positive if they have plus components in the direction of x So this current is the current component in the x direction And it should not be lost– maybe I didn't quite say it– that if you are dealing with a divergence of j, this is dj x dx plus dj y dy plus dj

Z dz And in the case of one dimension, you will have those, and you get this equation So it's certainly the reduction But here you see indeed, if the currents are positive– if the current at b is positive,

There is a current going out So that tends to reduce the probability That's why the sign came out with a minus On the other hand, if there is a current in a, that tends to send in probability, and that's why it increases it here

So the difference between these two currents determines whether the probability here increases or decreases

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