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Welcome to the Deep Dive.
Today we're tackling, well, a really fascinating chapter from Feynman, Chapter 21, Volume 3.
Yeah, the Schrodinger equation in a classical context.
It's all about superconductivity.
Right, and the goal here seems to be taking this notoriously weird quantum equation.
The Schrodinger equation, yeah.
And seeing how it explains something you can actually, you know, see or measure on a large scale.
Exactly.
We're moving from the fuzzy world of single particles to collective behavior, billions of electrons acting in concert.
Okay, let's unpack this.
So the mission is understanding how psi, the wave function, which usually just gives probabilities, how does that dictate something like a wire suddenly having zero electrical resistance?
That's the core question.
And Feynman frames it as a seminar focusing on the results, the physical intuition, rather than getting bogged down in, say, super complex math derivations.
So understanding what happens and why it happens conceptually.
Precisely.
Especially how things like the vector potential, A, influence the quantum state on this scale.
It's about translating the quantum machinery.
Okay, so where do we start?
The absolute basics.
Let's start with the quantum rules for how charged stuff flows.
The Schrodinger equation for a particle in a magnetic field is our jumping off point.
And Feynman points out something crucial right away about how electrons move between atoms.
Yeah, the amplitude, the chance for that jump to happen, it's directly affected by the vector potential, A, along the path the electron takes.
That seems subtle.
It is, but it's fundamental, and it connects directly to probability.
If we're talking about the chance of moving, we need to talk about where the probability is.
Which is psi squared, right?
Psi psi two steps.
The probability density, P.
Exactly.
The likelihood of finding the particle right here, right now.
And like any conserved quantity, if that probability decreases somewhere.
It has to have flowed somewhere else.
It can't just disappear.
That's the conservation principle.
The continuity equation.
Right.
The change in probability density over time, partial P partial T E, Tay, has to be balanced by the divergence of a probability current, J.
So partial P partial T dot.
Just like conserving charge or mass in classical physics.
It's the quantum version.
Probability flows like a conserved fluid.
This gives us a quantum way to think about electrical current.
Okay, that makes sense.
Now you mentioned magnetic fields.
This is where Feynman introduces a complication, or maybe a clarification, about a really important one, yeah.
Classically, momentum is just mass times velocity, MV simple.
But not in quantum mechanics with magnetic field?
We have to split it.
There's the kinematic momentum, MV, which is the actual physical movement, mass times velocity.
What you'd sort of measure directly.
Kind of, yeah.
But then there's the canonical momentum, or P momentum.
That's the one that appears in the Schrödinger equation as the momentum operator.
And they're math BFP plus Q math BFAA.
Why does this distinction matter so much?
We usually deal with the magnetic field B, not A.
Ah, but the wave function cares deeply about A.
Feynman stresses this.
A represents the potential energy landscape in a way.
Okay.
And here's the kicker.
If you suddenly change A, say by turning on a magnetic field, you instantly generate an electric field.
Right.
Faraday's law.
Exactly.
And that electric field immediately changes the current, which is related to MV.
The system responds to changes in A instantaneously, because A is part of that fundamental P momentum governing the wave functions phase.
So A isn't just a mathematical trick to get B.
It has direct physical consequences for the quantum state and the current.
That's the key takeaway.
A governs the phase and the phase governs the flow.
All right.
So we have conserved probability flow and momentum that depends on A.
How does this connect to the weirdness of superconductivity?
It all comes down to the charge carriers.
In a normal conductor, it's individual electrons.
But in a superconductor, it's Cooper pairs, right?
Electron pairs.
Exactly.
Electrons are fermions, normally obeying the Pauli exclusion principle.
They can't share the same quantum state.
They're antisocial.
Very.
But under the right conditions near absolute zero, they pair up mediated by lattice vibrations.
And these pairs, they act like bosons.
Which means they can all pile into the same quantum state, the lowest energy one.
Precisely.
They stop being individuals and form a single coherent macroscopic quantum state.
Like a giant collective wave function describing the whole system.
And the charge of this particle.
That's crucial.
Since it's a pair of electrons, the charge q is twice the electron charge.
2 2 ee.
Okay, 2 2 edu.
That feels like a detail that's going to be important.
Oh, it's huge.
It leads directly to the first big observable phenomenon, the Meissner effect.
Ah, the levitating magnets where the superconductor kicks out the magnetic field.
Perfect diamagnetism, yeah.
It actively expels magnetic flux.
And the math behind it, the London equation, shows that the superconducting current density, J, sets itself up to be proportional to negative A.
Jaj, it's called TransE.
Games is text constant math.
So the current automatically arranges itself to cancel any magnetic field trying to get inside.
Exactly.
It creates a counterfield.
The external field doesn't just stop dead at the surface, though.
Right, there's a penetration depth, lambda.
The field decays exponentially over that tiny distance lambda.
So for a thick superconductor, the inside is perfectly field free.
That's pretty dramatic.
But you said the q2 etia thing was really proven elsewhere.
Yes.
The Meissner effect is consistent with it, but the absolute confirmation, the sort of smoking gun, came from flux quantization.
Okay, what's that?
Does it involve a ring shape?
It does.
Imagine a ring, like a doughnut, made of superconducting material.
Now think about the wave function, psi i, for those cooper pairs within the ring.
The big collective wave function.
Right.
If you follow that wave function all the way around the ring and come back to your starting point.
The phase has to match up, doesn't it?
It has to be single valued.
Exactly.
It can't be ambiguous.
The phase must return to its original value or change by an integer multiple of two.
That's a fundamental requirement for any wave function describing reality.
Okay, so the phase has to behave nicely.
How does that constrain the magnetic field?
Remember, the phase is sensitive to the vector potential A.
The single -valued phase requirement forces the line integral of A around the loop.
Well, it can't be just any value.
It has to be quantized.
Yes.
It must be an integer multiple of two dollar bark -bark.
And since the integral of A around a loop is related to the magnetic flux, passing through that loop.
Ah, so the flux itself must be quantized.
Precisely.
The magnetic flux trapped in the ring can only exist in discrete packets, integer multiples of a fundamental unit of flux.
And that unit depends on the charge, Q.
It does.
The unit is H2.
And when experiments were done to measure this trapped flux - They didn't find the dollars.
They found integer multiples of two dollars.
Exactly what you'd expect if the charge carriers had a charge of two dollars.
Wow.
That's direct proof of the pairs.
Undeniable proof.
It beautifully ties the abstract phase requirement of quantum mechanics to a concrete measurable magnetic quantity, confirming the Cooper pair model.
Okay, that's incredible.
Phase coherence on a large scale, leading to quantized flux.
Feynman then pushes this phase idea even further with junctions, right?
He does.
He introduces the Josephson junction.
Picture two pieces of superconductor separated by a really, really thin insulating layer, maybe just 10 or 20 atoms thick.
Thin enough for quantum tunneling.
Exactly.
The Cooper pairs can tunnel across this barrier.
The system is described by two wave functions, PSI1 and PSI2, one on each side, and they're coupled.
Because of the tunneling.
Right.
And the phase difference between CS1 and TS2, let's call it delta, governs everything.
This leads to two amazing effects.
The Josephson effects.
What's the first one?
The DC Josephson effect.
Here's the weird part.
Even with zero voltage applied across the junction, a direct current, a DC current, can flow.
Just flows.
With no push.
Well, the push is the phase difference.
The current, J, is proportional to the sign of phase difference.
A current driven purely by quantum phase coherence.
That really highlights the reality of the wave functions phase.
It absolutely does.
Now, what happens if you do apply a voltage?
That's the AC Josephson effect.
Yes.
If you put a constant DC voltage, V, across the junction, something remarkable happens.
The phase difference, delta, doesn't stay constant anymore.
It starts changing linearly with time.
And since the current depends on Dac and delta, if delta is changing linearly, the current starts to oscillate.
An AC current from a DC voltage.
Exactly.
It oscillates at a very specific, very high frequency, omega, which is directly proportional to the voltage.
Omega, it feels too EVBO.
Notice the too big again.
It pops up everywhere, so the junction becomes a perfect voltage to frequency converter.
Incredibly precise.
It's actually used to define the standard volt now.
Amazing.
So we have current from phase, oscillating current from voltage.
How does this loop back to proving the vector potential, A, is real?
For that, Feynman brings in the squid,
the superconducting quantum interference device.
Sounds impressive.
It is.
Imagine taking two Josephson junctions and connecting them in parallel, forming a superconducting loop with two paths for the current.
Like a two -slit experiment, but for Cooper pairs.
That's a great analogy.
The total current that flows through this parallel setup depends on the interference between the wave functions taking the two paths.
And interference depends on phase differences.
Exactly.
And the phase difference between the two paths is determined by, you guessed it, the magnetic flux passing through the loop formed by the junctions.
Which is related to the integral of A around the loop.
Precisely.
So if you vary the magnetic field passing through the squid loop, you change the flux, which changes the phase difference, which changes how the currents interfere.
And the total current going through the squid changes.
Dramatically.
It oscillates up and down as you smoothly change the magnetic field, you get this beautiful interference pattern in the total current.
And this oscillation is measurable.
Very measurable.
And crucially, the current isn't responding directly to B, the magnetic field at the wires.
It's responding to the integral of A around the loop, which determines the flux inside the loop.
Even if B is zero along the path of the wires themselves.
Right.
This experiment provides direct measurable evidence that the vector potential A affects the quantum system's behavior, proving its physical reality beyond just being a mathematical tool for calculating B.
So wrapping this up, we started with Schrödinger's equation.
A fundamental quantum rule.
Applied it to a collective system, the Cooper pairs in a superconductor acting like bosons with charge $2.
That led to the Meissner effect kicking out magnetic fields.
And flux quantization, proving the $2 charge by showing that trapped flux comes in packets of 2E.
Then the Josephson effect showed current driven purely by phase difference.
And voltage creating oscillating currents.
Finally, the Scuderbude used interference to demonstrate undeniably that the vector potential A is physically real and directly influences measurable currents.
That's a good summary.
It's a beautiful arc, showing how core quantum principles manifest on a macroscopic scale.
It really is.
The abstract math translates directly into things we can build and measure.
Absolutely.
The quantum world isn't just hidden in atoms.
It governs these large scale electrical phenomena too.
Okay.
Here's a final thought for you to take away.
Those squids, born from understanding quantum interference and the reality of A, they are incredibly sensitive.
Extremely sensitive to magnetic fields.
Right.
They form the basis of some of the most sensitive magnetometers ever built.
We're talking about devices that can measure magnetic fields thousands of times weaker than the Earth's field.
Used in medical imaging, like magnetoencephalography to map brain activity, and in things like geological surveys.
So this deep dive into seemingly abstract quantum physics, understanding phase, and the vector potential.
It directly led to tools that let us explore everything from the human brain to the planet structure.
Quite the journey.
A fantastic example of how fundamental understanding drives innovation.
Indeed.
Well, thank you for joining us on this deep dive into Feynman's take on superconductivity.