Chapter 5: Spin One – Stern-Gerlach Experiment

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Welcome to the Deep Dive.

We're taking some really dense source material, trying to cut through the noise and just pull out the essentials you need.

Yep, the core ideas.

Today it's a big one.

We're diving deep into quantum mechanics, specifically how Richard Feynman introduces the, well, the mathematical language using the simplest possible system.

Right.

The mission really is to get a handle on this basic toolkit Feynman lays out, amplitudes, superposition, operators,

the real foundations.

And what's interesting, you know, is how he starts.

He doesn't really ease us in with classical ideas we might already grasp, like say angular momentum in the old sense.

No, he jumps straight into this thing called the spin one particle.

And that's the prototype.

It's kind of revolutionary because spin itself, it's purely a quantum thing.

No classical spinning top equivalent.

Not really, no.

The key is this particle exists in just three specific discrete states.

That's it.

Three possibilities relative to any axis you choose to measure along.

It's like the bare bones machinery of quantum mechanics.

Okay.

So if we want to see this machinery in action, we need a specific tool.

Let's talk about the Stern -Gerlach apparatus.

Ah, yes.

The quantum sorter.

So imagine a magnetic field, but it's not uniform.

It's specifically designed to be inhomogeneous, meaning it changes strength across its space.

And that gradient is key.

Exactly.

When you shoot a beam of atoms through it, this uneven field sorts them out based on their spin property along the direction of the field gradient.

Let's call that the z -axis or, well, the s -axis for this first setup.

Right.

And when we actually do this, you know, send a beam of these spin -one atoms through this s -apparatus like Feynman describes in section 5 -1.

What happens?

You don't get a smooth spread.

No, not at all.

Yeah.

Which classically you might expect maybe instead, bang, it splits into three separate distinct beams.

Three.

Why only three?

Because these spin -one particles, when measured by this s -apparatus, they only have three possible values for that z -component of their angular momentum.

That's it.

So we label them.

We label them the plus state, often written plus s -docs, the zero state, and the minus state where s.

And that discrete outcome, that sharp split, that's our first really clear sign we're not in Kansas anymore, classically speaking.

OK.

So this split is useful.

It means we can isolate one type of atom, right?

Create a pure state.

Precisely.

You can literally just put physical masks, you know, little blocks, in the path of two of the beams.

So if we block the zero and minuses beams...

We're left with just a stream of atoms, and every single one of them is in the plus s state.

It's a filtered beam, a pure state, relative to s.

And that's crucial for what comes next.

And how do we know it's really pure?

Ah, well, you check it.

You take that filtered plus s beam and immediately send it through a second s -apparatus, absolutely identical to the first one.

And what happens?

100%.

Every single atom goes straight through the plus path of that second device.

Nothing comes out the zero or minuses channels.

OK.

So that confirms it.

Once it's in the plus s state, it stays in the plus s state if you measure along the same s -axis again.

Exactly.

Pure state confirmed.

Now, you mentioned notation, because drawing magnets and blocks all the time would get, well, tedious.

Extremely tedious.

So Feynman introduces this shorthand, these bracket symbols.

You know, the vertical bars, the angle brackets.

The bracket notation.

Right.

And let's just talk about the state itself, like plus s, and the transition, say, from plus s to plus s without getting bogged down in the hardware.

It shifts focus to the quantum state.

And what happens to it?

OK, makes sense.

Keep it abstract.

Now, hold on, because here comes the quantum twist.

This is the experiment that really messes with classical thinking.

Uh -oh.

What happens if we take our pure, perfectly prepared Ni3 plus s little beam?

Yep.

Got that.

And we send it into a second Stern -Gerlach apparatus, let's call it t, but this t -apparatus is tilted by some angle, say alpha, relative to the first s -apparatus.

Wait a second.

We know the atom is in the L plus s state.

We just proved it's pure.

Why would tilting the second filter change anything?

Shouldn't it just, I don't know, maybe still come out a single path related to Pufferman plus?

Ah.

But that's the key.

The state plus s is pure only with respect to the s -axis.

Its purity is defined by that measurement direction.

So when it hits the t -apparatus, which is measuring along a different axis, that atom is suddenly not in a pure state relative to t anymore.

So it can come out any of the t -channels.

It now has a certain probability, a chance, of emerging in the plus t state, or the t -t state, or the t -state.

The originalized purity kind of dissolves the moment you measure along a different direction.

Okay, so purity is relative to the measurement axis.

That's a big takeaway.

And you said probability.

This must be where things get mathematically different from just counting coins.

Exactly.

This is where Feynman introduces the core concept, amplitudes.

In quantum mechanics, for any possible transition, like going from state A to state B, we calculate a complex number.

That's the amplitude.

A complex number, not just a percentage chance.

Not directly.

The amplitude, let's call it A A to B, is a complex number.

It has a magnitude and a phase.

The actual probability of that transition happening is found by taking the absolute value or magnitude of that complex amplitude and squaring it.

So probability D to B2.

Complex numbers.

Why the extra complexity, literally?

Why the phase?

Well, that phase part is absolutely essential.

It's what allows for quantum interference, which is coming up soon.

It means possibilities don't just add up.

They can interact, cancel each other out, or reinforce each other because of these phases.

It's not just about how much possibility, but how the possibilities relate.

Okay, so for our S to T transition, we started in one S state, say not plus S, and it could end up in any of the three T states, plus TT.

So there are three amplitudes there.

Right.

And since we could have started in plus S dollars, and each could potentially transition to any of the three T states.

Ah, that gives us nine possibilities in total.

Three starting, three ending, nine amplitudes.

Exactly.

And these nine complex numbers, they capture everything about what happens when you rotate the measurement axis by that angle alpha.

We can arrange them neatly into a three by three grid, a matrix.

A transformation matrix.

Precisely.

And this isn't just, you know, a convenient table.

That matrix is the mathematical representation of the physical process of rotating the measurement basis.

Its specific numbers depend entirely on that tilt angle alpha.

It dictates how an S state description transforms into a T state description.

Got it.

The matrix embodies the rotation, the physical change in perspective.

Okay, let's back up slightly and solidify this idea of base states from section 5 -4.

You're saying the set of states we get from one Stern -Gerlach device, like our plus seller dollar states along the S axis, they form a complete set.

Yes, a complete set of base states for describing the spin of this particle along that axis.

The really powerful idea is that any possible spin state, no matter how weird or mixed up it seems, can be written down mathematically as a combination, a superposition of just these three base states.

Like mixing primary colors to get any other color.

Kind of like that, yeah.

But here, the mixing coefficients are those complex amplitudes we just talked about.

And that leads us straight into the really non -intuitive part.

Quantum interference, covered in section 5 -5.

Imagine a slightly more complicated experiment now.

Okay, laid on me.

We start with our pure plus S beam again.

We send it into a T apparatus, the tilted one.

But this time, we make the T apparatus wide open.

Wide open.

What does that look like?

Do we just take the blocks away?

Yeah, essentially.

The T apparatus still splits the incoming beam into the three paths, the plus key and T channels.

But we don't stop any of them.

We let all three beams travel a bit, and then crucially, we recombine them back into a single beam before doing anything else.

So split, then rejoin, without checking which path any atom took.

Exactly.

No checking.

Then, after recombination, we send this beam into a final filter, say, another S apparatus.

Maybe labeled S prime, S app.

Okay, so the path is, pure your complex,

split by T, recombine measure with S.

What's the point?

This setup perfectly demonstrates superposition.

An atom, starting as phi, could potentially reach the final S detector by going through the D plus T intermediate path, or the DT path, or the T path.

Three possible routes inside the T apparatus.

Right.

Now, how do we calculate the total chance of, say, starting at plus S, and ending at plus SS?

In classical physics, if you had probabilities for each path, you just add them up.

Probability path one, plus probability path two, plus probability path three.

Makes sense.

More paths, more chances.

But not in quantum mechanics.

Here, the rule is, the total amplitude for the overall transition, S to S, is the sum of the amplitudes for each individual path.

S to plus T to S, S to zero, T to S, is S to T to S.

We're summing the complex amplitudes again, not the probabilities.

And because amplitudes have phases, they can interfere.

The amplitude for the plane plus T path might partially cancel the amplitude for the TD graph, for example, or they might reinforce each other.

So the final probability, which is the square of the total summed amplitude,

could be much larger or much smaller, even zero, than you'd expect just by adding the probabilities for each path.

Precisely.

That difference is quantum interference.

It arises directly from summing amplitudes instead of probabilities.

But there's a catch, right?

You said no checking.

A huge catch.

This whole interference thing, the summing of amplitudes, it only works if those intermediate paths, the plus T channels,

are fundamentally indistinguishable.

If you put any kind of detector in there, even a hypothetical perfect one, to see which path the atom actually took.

The interference vanishes.

Instantly.

The moment you gain which path information, the superposition is destroyed, the atom is forced to choose a path, and then the probabilities just add up classically.

The quantum weirdness disappears.

Measurement collapses the possibilities.

Okay, so let's try to pull this all together.

Feynman uses this simple spin -one system to build up the essential rules, the real machinery of quantum mechanics, as laid out in 5 .6.

Right.

We've basically uncovered three core components.

First,

the state.

The quantum state of our particle isn't just one number.

It's represented by that set of three complex amplitudes.

The potential for finding it in each of the three base states, plus s, ss, those three numbers define the state.

Think of them as coordinates in a state space.

Second, the apparatus or operator.

Any physical thing we do to the system, a filter, a rotation, interacting with the field, is represented mathematically by a matrix, a transformation matrix.

And this matrix acts on the state vector, those three amplitudes, to transform them into a new set of amplitudes representing the state after the interaction.

It's an operator changing the state.

Okay.

And third, compound apparatus.

What if we do things in sequence?

If you have one operation, say, apparatus A, followed by another, apparatus B, the overall effect, the transformation for the combined system C, is found by multiplying their matrices.

Matrix multiplication.

But you have to be careful about the order, right?

Absolutely critical.

The order matters.

The matrix for the combined operation C is the matrix for B multiplied by the matrix for A.

So D $E is B times A way.

The operation that happens last, B, comes first in the multiplication.

Why is that order so important?

In everyday life, putting on socks then shoes is the same result.

No way it's not.

Shoes then socks is messy.

Hey, good analogy, actually.

In quantum mechanics, if the matrix is for A and B don't commute, meaning if A $B gives a different result than D $A, then the physical outcome genuinely depends on the sequence you perform the operations or measurements in.

So measuring spin along X, then spin along Z gives a different result than measuring Z than X.

Generally, yes.

That non -commutativity is fundamental.

It's baked into the mathematical structure, and it reflects a deep physical reality about quantum measurements interfering with each other.

It prevents us from knowing everything simultaneously with perfect certainty.

And one last piece, changing bases.

Section five to seven.

We chose S and T axes kind of arbitrarily.

The physics shouldn't depend on our choice of coordinates, right?

Exactly.

Physics has to be consistent.

So there must be and is a specific transformation matrix that lets us convert our description of a state from the S -based state amplitudes to the T -based state amplitudes.

So we can translate between perspectives.

Right.

It ensures the underlying quantum state is the same reality, just described using a different equally valid set of base states.

So let's recap the huge foundation Feynman built just using this spin -one particle.

Three massive ideas emerge.

One, quantum states are fundamentally described by complex amplitudes relative to a chosen set of base states.

Two, when multiple pathways are possible and unobserved, the amplitudes for those paths add together, leading to interference.

It's amplitude summation, not probability summation.

And three, physical processes, measurements, interactions, they're all represented by operators, mathematically matrices, which trans from the state amplitudes.

And importantly, these operators often don't commute.

The order matters.

So for you, the learner grappling with quantum mechanics, this spin -one example, though abstract, gives you the essential mathematical toolkit.

Amplitudes, superposition, operators, matrices, these are the core tools needed for any quantum system, however complex it gets later.

It's the fundamental grammar of the quantum language.

OK, that brings us to a final thought to leave you with.

This whole discussion, all these rules about summing amplitudes, it relied on starting with a pure state, like R perfectly filtered plus S beam.

Right, a state where we had definite knowledge relative to the S axis.

But what if we didn't have that?

What if our source just produced atoms with completely random spin orientations, a statistical mixture and not a pure state?

How would that change things?

How would the rules of just adding classical probabilities for that initial random mix compare with the quantum rules of adding amplitudes that we've just painstakingly derived?

That difference between dealing with a known pure state versus an unknown statistical mixture highlights the boundary between where quantum amplitude rules dominate and where classical probability rules might seem sufficient, but are ultimately incomplete.

Something to think about.

Definitely something to chew on.

Thank you for joining us for this deep dive into Feynman's foundations.

We hope this makes that quantum machinery feel a little less daunting.

Yeah.

Thanks for listening.