Chapter 6: Spin One-Half & Rotations in Quantum Mechanics

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Welcome to the Deep Dive.

We're the show that takes on challenging texts and pulls out the core ideas for you.

And today, we're jumping into something truly fundamental in quantum mechanics,

particle spin.

Spin is this intrinsic property like charge, but it's angular momentum.

It defines particles like electrons.

Our mission is to really dig into chapter six of the Feynman lectures on physics,

volume three, the part called spin one half.

We want to understand the logic, you know, not just look at equations.

We're aiming to unpack Fahman's core mathematical reasoning for how particles quantum state.

Well, how it changes when you simply rotate the coordinate system you're using to look at it.

Right.

And this chapter, it's just a masterpiece of abstract thinking.

It shows how basic ideas about space like symmetry are incredibly deeply tied to the quantum properties of these tiny particles.

We're going to try and visualize how these transformations actually work, starting with the basic idea of transforming amplitudes.

So what are we actually hunting for here?

Is it this transformation matrix?

Exactly.

That's the goal.

The matrix that lets us convert these probability amplitudes from one set of base states, say, are starting set up to another set defined by the rotated system.

Okay.

So to get there, we have to start with the absolute basics of quantum mechanics, the idea of base states.

Think of them as the fundamental sort of orthogonal options for a particle's property.

For spin one half, it's simple.

Spin up or spin down along some axis.

For two possibilities.

For a given axis, yes.

And any actual state the particle is in, its complex reality is described as a combination of those base states.

Each base state has a weighting factor called an amplitude.

And that amplitude tells us the chance of finding it in that state.

Almost.

It's a complex number.

The square of its magnitude gives you the probability, but the amplitude itself with its phase is crucial for quantum effects.

Okay.

I think I follow.

So if a particle starts in some state, let's call it S, and we want the amplitude for it ending up in state T, how do we calculate that?

Do we go straight from S to T?

That's the key quantum rule.

You can't just jump directly.

You have to consider all the possible intermediate base states it could sort of pass through.

The total amplitude from S to T is the sum of all paths.

You take the amplitude to go from S to an intermediate state, multiply it by the amplitude to go from that state to T, and then sum those products up for all the intermediate states.

Okay.

Summing over all possibilities.

That sounds fundamental.

So if we have our original base states from system S and a new set from the rotated system T, we need something to connect them mathematically.

Precisely.

And that's where these transformation coefficients come in.

They're often written as R sub i.

For spin one half, there are basically four key numbers.

Four numbers.

Why four?

Well, you have two base states in S, up down, and two in T, up down.

You need to know how much of S up is in T up, how much of S on is in T up, and how much of S down is in T down.

Four connections.

Got it.

And these four numbers form the rotation matrix we're looking for.

Exactly.

They were the rotation matrix for this system.

And the reason this whole thing matters is symmetry.

If we rotate our lab equipment, the physics shouldn't break.

The math has to follow that physical rotation smoothly.

That's the core idea.

We're using the symmetry of space itself to figure out the rules of quantum mechanics for spin.

Okay.

Let's make this more concrete.

Feynman uses the improved Stern -Gerlach apparatus.

Can you describe that for us?

Sure.

Imagine a device, basically a special magnet, creating a non -uniform magnetic field.

When you send a beam of spin one half particles, like silver atoms, through it, the field grabs onto the particle's spin.

It splits the beam cleanly into two distinct paths.

One path corresponds to particles with spin up along the field's direction.

We call that the plus plus state.

The other is spin down, the minus state.

So it's like a spin filter.

Exactly.

A perfect filter and sorter based on spin orientation relative to the apparatus's axis.

Now the thought experiment, we put two of these in a row, right?

Let's call them s and t.

Yep.

Apparatus s goes first.

Let's say we set it up to only let the upstate particles through.

We block the down channel.

So coming out of s, we have a pure beam.

All particles are spin up relative to s's axis.

Okay.

A prepared state.

A prepared state.

Then this pure beam immediately enters the second apparatus, t.

But here's the key.

We physically rotate apparatus t relative to s, maybe tilt it sideways or spin it around.

Ah, okay.

So t is measuring spin along a different axis than s prepared it along.

Precisely.

And the question is,

when those s prepared up particles go into the rotated t apparatus,

what are the amplitudes and thus the probabilities for them to come out in t's up channel versus t's down channel?

So the measurement t makes is essentially finding how much the original s upstate overlaps with the t up and t down states.

You got it.

And since we started with s up, and implicitly s down was zero, and we have the two outcomes in t, figuring out these outcomes experimentally or theoretically gives us two of those transformation coefficients.

If we'd started with s down, we'd find the other two.

The physics of the split beams directly tells us the numbers in that rotation matrix.

Very clever.

Now let's simplify.

Let's rotate t just around the z axis, the same axis s was aligned with.

We rotate it by an angle.

Let's call it fymify.

Okay, that seems like the simplest possible rotation.

What happens then?

Does it mix up and down?

No, intuitively it shouldn't, right?

If you're measuring along the same direction, up should stay up and down, should stay down.

And it does, mostly.

The main effect of this z rotation, as Feynman shows, is on the phase of the amplitudes.

The magnitudes don't change, meaning the probabilities p plus and p stay the same.

So rotating the detector around the beam axis doesn't change the chance of finding it up or down.

That makes sense.

But the phase changes.

Yes.

The amplitude c plus gets multiplied by a phase factor, like e lambda plus and c by e lambda.

And Feynman finds this phase shift, lambda is directly proportional to the rotation angle handful,

specifically lambda m phi, where m is the spin quantum number along z, which is plus 12 for the upstate and negative 12 for the downstate.

Lambda pm frac one two.

Okay, seems straightforward so far.

But here is where quantum mechanics throws us a curveball, a really big one.

Think about rotating something in our everyday world by 360 degrees.

It comes back to where it started, looks exactly the same.

Exactly.

You'd expect the physics, the description, everything to be identical.

But what happens to the spin one half amplitudes if we rotate the t apparatus by phi 360 circ?

Let's say using that formula, lambda pm frac one two times 360 circ, the phase factor is e p m i 180 plus i sin and that's minus one.

Precisely.

After a full 360 degree rotation of the coordinate system, the amplitude c plus becomes e lambda plus and c becomes c at i.

They flip their sign.

Whoa.

Okay, wait.

The amplitudes flip sign after a 360 rotation.

How can that be?

Doesn't that change the physics?

That's the crucial point.

It doesn't change the measurable physics because the probability depends on the amplitude squared or technically the amplitude times its complex conjugate two c toss two.

Ah, right.

And c two toes is the same as t two so two.

So the probability of measuring up or down is unchanged after 360 degrees.

The physical outcome is the same, but the mathematical object describing the state, the amplitude itself is fundamentally different.

It's picked up a minus sign.

So to get the amplitude itself back to its original positive value, you'd need to rotate another 360 degrees.

Exactly.

You need a full $720 rotation to bring the amplitude back to its starting point.

This $720 six to two tapody is a hallmark, a defining characteristic of spin one half particles.

It's deeply weird and has no classical analog.

It's like the particles internal reference frame doesn't match up with our 360 degree space in the expected way.

That's a good way to put it.

It stems directly from that factor of 12 in the spin quantum number.

Okay.

Mine's slightly blown by the $720 thing.

Let's change the rotation axis.

What about rotating apparatus t around the y axis perpendicular to the initial z axis?

It starts simple.

180 degree rotation around half.

Okay.

Visualize that.

If s defines the z axis as up rotating t by 180 degrees around the perpendicular y axis means t's up direction.

It's c's axis is now pointing exactly opposite to s z axis.

So what was up for s is now down for t and what was down for s is now up for t.

Exactly.

The physical transformation is clear.

So the new amplitude for being up in t plus plus must be related to the old amplitude for being down in s and the new down must relate to the old up plus plus.

So the transformation matrix would have zeros on the diagonal and non -zero terms off diagonal.

Correct.

Feynman shows the relations are $6 plus msc2 and $c plus dollar.

Notice that minus sign crept in again.

It's required by the consistency of rotation rules.

Okay.

The 180 flip is manageable.

What about the trickier one?

90 degrees around the y axis.

Now t's axis is pointing along what s called the x axis.

Right.

Here simple swapping doesn't work.

You can't just say up becomes sideways.

We need to actually calculate those for our coefficients.

A B C D D as Feynman legals them using the mathematical properties of rotation.

It involves solving a system of equations based on how rotations combine, right?

Yes.

You use the fact that doing two 90 degree rotations should equal the 180 degree rotation we just figured out plus other consistency conditions.

When you grind through the math.

What falls out?

What are the coefficients for a 90 degree y rotation?

The result is really elegant and important.

All the coefficients A B C D D turn out to have a magnitude of $1 score.

Specifically $1 score and $1 C1 score too.

Or variations depending on sign conventions, but the magnitude is key.

$1 score.

What does that mean physically?

If we sent in a pure s upstate, $1 C $2.

Then the new amplitudes in t are C $ plus A $ plus A $0 .001 score.

So the amplitude is $1 for t up and $1 scored it for t down.

No.

But the probabilities are $1 scored at 12 .2.

Exactly.

If you prepare a particle spin up along z and then measure it along the x axis, which is z rotated 90 degrees around y, you have a 50 % chance of finding it up along x and a 50 % of finding it down along x.

The rotation directly creates a perfect quantum superposition.

That's a direct consequence of the geometry linking these states.

So we've handled z rotations and specific i rotations, but what about rotating by any angle around any arbitrary axis?

How do we generalize?

Yeah, like rotate 23 degrees around x, then 50 degrees around i, then 10 degrees around z.

How does that work?

That's where the full power of the mathematical framework comes in.

We use Euler angles.

Any arbitrary orientation of apparatus t relative to s can be described by a sequence of three specific rotations.

Usually one around z, then one around the new y -axis, then one around the final z -axis.

Let's call the angles alpha, beta, gamma.

Okay, so we can build up any complex rotation from simpler ones we already understand.

Precisely.

By mathematically combining the rotation matrices for these individual steps, which we derive the principles for, we can construct the single transformation matrix R that represents the total arbitrary rotation defined by alpha, beta, gamma.

And Feynman derives this general formula, equation 6 .36, in the text.

When you look at that final general result for the transformation amplitudes, six of dollars plus and six, what's the most striking feature?

The thing that immediately jumps out is that the formulas involve trigonometric functions, not of the rotation angles themselves, alpha, beta, gamma, but of half those angles.

That connects directly back to the 720 degree rotation, doesn't it?

Absolutely.

It's the ultimate confirmation that half -angle dependence is the mathematical signature of spin one -half systems under rotation.

If you plug in a rotation angle of 360 degrees for beta, you're dealing with ages again, which brings in that crucial factor of minus one.

You need betas of 120 circuit plus little one.

The entire structure holds together.

The half -angle dependence mathematically requires the 720 degree property for the amplitudes to return to their original state.

We started with basic symmetry arguments and ended up with a complete mathematical description.

The rotation matrix R linking physical space rotation to the quantum state transformation.

Those final tables in the chapter, table six one and six two,

they just summarize these definitive rotation matrices for X, Y, and Z axis using these half -angle formulas.

Yes, they are the culmination.

They provide the explicit rules for how spin one -half states transform under any rotation.

That's the payoff for the whole chapter's logical development, hashtag tag outro.

Stepping back, what does all this really tell us?

I think this dive shows that the mathematical rules for spin one -half aren't just made up.

They are the necessary consequence of demanding that quantum mechanics respect the symmetries of ordinary space.

The way spin sees rotation is fundamentally tied to the geometry of rotation itself, but in this peculiar quantum way that's completely alien to our classical experience.

Yeah, we really went from basic concepts like amplitudes and base states all the way to this incredibly counterintuitive yet mathematically solid description involving half angles and the 720 degree cycle.

It connects the abstract math to the physical act of rotation.

And it leaves you thinking.

We established that the 360 degree rotation flips the sign of the amplitude, but the probability is okay.

Right.

But quantum mechanics is full of interference effects where amplitudes add or subtract.

These effects depend critically on the relative phases, the signs of the amplitudes.

So a really interesting question to ponder is could you design an experiment maybe recombining two beams of spin 12 particles where one beam has undergone a 360 degree rotation relative to the other.

Would that sign flip, that phase difference show up as a measurable change in the interference pattern, even though the probabilities for each beam separately are unchanged?

Exactly.

What are the physical consequences of this fundamental phase ambiguity inherent in spin 12 rotations?

It's a deep question about the nature of quantum reality.

Definitely something fascinating to chew on.

Well, thank you for joining us on this deep dive into the truly unique world of quantum spin and how it experiences rotation.

We hope this walk through Feynman's logic helps you connect those dots.