Chapter 7: Time Dependence of Amplitudes

Welcome to Last Minute Lecture.

This free chapter overview is designed to help students review and understand key concepts.

These summaries supplement, not replace, the original textbook and may not be redistributed or resold.

For complete coverage, always consult the official text.

Okay, let's unpack this.

Imagine trying to find like the ultimate rule book for how everything in the universe behaves over time.

Not just where a particle is, but the, I guess the precise rhythm of its potential existence.

That's pretty much our mission today.

Exactly.

We're diving deep into the dependence of amplitudes on time from, well, a really foundational text on quantum mechanics.

We're zeroing in on that probability amplitude, you know, the complex number, square it, and you get the probability of finding the particle.

We want to know how it evolves.

How it changes.

Yeah, how energy, momentum, potential, even spin kind of dictate this flow through time.

And for you listening, what we're aiming for here is a clear step -by -step breakdown.

We're going to try and translate the logic of the equations into something more conversational.

Without getting totally bogged down in the math symbol.

Exactly.

Focusing on how change happens at this fundamental level.

We have to start with the most basic thing.

The link between a system having a definite energy and how its phase just constantly varies.

That's the bedrock you're saying.

It really is.

That single idea.

It pretty much governs everything else that happens dynamically in the quantum world.

Okay, so let's start simple.

Stationary states.

What are we talking about?

Right, so picture maybe an electron stuck in an atom.

It's stable.

It has one specific definite energy.

Let's call it Eli.

Okay.

Now, the really remarkable thing is if you go looking for that electron,

the probability of finding it at any particular spot,

it just doesn't change over time.

It's constant.

Independent of time.

Yeah, so it sounds like nothing's happening, right?

Seems static.

It does.

But if you can peek underneath, at the amplitude itself, oh, it's buzzing with activity.

Okay, so the probability is constant, but the amplitude isn't.

Exactly.

The amplitude at any time is just the original amplitude multiplied by this little spinning factor.

That's the fundamental time dependence phase factor.

Okay, E dollar to the minus eight RG over H bar times two hours.

And if the probability isn't changing, that must mean this factor only fiddles with the phase, the angle of the complex number, not its size.

You got it.

Yeah.

The magnitude stays the same, only the phase rotates.

The energy dollars basically sets the speed of this rotation.

It's like an internal clock for the system.

An internal phase clock.

That's a good way to put it.

And when you square the amplitude to get the real world that rotation, that phase part, it cancels itself out.

Poof.

Leaving the probability distribution fixed in space.

Hence, stationary state.

Okay, that makes sense.

But speaking of energy,

what about rest mass energy?

This is usually a huge number.

It is.

But in atomic physics, we often just ignore it.

Seems like a pretty big omission.

How do we get away with that?

It's a great question.

And it actually shows off how this phase factor works.

Look, if including mixed U2N just adds a constant energy chunk to everything,

then it means all the amplitudes of your system get multiplied by the same extra phase factor.

EIMC2's probability only cares about the relative phases or the absolute square.

If everything gets the same twist, it doesn't change the interference pattern or the final probabilities.

Precisely.

It's like deciding sea level is your zero point for height versus the center of the Earth.

Your height differences, the physics,

stay the same.

We just shift the zero point of energy for convenience in atomic stuff.

Exactly.

Doesn't change the physics.

But that neat trick only works if there's one energy.

Right.

So the dynamics, the visible change, must come back when you have more than one energy state involved.

Now you're talking.

What if our particle isn't in just one state, but it's in a super position?

A mix, say, of two states with different energies.

E $1 and E210.

Okay, so now we have two different amplitudes.

Each with its own phase clock.

One taking it according to E $ or the other according to T $2.

They're running at different rates.

And the total amplitude is the sum of these two.

Yes.

And when you add two waves that are oscillating at different frequencies.

Interference.

Interference.

You get beats.

The total probability isn't constant anymore.

It oscillates over time.

So the probability of finding the particle actually shifts around.

It does.

The system is visibly changing.

Sort of sloshing back and forth between the properties of state one and state two, because their phases are going in and out of sync.

Okay, so that's time evolution for something kind of stuck, like in an orbit.

What about a particle just moving, you know, flying across space?

Good transition.

We need to bring in momentum.

How does that basic phase factor, the e -bow thing, handle motion?

Does it still connect to like regular classical velocity?

It does, but it needs help.

It adapts by combining the time variation with a spatial variation.

The amplitude now has to vary in space, too, not just time.

Ah, so it depends on position.

Six dollars.

Right.

We essentially add a term related to momentum, pairs, into the phase.

So the whole factor looks something like EI.

It weaves together time and space variation.

Okay, so the E $ part still gives the frequency in time, the mega.

What does the pay -a -part do for space?

The pay -a -part gives it a wave number.

It determines how rapidly the phase changes as you move through space.

Basically, it sets the wave length.

Pay -al is taller world.

Got it.

Frequency from energy, wave number from momentum.

But here's something that always felt a bit weird.

Quantum mechanics says particles are waves, right?

And mathematically, a pure wave with a single frequency and wave length.

It's infinite.

It fills all of space.

That's true.

But, you know, a real particle, like an electron we shoot, it's localized.

It's here, not everywhere.

How does an infinite wave description turn into a localized moving thing?

That's the crucial point.

We never actually deal with a single, pure, infinite wave in reality.

What we observe, what behaves like a particle, is always a group of waves.

A wave packet.

A packet.

How's that formed?

It's formed by adding up, superimposing, many different amplitudes.

They all have slightly different energies and slightly different momenta, clustered around some average values.

Okay, so you mix a bunch of similar waves.

And through interference, they mostly cancel each other out everywhere except in one small region.

That localized disturbance is the wave packet.

That's our particle.

That packet moves.

Yes.

And here's the really neat part.

You can calculate the speed of that packet, the speed of the overall envelope.

It's called the group velocity, five velocities.

And when you do the math, you find that the group velocity, V dollars, which is how the frequency changes with the wave number, turns out to be exactly equal to P in car.

PM now, wait, that's the classical velocity?

Bingo.

Incredibly satisfying result.

It shows that the wave description naturally leads to the correct classical motion.

But only if you think in terms of these localized wave groups, not single infinite waves, it bridges the quantum and classical views perfectly.

Alright, we've got time evolution, we've got motion.

Now let's bring in forces, or really potential energy, V dollars.

Right, because particles usually move in some kind of field or landscape.

So the big picture is always about total energy conservation, isn't it?

The total energy, let's call it E dollars, which is the kinetic energy, plus the potential energy, five dollars, that's always constant for the particle.

Always.

And critically, the time variation of the amplitude, that phase clock speed, it's always governed by this total conserved energy, two dollars.

Not just the kinetic part.

Okay, so two dollars, that's the master clock rate?

Exactly.

That's the universal rule.

Now think about what happens when a particle moves into a region where the potential energy dollar changes.

Well, if dollar has to stay constant and dollar goes up, then the kinetic energy must go down.

Right, and lower kinetic energy means lower momentum, two dollars.

Okay, so the particle slows down, how does the wave see that?

Lower momentum, two dollars, means a smaller wave number.

And a smaller dollar means a longer wavelength.

Ah, so the wave literally stretches out as the particle slows down in a higher potential region.

That's right, the wave physically adapts to the potential landscape.

Okay, but what happens if the potential energy dollar gets really high?

Like, higher than the particle's total energy, two dollars.

Now we're getting to the really quantum weirdness.

Classically, that's impossible, right?

Yeah, kinetic energy would have to be negative.

If you roll a ball up a hill and it doesn't have enough total energy to reach the top, it just stops and rolls back.

It can't go into the negative kinetic energy zone.

Classically, it hits an impenetrable wall,

but quantum mechanically.

What happens to the math?

One dolly equals P2 to M plus V.

If violer, then P2 objects be negative.

Which means the momentum dollars has to become an imaginary number.

Imaginary momentum?

What does that even mean for the wave?

Well, remember the spatial part of the wave goes like eat by.

If paytowers is imaginary, say P, P I call their cap is real, then that factor becomes P tap 4.

Both eight.

That's not an oscillation anymore, that's an exponential decay.

Exactly.

Inside the barrier, where the wave amplitude doesn't oscillate, it just dies off exponentially.

It gets weaker and weaker the deeper it goes.

But it doesn't instantly go to zero at the edge.

No.

It penetrates into the barrier, decaying as it goes.

So, if the barrier is thin enough… Then the amplitude might not decay all the way to zero before it reaches the other side.

There could be a tiny, but non -zero amplitude left over.

Which means a non -zero probability of finding the particle on the far side of the barrier.

Even though it classically didn't have enough energy to get over it.

That's quantum tunneling.

It just leaks through.

Wow.

Is there a real -world example of this?

Absolutely.

The classic one is alpha decay in radioactive nuclei, like uranium.

An alpha particle inside the nucleus is trapped by a huge potential energy barrier created by the nuclear forces.

Classically, it should be stuck forever.

Right.

But quantum mechanically, there's an incredibly small but non -zero probability that the alpha particle's wave function will tunnel out through that barrier.

And because the probability is so tiny, it takes a very long time on average for it to happen.

Billions of years for uranium.

Yeah.

That's why these nuclei are radioactive over such long timescales.

They're constantly, slowly leaking via tunneling.

Okay.

That's mind -bending.

Let's shift gear slightly.

How do classical forces fit into this wave picture?

We talked about potential vive dollars.

How does a change in vive of dollar create a force quantum mechanically?

Let's think about a particle moving along, say, the x direction.

But imagine the potential energy dollar isn't uniform sideways.

Maybe it increases as you go in the wave direction.

Okay.

So there's a gradient, partial v, partial ea.

Classically, that gradient is the force in the a direction, five f high of particle v, partial ea.

They'll push the particle sideways.

Right.

Now, quantum mechanically, think about the wave front of our particle wave.

It's moving mostly in x, but it has some width in y.

Okay.

On the side where vive is higher, the total energy dollars is the same, but the local momentum dealer must be slightly lower, remember?

Right.

Higher dollar means lower kinetic lower dollars.

So the part of the wave on the higher v side travels effectively a tiny bit slower than the part on the lower v side.

Ah.

It's like one side of a tank track slowing down, making the whole tank turn.

That's a fantastic analogy.

The wave front tilts.

The surfaces of constant phase get refracted.

So the direction of the wave changes, it gets deflected.

Exactly.

And the amount of this tilt, the angle of deflection, delta theta, can be calculated directly from how much the phase accumulation differs across the wave front due to that potential gradient.

You don't even need to talk about force.

It just comes from the wave adapting its phase and momentum locally to conserve total energy.

Precisely.

It's purely a wave phenomenon driven by phase shifts.

And does this wave picture give the same answer as the classical force calculation?

Yes.

And this is another beautiful connection.

When you work through the math, the angle of deflection derived from these wave phase arguments turns out to be exactly the same as the deflection you'd calculate using Newton's laws in the classical limit.

What's the classical limit here?

It's when the wavelength of the particle is very, very small compared to how quickly the potential dollars is changing in space.

So for macroscopic objects where wavelengths are absurdly tiny, the quantum calculation just melts perfectly into the classical one.

It does.

It shows that quantum mechanics doesn't just replace classical mechanics.

It contains it as a special case.

It's the deeper, more fundamental theory.

All right.

We've covered position, motion, potential barriers, forces.

What about intrinsic properties like spin?

Good point.

Let's consider spin, probably the most famous quantum property besides position and momentum.

Take a simple case,

a spin one -half particle like an electron or a muon.

Okay.

Let's put it in a uniform magnetic field, Dural dollar pointing, say, along the z -axis.

What does the magnetic field do to the electron's energy?

It adds a potential energy, Dor, that depends on the spin orientation relative to the field.

If the spin is up, aligned with B, the energy is shifted one way.

If it's down, anti -lined, it's shifted the other way.

One dollar, where Wulmus is the magnetic moment.

Okay.

So spin up state has one energy.

Let's call it eibow, and spin down has a different energy, eibow.

A two distinct energy states based purely on spin orientation.

Wait a minute, two distinct energy states.

This sounds familiar, like section one.

It is.

We're back to having two different phase clocks running at different rates.

The amplitude for being spin up evolves with frequency ebar, and the amplitude for spin down evolves with frequency ebar bar.

So what happens if the particle starts out not purely up or down, but in a superposition?

Like maybe its spin is pointing sideways in the x -direction?

That x -direction state is a superposition of up and down.

It's a specific mix of the two.

Okay.

So we start with that mix.

Then the up part starts evolving its phase at one rate, and the down part evolves its phase at a different rate.

Right.

They drift out of sync with each other.

Which means the specific combination that corresponds to spin in the x -direction won't stay stable.

Exactly.

The relative phase between the up and down components changes over time.

And if you calculate the probability of finding the particle still in that initial spin x state.

It must oscillate.

It does.

It goes up and down sinusoidally.

The probability of being spin plus x varies as Ducospeedy part.

So the particle spin isn't fixed sideways anymore.

What is it doing physically?

It's precessing.

The spin axis is actually rotating around the direction of the magnetic field, the z -axis.

This oscillation in the probability of finding it along x is the mathematical signature of that physical rotation.

And the frequency of that oscillation or that precession depends directly on the magnetic field strength?

Yes.

It's a direct prediction from the energy difference between the spin states and the fundamental time evolution rule.

And this isn't just theoretical, right?

This precession is something we measure and use.

Oh, absolutely.

It's fundamental to technologies like magnetic resonance imaging, MRI.

And in particle physics, measuring precession frequencies very precisely, like for muons, is a way to test the standard model for tiny deviations.

It connects this basic phase evolution directly to cutting edge experiments.

Amazing how it all ties back to that energy dependent phase factor.

Hashtag tag tag outro.

It really does.

Okay, let's try and wrap this up.

We started with that foundational idea, the phase factor, a state with definite energy.

Our basic phase clock.

Right.

And we saw how that one single rule incredibly seems to govern everything else.

It dictates how a wave packet moves to give us classical velocity.

Through the group velocity.

It explains how particles can sneak through barriers they shouldn't be able to cross.

Quantum tunneling driven by that exponential decay when momentum becomes imaginary.

And it even describes the intricate dance of an internal property like spin rotating in a magnetic field.

Spin precession, yeah.

All driven by energy differences affecting the phase evolution.

So the really big takeaway here seems to be the absolute centrality of the total energy dollars.

That's the unifying principle.

Whether it's kinetic energy, potential energy from fields, or even internal energy related to spin, it all gets bundled into that total energy dollars.

And that dollar sets the rate of the phase clock, which dictates all the dynamics.

It provides this incredibly unified and in a way simple framework for how quantum systems change in time.

It's quite profound.

So maybe a final thought for you listening.

Consider that the quantum world isn't really about things being in places and moving in the classical sense.

It's more like it's defined by these internal energy clocks ticking at different rates.

And all the phenomena we see, motion interaction change, are really just the result of the interference patterns created.

As these different clocks drift in and out of phase with each other, the universe is this complex web of oscillating amplitudes.

And measurement kind of freezes one frame of that interference pattern, giving us a static probability.

Yeah, it's a constant intricate oscillation underpinning reality.

Well, that covers the core ideas of how amplitudes depend on time, according to Feynman.

It's the rule book for change.

Indeed.

Thank you for joining us on this deep dive.

Yes, thanks for tuning in.

We hope this walkthrough helps solidify how these fundamental quantum rules play out.