Failure of the classical model. (a) What is the spin angular...

Failure of the classical model. (a) What is the spin angular momentum of an electron along the $z$ direction, in SI units? (b) In the rest of this problem, we shall try to understand the behavior of an electron by modeling it as a classical ball of matter. We shall also use the nonrelativistic formulas, although a thorough treatment would require use of the more complicated relativistic ones. If we model this electron classically as a solid uniform sphere of diameter $1.0 \times 10 ^ { - 15 } \mathrm { m } ,$ what would be its angular velocity in $\mathrm { rad } / \mathrm { s } ?$ (c) What would be the speed of the surface of this classical sphere? Is there anything suspicious about this result? (d) In part (c), the result cannot be correct because the surface of the electron cannot move faster than the speed of light. Suppose we try to "correct" this model by assuming that the surface is moving at the speed of light the upper limit of its speed. What would have to be the diameter of an electron in that case? Given that atomic nuclei are about $10 \times 10 ^ { - 15 } \mathrm { m }$ in diameter, does this result seem plausible for the size of an electron? (e) Would the bizarre results in parts (c) and (d) be affected appreciably if we instead modeled the electron as a hollow spherical shell instead of a solid? Why? Notice that we encounter results that contradict observation when we try to think of the electron simply as a spinning sphere. We simply cannot understand the behavior of an electron (and other subatomic particles) by using our ordinary classical models.

Failure of the classical model. (a) What is the spin angular momentum of an electron along the $z$ direction, in SI units? (b) In the rest of this problem, we shall try to understand the behavior of an electron by modeling it as a classical ball of matter. We shall also use the nonrelativistic formulas, although a thorough treatment would require use of the more complicated relativistic ones. If we model this electron classically as a solid uniform sphere of diameter $1.0 \times 10 ^ { - 15 } \mathrm { m } ,$ what would be its angular velocity in $\mathrm { rad } / \mathrm { s } ?$ (c) What would be the speed of the surface of this classical sphere? Is there anything suspicious about this result? (d) In part (c), the result cannot be correct because the surface of the electron cannot move faster than the speed of light. Suppose we try to "correct" this model by assuming that the surface is moving at the speed of light the upper limit of its speed. What would have to be the diameter of an electron in that case? Given that atomic nuclei are about $10 \times 10 ^ { - 15 } \mathrm { m }$ in diameter, does this result seem plausible for the size of an electron? (e) Would the bizarre results in parts (c) and (d) be affected appreciably if we instead modeled the electron as a hollow spherical shell instead of a solid?
Why? Notice that we encounter results that contradict observation when we try to think of the electron simply as a spinning sphere. We simply cannot understand the behavior of an electron (and other subatomic particles) by using our ordinary classical models.

Key Concepts

Relativistic Speed Limit and its Implications

Relativity imposes a fundamental speed limit, the speed of light, which no object or part of an object can exceed. When classical calculations for a spinning electron suggest surface speeds greater than the speed of light, it indicates a breakdown of the classical model. This discrepancy highlights the need for relativistic and quantum mechanical treatments for subatomic particles.

Failure of Classical Models in the Quantum Realm

The attempt to model electrons as spinning classical spheres leads to unphysical results, such as surface speeds surpassing the speed of light or implausibly small electron dimensions. This failure underscores the inadequacy of classical mechanics in describing phenomena at the quantum scale and reinforces the necessity of quantum theory to properly understand the behavior of particles like electrons.

Intrinsic Electron Spin

Electron spin is a fundamentally quantum-mechanical property that gives rise to an intrinsic angular momentum, independent of any classical rotation. Unlike a spinning macroscopic object, electron spin does not imply that the electron is physically rotating, and its magnitude is quantized in terms of the reduced Planck’s constant.

Classical Angular Momentum and Moment of Inertia

Classically, the angular momentum of a rotating rigid body is computed using its moment of inertia and angular velocity. For a spherical object, the moment of inertia is derived from its mass distribution. When this classical approach is naively applied to an electron (modeled as a tiny spinning sphere), it leads to inconsistencies with observed quantum behavior.

Transcript

00:01 Question number 30 is a bit lengthy, but we'll try to cover it as succinctly as possible.

00:07 So, start looking at it in chunks with part a.

00:12 We want to find the spin -angle momentum of an electron along the z direction in si units.

00:21 Well, if we have a classical model, then a sphere with some axis that it's rotating around, say in that direction maybe it will have a moment of inertia which is equal to 2 over 5 m r square and the angular momentum is going to just be equal to i omega and then just in case it comes in handy which it might we want to take a point on the surface of that sphere and that will be our tangential speed v, which is simply given by r, omega.

01:19 Okay, so let's get into a.

01:22 What is the spinning momentum of an electron in a classical picture? or sorry, in the quantum picture, it's going to be sz, plus or minus, hbar over two.

01:47 And in si units, that is plus or minus, 5 .27 times 10 to the minus 35 kilogram meter squared per second.

02:07 Well, that was easy.

02:08 We just plugged in for h -bar.

02:10 But b wants us to try to get at what the classical model will give us if we were to assume that the electrons is a spinning ball.

02:21 So we are going to use the non -relativistic stuff just because it's much easier.

02:30 And one more thing that i should write down here.

02:33 The radius, rather the diameter, but let's see, won't have this, is 1 .0 times 10 to the minus 15 meters.

02:52 So the angular velocity, in that case, we'll still try to fit this on this page, but we will move to a new page soon.

03:00 But you can go back and forth in the video.

03:03 So b, if we first work out our moment of inertia, which i have it written down here, we have a mass because this is the mass of the electron, and then we have the radius, which is here.

03:28 We can see that our moment of inertia is going to be 9 .11 times 10 to minus 62 kilogram meter squared.

03:43 And then our angular velocity is simply the angular momentum divided by that moment of inertia, which gives us 5 .78 times 10 to the 26 radiance per second...

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