Electron Energy in a Nucleus. The radii of atomic nuclei are...

Electron Energy in a Nucleus. The radii of atomic nuclei are of the order of $5.0 \times 10^{-15} \mathrm{m}$ . (a) Estimate the minimum uncertainty in the momentum of an electron if it is confined within a nucleus. (b) Take this uncertainty in momentum to be an estimate of the magnitude of the momentum. Use the relativistic relationship between energy and momentum, Eq. $(37.39),$ to obtain an estimate of the kinetic energy of an electron confined within a nucleus. (c) Compare the energy calculated in part (b) to the magnitude of the Coulomb potential energy of a proton and an electron separated by $5.0 \times 10^{-15} \mathrm{m}$ . On the basis of your result, could there be electrons within the nucleus? (Note: It is interesting to compare this result to that of Problem $39.80 . )$

Electron Energy in a Nucleus. The radii of atomic nuclei are of the order of $5.0 \times 10^{-15} \mathrm{m}$ . (a) Estimate the minimum uncertainty in the momentum of an electron if it is confined
within a nucleus. (b) Take this uncertainty in momentum to be an estimate of the magnitude of the momentum. Use the relativistic relationship between energy and momentum, Eq. $(37.39),$ to obtain
an estimate of the kinetic energy of an electron confined within a nucleus. (c) Compare the energy calculated in part (b) to the magnitude of the Coulomb potential energy of a proton and an electron
separated by $5.0 \times 10^{-15} \mathrm{m}$ . On the basis of your result, could there be electrons within the nucleus? (Note: It is interesting to compare this result to that of Problem $39.80 . )$

Key Concepts

Heisenberg Uncertainty Principle

This principle states that the product of the uncertainties in position and momentum for a particle cannot be smaller than a certain value on the order of Planck's constant. In contexts where a particle is confined to a very small region, such as a nucleus, this leads to a correspondingly large uncertainty in momentum, which has significant physical implications.

Quantum Confinement

Quantum confinement refers to the effect observed when a particle is restricted to a small spatial region, leading to quantization of its energy levels and an increase in momentum uncertainty. This concept is crucial for understanding phenomena in atomic and nuclear physics, where the scale of confinement is comparable to fundamental quantum limits.

Relativistic Energy-Momentum Relationship

In scenarios where particle momenta are very high, the classical expression for kinetic energy becomes inadequate. Instead, the relativistic relationship between energy and momentum, which combines the effects of mass and the speed of light, must be used to accurately estimate the kinetic energy of particles confined in small regions.

Coulomb Potential Energy

Coulomb potential energy describes the energy stored in the electrostatic interaction between charged particles, which is inversely proportional to their separation. It provides a baseline for comparing other forms of energy in atomic systems, thereby playing a key role in determining whether certain configurations, like electrons being confined within a nucleus, are energetically favorable.

Energy Scale Comparison in Quantum Systems

Comparing different energy scales, such as the kinetic energy derived from quantum uncertainty and the Coulomb potential energy, is essential for assessing the stability and possible arrangements of particles in atomic and subatomic systems. This analysis helps in understanding why some configurations, like electrons within a nucleus, may or may not occur naturally.

Transcript

00:01 In this question, we want to use the uncertainty principle to kind of estimate the energy of electron if it were to be bounded inside the nucleus.

00:15 So given that the size of the nuclei is about 5 times 10 power minus 15, we can use that to as our estimate of the uncertainty in the position.

00:30 Given the uncertainty principle is that delta x times delta p greater equals to hb over 2, we can find the uncertainty in the momentum.

00:42 It's greater or it goes to hbauer over 2 times 5 times 10 power minus 15.

00:56 Right.

00:58 And this should give us approximately 1 .1 to 10 power minus 20 terms of the sr units.

01:11 And we are asked to approximate this to be the momentum of the electron.

01:20 Now we can find what is the kinetic energy of the electron that is confined with this amount of momentum.

01:33 So the kinetic energy is just the total energy minus away the rest energy.

01:43 In this case the total energy we have to use the relativistic equation.

01:47 That is p square plus m square c4 right minus away the rest energy so we can substitute in the values so for p this is the value we'll substitute in ccc is just the speed of light m we'll use the mass of the electron right and that's all the unknowns, sorry, all the constants inside this expression which we can substitute in.

02:30 Evaluating this in the calculator, we should get about 3 .084, 10 power of minus 12 joules.

02:46 Now since you want it in terms of electron volts, we can actually divide this...

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