An electron of momentum p is at a distance r from a...

An electron of momentum $p$ is at a distance $r$ from a stationary proton. The electron has kinetic energy $K=p^{2} / 2 m_{e} .$ The atom has potential energy $U_{E}=-k_{e} e^{2} / r$ and total energy $E$ $=K+U_{E} .$ If the electron is bound to the proton to form a hydrogen atom, its average position is at the proton but the uncertainty in its position is approximately equal to the radius $r$ of its orbit. The electron's average vector momentum is zero, but its average squared momentum is approximately equal to the squared uncertainty in its momentum as given by the uncertainty principle. Treating the atom as a one-dimensional system, (a) estimate the uncertainty in the electron's momentum in terms of $r$. Estimate the electron's (b) kinetic energy and (c) total energy in terms of $r$. The actual value of $r$ is the one that minimizes the total energy, resulting in a stable atom. Find (d) that value of $r$ and (e) the resulting total energy. (f) State how your answers compare with the predictions of the Bohr theory.

Key Concepts

Heisenberg Uncertainty Principle

This principle states that there is an inherent limit to the precision with which certain pairs of physical properties, such as position and momentum, can be known simultaneously. In the context of the hydrogen atom, it is used to relate the uncertainty in the electron's position (on the order of the orbital radius) to an uncertainty in its momentum, forming the basis for estimating its kinetic energy in a quantum-mechanical framework.

Quantum Kinetic Energy Estimation

In quantum mechanics, the kinetic energy of a particle can be approximated by using its momentum uncertainty. Since the electron’s momentum is not sharply defined but spread over an uncertainty range given by the uncertainty principle, its kinetic energy is estimated as being proportional to the square of this uncertainty divided by twice its mass. This approach provides insight into the energy contribution from the electron’s motion in an atom.

Coulomb Potential Energy

The Coulomb potential energy describes the electrostatic interaction between charged particles, such as the attractive force between the negatively charged electron and the positively charged proton in a hydrogen atom. It is inversely proportional to the distance between the charges, and its negative sign indicates that the interaction is attractive. This energy term plays a crucial role in the total energy of the system and is essential for understanding atomic binding.

Variational Principle and Energy Minimization

The variational principle involves estimating the lowest energy state of a system by considering trial wavefunctions or parameters and then minimizing the total energy with respect to these parameters. In this context, it is used to determine the orbital radius that minimizes the sum of the quantum kinetic energy and the Coulomb potential energy, resulting in a stable bound state for the electron in the hydrogen atom.

Bohr Model Comparison

The Bohr theory of the hydrogen atom postulates that electrons travel in quantized orbits with specific radii and energies, derived from classical assumptions enhanced by quantum conditions. Comparing the estimates obtained from applying the uncertainty principle and energy minimization to those predicted by the Bohr model helps to illustrate both the similarities and differences between semi-classical and more refined quantum-mechanical approaches to describing atomic structure.

Transcript

00:01 In this problem, we're using heisenberg's uncertainty principle to reconstruct the energy equation for the bore model of the hydrogen atom.

00:09 And this works because the uncertainty principle is essentially implicitly just encoding the wave properties of matter, and that uncertainty relation is implicit within the wave structure matter.

00:25 So we're implicitly doing a kind of wave -like model for the hydrogen atom by expressing it, the uncertainties in position and momentum.

00:39 You can, in a sense, prove the uncertainty principle purely mathematically with fourier transforms, but that's a little bit more advanced.

00:50 So the problem tells us to treat the problem as a 1d, one -dimensional situation.

00:59 We're treating uncertainty exposition as approximately the same thing as the radius, so we're just going to ignore rotation effectively.

01:08 It refers to the average speed is zero, but the average of the speed squared is proportional to the square of the uncertainty of the momentum.

01:20 It gives us potential energy that's just the hydrogen atom radially symmetric k -e -square divided by r total energies connect plus potential energy so the proper uncertainty principle is the standard deviation in x and the momentum in the x direction is greater than or equal to h bar divided by two that's the reduced uncertainty principle the original that heisenberg had was delta x delta p for moment momentum in the same direction.

02:03 The relationship, the uncertainty relationship has to be on the same axis that was proportional to age bar.

02:11 I'm going to use this one just because that factor of two in that one, it creates a factor of four difference in the final result.

02:25 And if we use this one, then we will get the exact same energy as the bore.

02:32 Model, so that's the one i'm going to use.

02:36 So, first thing we have to do is find momentum in terms of r.

02:42 So we've let, i'm just going to let it equal, h -bar.

02:57 Just not going to bother with the greater -than -symbols.

02:59 We're just going to do equal, because that's a minimum ideal case.

03:04 So that becomes r because that's just the, one -dimensional situation we're in.

03:16 So that means delta p and turn the uncertainty in momentum in terms of r.

03:23 A's bar divided by r.

03:31 So then we want to estimate the kinetic energy of the electron.

03:35 So kinetic energy can be defined as momentum squared divided by 2m.

03:42 That's the same thing as one half mb squared, which is written differently.

03:47 So so that becomes delta p squared, which the problem stipulated, that was the average velocity squared was proportional to that, which is in non -zero, which was the relevance of stipulating that in the problem.

04:12 So that becomes h -bar squared r squared times one -half m.

04:17 So that becomes h -bar squared 2m -r -r -squared equals connecticut.

04:30 So then we want to find the, we have to estimate the total energy in terms of the radius r.

04:40 So e equals k plus u -e...

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