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 $U=-k_{e} e^{2} / r$ and total energy $E=K+U .$ 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.

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 $U=-k_{e} e^{2} / r$ and total energy $E=K+U .$ 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

Coulomb Potential Energy

The Coulomb potential energy describes the interaction energy between charged particles, such as an electron and a proton. It is typically characterized by an inverse dependence on the separation between the charges. This concept is critical in determining the overall energy of systems where electromagnetic forces play a significant role, particularly in atomic models.

Energy Minimization and Variational Methods

Determining the stable configuration of a quantum system often involves finding the state that minimizes the total energy, which includes both kinetic and potential energy contributions. This approach, common in variational methods, is essential for estimating the most probable parameters (such as orbital radius) that characterize bound states or stable configurations in quantum systems.

Heisenberg Uncertainty Principle

This principle states that one cannot simultaneously know the exact position and momentum of a particle. In quantum systems, the uncertainties in these quantities are inversely related, which implies that if a particle is confined to a small spatial region, there is an intrinsic uncertainty or spread in its momentum. This concept is foundational to estimating properties like the kinetic energy of particles in confined systems.

Quantum Kinetic Energy

In quantum mechanics, the kinetic energy of a particle is often expressed in terms of its momentum uncertainty, as the average momentum may be zero while its fluctuations contribute to a non-zero kinetic energy. By relating momentum uncertainty to confinement size using principles like the uncertainty principle, one can estimate the kinetic energy in bound quantum systems.

Bohr Model Comparison

The Bohr model of the atom introduced the idea of quantized orbits for electrons, where energy levels and orbital radii are determined by balancing classical terms with quantum conditions. Comparing quantum estimates from the uncertainty principle and energy minimization to the predictions of the Bohr theory allows one to assess the validity and limitations of classical approximations when describing atomic systems.

Transcript

00:01 So in this very long question, we have quite a few bits of things to digest first.

00:08 We're looking at the uncertainty principle over here and how we use that to actually find our boss radius and also the total energy.

00:24 Now what it states in the question is first the average position, for the electron, right, says the uncertainty in that position is approximately equals to the radius of its orbit.

00:43 So basically delta x, the uncertainty in position is approximately the radius r.

00:55 Next we also have that the average squared momentum is approximately equal to square of the unsycephemy.

01:05 In this momentum.

01:07 So uncertainty in momentum squared is approximately p squared.

01:15 I'm going to use all of these properties to find different properties such as the energy and the radius.

01:28 So first off we know that the uncertainty principle is delta x, delta p, must be greater or equals to h power 2.

01:41 So if you are given that our delta x is approximately r, then delta p must be greater equals to h bar over 2r.

01:58 This is our first part of the question, certainty of the electrost momentum.

02:04 Next to find the kinetic energy.

02:08 Now we know that delta p must be greater than hb over 2r.

02:13 We can choose a value that's greater than h hb r and we're going to do we're going to choose delta p equals to approximately h power of r so we can ignore the numerical factor half so we'll make things easier and from here we can find the kinetic energy which is p square over 2m 2m e, and so the electron, which is approximately equals to delta p square over 2m.

02:54 Since p square is approximately delta p square.

02:59 So give us hbb square over 2m e r square...

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