Prove that the potential energy of a central force 𝐅=-k r^n...

Key Concepts

Conservative Forces

A conservative force is a force field in which the work done in moving an object between any two points is independent of the path taken. This property implies that the force can be expressed as the negative gradient of a potential energy function. Conservative forces are associated with energy conservation, allowing for a potential energy function to be defined such that the total mechanical energy remains constant.

Central Forces

A central force is one that acts along the line joining the object and a fixed center, and its magnitude depends only on the distance from that center. Due to its radial symmetry, the analysis and integration of central forces often simplify when transitioning to spherical or polar coordinates. These forces are integral in the study of orbital mechanics and systems with spherical symmetry.

Potential Energy

Potential energy in the context of conservative forces represents the stored energy of a system due to its configuration. For a force defined as the negative gradient of the potential, the potential energy function is obtained by integrating the force. This integration often involves a constant of integration, which can be chosen based on a defined reference point where the potential energy is set to zero.

Integration of Force to Determine Potential Energy

When a force depends on the distance raised to a power, the potential energy function is derived by integrating the force with respect to the distance variable. For a force expressed as F = -kr^n, under the condition that n ? -1, the integration leads to U = k r^(n+1)/(n+1), ensuring that the force is indeed the negative derivative of the potential energy with respect to distance. This method is fundamental in classical mechanics for relating force fields to their corresponding energy functions.

Simple Harmonic Oscillator

The simple harmonic oscillator is a specific instance in which the restoring force is directly proportional to the displacement from equilibrium. When the exponent n is set to 1, the central force becomes F = -kr, characteristic of a harmonic oscillator. The corresponding potential energy function is U = (1/2) k r^2, which is the well-known parabolic potential that describes many oscillatory systems in classical mechanics.

Transcript

00:01 Okay, in this problem we have a hydrogen atom in its n -equals -1 ground state.

00:07 We're asked to find in part a the kinetic energy, potential energy, and total energy of this ground state for an electron orbiting simple nucleus, hydrogen nucleus.

00:18 Part b, we're asked to show this relationship that the potential energy for any arbitrary state level is equal to negative two times the kinetic energy of that level.

00:28 We're then asked to show the relationship that the potential energy that the energy, the kinetic energy of a particular level is simply the negative of the total energy.

00:36 So we're going to need various aspects of the bore model for this problem.

00:39 We're going to need the kinetic energy for an arbitrary level, 1 half mv squared.

00:44 V is dependent on the energy level.

00:47 We have that expression in terms of various known quantities.

00:53 Similarly, we can use the potential energy for an electron orbiting a proton at some radius rn away.

01:04 And we can express rn in terms of known quantity.

01:07 Using board model.

01:09 These all depend on the n index.

01:13 And we have the total energy in terms of known quantities and n.

01:17 So let's get started.

01:18 Let's jump into it.

01:20 So part a, we have the ground state n equals 1.

01:25 So we simply need to take our expressions for k, u, and e, and make n equals 1.

01:33 That should be relatively straightforward.

01:34 So in this case, for k, we have 1 half m v1 squared.

01:44 Combining the k expression with v, we can end up with 1 over epsilon not squared, m .e to the 4th over 8, h squared.

02:03 Similarly, u1 is equal to 1 over 4 pi epsilon knot, e squared over the bore radius for level 1.

02:17 Squared, setting n equal to 1 in the bore radius and combining with this expression for potential energy, we're left with, oh, negative 1 there.

02:31 We're left with 1 over, negative 1 over 4 pi epsilon not, as before.

02:36 But the 4 pi cancels out.

02:39 So we're actually left with 1 over epsilon not, and me to the 4 power over 4 h squared.

02:54 And finally, for the total energy, e1, we have.

02:59 Negative 1 over epsilon 0 squared, m e to the fourth power, all over 4 or 8, 8 squared...

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