CALC A particle with mass m moves in a potential...

CALC A particle with mass $m$ moves in a potential energy$U(x)=A|x|,$ where $A$ is a positive constant. In a simplified picture, quarks (the constituents of protons, neutrons, and other particles, as will be described in Chapter 44 ) have a potential energy of interaction of approximately this form, where $x$ represents the separation between a pair of quarks. Because $U(x) \rightarrow \infty$ as $x \rightarrow \infty,$ it's not possible to separate quarks from each other (a phenomenon called quark confinement). (a) Classically, what is the force acting on this particle as a function of $x ?$ (b) Using the uncertainty principle as in Problem 39.78 , determine approximately the zero-point energy of the particle.

CALC A particle with mass $m$ moves in a potential energy$U(x)=A|x|,$ where $A$ is a positive constant. In a simplified picture, quarks (the constituents of protons, neutrons, and other particles, as will be described in Chapter 44 ) have a potential energy of interaction of approximately this form, where $x$ represents the separation between a pair of quarks. Because $U(x) \rightarrow \infty$ as $x \rightarrow \infty,$ it's not possible to separate quarks from each other (a phenomenon called quark confinement).
(a) Classically, what is the force acting on this particle as a function of $x ?$
(b) Using the uncertainty principle as in Problem 39.78 , determine approximately the zero-point energy of the particle.

Key Concepts

Absolute Value Function in Potentials

When the potential energy is expressed using the absolute value function, such as U(x) = A|x|, the derivative involves a sign function. This leads to a piecewise definition of force that changes its sign depending on the direction of displacement, illustrating how non-smooth potentials impact force calculations.

Uncertainty Principle

The uncertainty principle is a cornerstone of quantum mechanics, stating that there is a fundamental limit to the precision with which position and momentum can be simultaneously known. This principle is used to estimate minimum energy levels in quantum systems, such as the zero-point energy of a confined particle.

Potential Energy Function

A potential energy function represents the position-dependent energy of a system. It is a fundamental concept in both classical and quantum mechanics, providing a framework to analyze forces and particle dynamics by considering how energy is stored and transformed as the particle moves.

Force from the Potential Energy

In mechanics, the force acting on a particle is derived as the negative derivative of the potential energy with respect to position, F(x) = -dU/dx. This relation connects the spatial variation of energy to the acceleration and dynamics of the particle in a given force field.

Zero-Point Energy

Zero-point energy refers to the lowest possible energy that a quantum mechanical system can possess, even when it is at its ground state. It arises from the inherent quantum fluctuations dictated by the uncertainty principle, and is a key concept for understanding the energy constraints of bound systems.

Transcript

00:01 In this question, we're told that we have a particle which has mass m, and it's in some kind of potential, which looks like this.

00:09 So its potential energy is relative to x by a times absolute value of x.

00:14 And in question a, they ask us, what is the force acting on the particle? so we don't know anything about the actual nature of the force, but we can come up with a function for it because we now have force is related to potential energy.

00:28 Specifically, you knew that u of x is the opposite of the integral of f.

00:37 So to find f, we're just going to take the derivative of the integral.

00:44 So f is d d dx of u of x minus 1.

00:51 All right, so it's kind of tough to work with absolute values.

00:57 So i like to rewrite these as piecewise functions.

01:01 So i'm going to rewrite u of x is equal to, and then it can either be positive ax, or it can be minus ax.

01:15 So it's positive ax if x is greater than equal to zero, and it's minus ax if x is plus and not equal to zero.

01:25 Okay, hopefully that makes sense what i'm doing here.

01:29 So now i would just need to take the derivative and then we'll have that.

01:31 All right, so taking the derivative of the top portion first, we're just going to get a if x is greater than equal to zero.

01:45 So they didn't look quite like like a greater than equals t sign.

01:50 And then it's going to be minus a if x is plus than or equal to zero.

01:57 This is our answer for part a.

02:03 It's going to have to be in a piecewise format since there isn't really a good way to represent percent the absolute value otherwise.

02:12 And in question b, they ask us what is the zero point energy of this particle? so what we're looking for is the minimum energy a particle can take on.

02:26 So typically how we find a minimum the function is we take the derivative, in this case with respect to x, and then we set that equal to zero, and then we get x -min.

02:41 And then we plug that into our values that we're trying to minimize.

02:47 So to find the minimum energy, first we're going to write out the equation for the energy, take the derivative with respect to x, set equal to 0, and then we'll find that x -min plug again.

02:59 All right, so of course we need to start with a function, and we know that e is equal to p squared over 2m plus u.

03:11 And we of course know that u of x is equal to a times at a of suit value of x so we'll sublate in.

03:19 We're also told that the following relation holds px is equal to h.

03:25 This is sort of an approximation of the heisenberg uncertainty principle.

03:31 We're just kind of dropping the 4 pi.

03:35 Yeah, and this might be a good approximation for the force of of glue ons, so the strong force.

03:45 All right, so we know that this is true and we know this is a function for e.

03:51 And since we need to take a derivative to x, we're going to need to use this formula to solve in momentum for something in terms of x...

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