Where does exp(-4pi^2) appear in physics?

  • Thread starter franoisbelfor
  • Start date
  • Tags
    Physics
In summary, the OP is asking for a problem in which the number exp(-4pi^2) appears. François provides a potential equation and solution, which predicts that the temperature at the midpoint after 1 second is exp(-4\pi^2).
  • #1
franoisbelfor
42
0
This is a somewhat unusual question.
I am looking for any physics problem
in which the number or factor exp(-4pi^2)
appears.

If you know one, I'd like to hear about it!

François
 
Physics news on Phys.org
  • #2
Given a potential described as -exp(-x), what is the work done against the potential when moving a unit mass from infinity to x = 4\pi^2 ?
 
  • #3
Introduce the new mass unit, [itex]\hat{m}\equiv{m}{e}^{4\pi^{2}}[/itex], where m is the standard unit of mass.

Then, Newton's second law of motion reads:
[tex]F=e^{-4\pi^{2}}\hat{m}a[/tex]
 
  • #4
Here's one that's less arbitrary: the solution of

[tex]\frac{\partial u(x,t)}{\partial t}=\frac{\partial^2 u(t,x)}{\partial x^2}[/tex]

is

[tex]u(x,t)=\sum_{n=1}^\infty A_n\exp^{-(n\pi/L)^2t}\sin\frac{n\pi x}{L}[/tex]

where [itex]A_n[/itex] is calculated from the initial conditions. But the upshot is, because the above equation governs heat diffusion by conduction, if you had a bar of material with length 0.5 m, thermal diffusivity 1 m2 s-1, end temperatures of 0°C (i.e., [itex]u(0,t)=u(L,t)=0[/itex]), and an initial sinusoidal temperature distribution with maximum temperature 1°C (i.e., [itex]u(x,0)=\sin(\pi x/L)[/itex]), the temperature at the midpoint after 1 second is predicted to be [itex]\exp(-4\pi^2)[/itex], which is the term you're looking for.
 
  • #5
How is that less arbitrary? You still have to pick a material with the right size and properties.
 
  • #6
Vanadium 50 said:
How is that less arbitrary? You still have to pick a material with the right size and properties.

It's less arbitrary because the exponential function and the [itex]\pi^2[/itex] term come out of the physics rather than being input as variables or functions, as in the other two cases (although we do need to specify a sinusoidal initial temperature distribution). And the values aren't fixed; it could be a [itex]10\,\mu m[/itex] long microfabricated silicon beam (thermal diffusivity [itex]8\times 10^{-5}\,\mathrm{m^2}\,\mathrm{s^{-1}}[/itex]) after [itex]5\,\mu s[/itex], for example. Don't you think it's interesting that the [itex]\pi^2[/itex] emerges naturally here?
 
  • #7
Mapes said:
It's less arbitrary because the exponential function and the [itex]\pi^2[/itex] term come out of the physics rather than being input as variables or functions, as in the other two cases (although we do need to specify a sinusoidal initial temperature distribution). And the values aren't fixed; it could be a [itex]10\,\mu m[/itex] long microfabricated silicon beam (thermal diffusivity [itex]8\times 10^{-5}\,\mathrm{m^2}\,\mathrm{s^{-1}}[/itex]) after [itex]5\,\mu s[/itex], for example. Don't you think it's interesting that the [itex]\pi^2[/itex] emerges naturally here?

Not really. Pretty much any time-harmonic or complex number system you can probably easily massage out an exp(\pi) and exp(\pi^2) dependence of some kind. I could specify a transmission line and given a certain length and loss I could get you an attenuation of exp(-4\pi^2).

The OP is just posing a really bad question. It is completely arbitrary because, like arildno shows, when it comes to picking out a constant you can get it from just about any kind of equation from judicious choice of your units, scale, or choice of parameters.
 
  • #8
Born2bwire said:
I could specify a transmission line and given a certain length and loss I could get you an attenuation of exp(-4\pi^2).

Go for it! That's what the poster seems to be looking for: physical circumstances in which the term arises naturally. Maybe he or she saw the term on a blackboard once, or a poster (or a tattoo!), and wants to know what the context might have been. I don't know. But I don't think it's a bad question; in fact, I was looking forward to comparing the responses to get a sense of the... grand interconnectedness of physics. Consider it a challenge: what's the least amount of massaging needed to get [itex]\exp(-4\pi^2)[/itex], without inputting it directly?
 
  • #9
"physical circumstances in which the term arises naturally"

What is natural, or unnatural about a particular choice of length scale??

Yet, they are also part of physics..:smile:
 
  • #10
Getting a pi^2 is not horribly difficult; have a 4 or 5 dimensional volume in phase space. You'll have to turn a 1/2 or an 8/15 into a 4, but that shouldn't be hard. Then find a reason to exponentiate it...e.g. as a partition function.

This is contrived, of course, but no more so than a length of one meter, time of one second, capacitance of one farad, etc.

As far as the context, I fear it's not a blackboard, poster or tattoo. A search of other messages here will perhaps provide some enlightenment.
 
  • #11
Vanadium 50 said:
As far as the context, I fear it's not a blackboard, poster or tattoo. A search of other messages here will perhaps provide some enlightenment.

Ah, got it. I was originally thinking it was something innocuous like https://www.physicsforums.com/showthread.php?t=257304".
 
Last edited by a moderator:

1. What is the significance of exp(-4pi^2) in physics?

Exp(-4pi^2) is an important mathematical function that appears in various areas of physics, such as quantum mechanics and wave mechanics. It is used to describe the behavior of oscillating systems and is essential in understanding the properties of waves.

2. How does exp(-4pi^2) relate to the Schrödinger equation?

The Schrödinger equation, which is used to describe the behavior of quantum particles, involves the use of exp(-4pi^2). This function appears in the solutions of the equation and helps to describe the wave-like nature of particles.

3. Can you give an example of where exp(-4pi^2) is used in physics?

One example is in the study of electromagnetic waves. Exp(-4pi^2) appears in the mathematical description of these waves and helps to determine their properties, such as wavelength and frequency.

4. How does exp(-4pi^2) relate to the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle states that it is impossible to know both the position and momentum of a particle with absolute certainty. Exp(-4pi^2) plays a role in this principle by describing the probability distribution of a particle's position and momentum.

5. Why is exp(-4pi^2) important in Fourier analysis?

In Fourier analysis, exp(-4pi^2) is used as a basis function to decompose a complex signal into simpler components. This allows for the analysis of a signal's frequency components and is used in many areas of physics, such as signal processing and quantum mechanics.

Similar threads

Replies
1
Views
466
Replies
1
Views
589
  • Classical Physics
2
Replies
42
Views
1K
  • Introductory Physics Homework Help
Replies
5
Views
265
Replies
2
Views
506
Replies
8
Views
1K
Replies
5
Views
677
Replies
7
Views
1K
  • Classical Physics
Replies
3
Views
588
  • Classical Physics
Replies
6
Views
462
Back
Top