Does Exponential Decay Fail at the Quantum Level?

Click For Summary

Discussion Overview

The discussion revolves around the applicability of exponential decay models in real-world phenomena, particularly in the context of the discharging of a parallel plate capacitor. Participants explore whether these models hold true at the quantum level, especially when dealing with small quantities of charge.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose that exponential decay formulas may fail when the charge being measured approaches the elementary charge, suggesting a limitation in the model's applicability.
  • Others argue that the exponential decay formula is valid for large numbers of electrons, but its accuracy diminishes as the number of electrons decreases.
  • A participant questions the definition of "failure" in this context, suggesting that the error associated with measuring small quantities of charge can be quantified and may not necessarily indicate a failure of the model.
  • It is mentioned that for small numbers of electrons, the Poisson distribution may be more appropriate for modeling the decay process.
  • There is a suggestion that at low electron counts, quantum mechanical considerations might need to be taken into account, although this remains uncertain.

Areas of Agreement / Disagreement

Participants express differing views on the limitations of exponential decay models at low charge levels, with some agreeing that the model becomes less reliable, while others emphasize the need for a clear definition of failure and the potential use of statistical methods.

Contextual Notes

The discussion highlights the dependence on definitions of failure and the statistical nature of measurements at low electron counts, indicating that the applicability of exponential decay may vary based on specific experimental setups and measurement techniques.

aftershock
Messages
106
Reaction score
0
Hi everyone,

There's something that's kind of been bugging me about applying exponential decay formulas to real world phenomena. For example let's say the discharging of a parallel plate capacitor. Let's consider the negative plate. As it discharges excess electrons leave the plate. The charge falls off exponentially and we model this mathematically by an exponential decay formula.

But wouldn't there be a time while the amount of charge leaving is less than the elementary charge? We know energy is quantized and it seems to me that the exponential decay model completely fails when we get around to the capacitor holding a charge of 1e.
 
Physics news on Phys.org
aftershock said:
Hi everyone,

There's something that's kind of been bugging me about applying exponential decay formulas to real world phenomena. For example let's say the discharging of a parallel plate capacitor. Let's consider the negative plate. As it discharges excess electrons leave the plate. The charge falls off exponentially and we model this mathematically by an exponential decay formula.

But wouldn't there be a time while the amount of charge leaving is less than the elementary charge? We know energy is quantized and it seems to me that the exponential decay model completely fails when we get around to the capacitor holding a charge of 1e.

That's right. The exponential decay formula only holds for a large number of electrons.
 
Rap said:
That's right. The exponential decay formula only holds for a large number of electrons.

That's interesting. Can anyone further elaborate on this? When does it start to fail, and what do we use instead when it does. Does it become a quantum mechanical problem?
 
Before you can talk about "failure", you have to talk about the definition of failure. The quantization produces an error from the exponential decay. If you are measuring n electrons/second, the error will be about sqrt(n) electrons/second. So if you are measuring 1 amp, that's like 10^16 electrons/sec with an error of 10^8 electrons/sec or about 10^-8 amp or 10^-6 percent. The exponential decay will be good. If you are measuring 100 electrons/sec the error will be 10 electrons/sec or 10 percent. The exponential decay is not so good. Pick a percentage error that you call "failure" and you can figure out at what current that error will occur. For high error rates, it becomes a statistical problem. I think (not sure) that the electrons will have a Poisson distribution and you have to talk about the probability of measuring a certain number of electrons per second. It will depend on your measuring device too - if it cannot count individual electrons, then you have to take that into account. Depending on your particular setup, this might be enough, but maybe not, you may have to start doing QM calculations as well.
 
Yes, for small numbers, the Poisson distribution is the appropriate one to use.
 

Similar threads

Undergrad Does a Fabry Perot cavity charge in discrete steps?
  • · Replies 12 ·
Replies
12
Views
4K
Undergrad Electromagnetic radiation, photons, quantized energy levels
  • · Replies 1 ·
Replies
1
Views
2K
Why can we model spherical capacitor with two dielectrics as two capacitors in series?
  • · Replies 4 ·
Replies
4
Views
2K
How does the electroscope lose electrons to the Earth?
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Avalanche photon detectors
  • · Replies 0 ·
Replies
0
Views
881
Graduate How Does Decoherence Affect a Single Two-Level System in Hilbert Space?
  • · Replies 9 ·
Replies
9
Views
2K
Determining electric permittivity from time constant decay
  • · Replies 1 ·
Replies
1
Views
3K
RLC Circuits Problem: Find Frequency, Decay Time & Current
  • · Replies 10 ·
Replies
10
Views
3K
Graduate What exactly is a "rare event"? (Poisson point process)
  • · Replies 12 ·
Replies
12
Views
5K
Final project ideas using Noether's theorem in simulation class
  • · Replies 4 ·
Replies
4
Views
2K