Why Does Using Power Series Help Approach the Classical Limit in Physics?

In summary, the conversation discusses the use of power series expansion to show that the Planck radiation formula and the equation for kinetic energy in special relativity reduce to their classical counterparts in the appropriate limit. The use of a power series approximation is effective because it accurately approximates a function when the argument is small. The concept of "classicality" can be linked to the smallness of the argument in these cases.
  • #1
CAF123
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Homework Statement


I have to show that the Planck radiation formula reduces to the Rayleigh-Jeans formula in the classical limit for blackbodies.

The Attempt at a Solution


I can easily show it using power series expansion of [itex] e^{(hc/\lambda kT)}[/itex] but I don't understand really why using a power series approximation makes something tend to the classical limit?

Similarly, for [itex] E_k = mc^2(\gamma -1) [/itex] tending to [itex] \frac{mv^2}{2} [/itex], in the classical limit. The results are clear, I just don't understand why using a power series actually works.

Many thanks.
 
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  • #2
The first term (or terms) of a power series is a good approximation of a function only when its argument is small. Use any estimate of the approximation error to show that formally.

If you can cast "classicality" as a smallness of some argument to some function, then a power series (polynomial, actually) approximation would describe the phenomenon "classically". See how that applies to these two cases.
 
  • #3
voko said:
The first term (or terms) of a power series is a good approximation of a function only when its argument is small. Use any estimate of the approximation error to show that formally.
What do you mean by the word 'argument' here?
 
  • #4
f(x) is function f of argument x.
 
  • #5



The use of power series in physics is a common mathematical tool to approximate complex functions. In the case of the Planck radiation formula, using a power series expansion of e^(hc/lambda kT) allows us to simplify the formula and make it more manageable for calculations.

In the classical limit, we are dealing with large values of temperature and wavelength, which results in the exponential term becoming very small. This allows us to neglect higher order terms in the power series expansion, resulting in the Rayleigh-Jeans formula.

Similarly, in the case of E_k = mc^2(\gamma -1) tending to \frac{mv^2}{2}, the use of a power series allows us to approximate the relativistic energy equation and simplify it to the classical kinetic energy formula. This is because at low velocities, the relativistic factor \gamma approaches 1, making the higher order terms negligible.

In general, the use of power series allows us to approximate complex functions and make them more manageable for calculations in the classical limit. However, it is important to note that power series are only valid for a specific range of values and may not accurately represent the behavior of a function outside of that range.
 

Related to Why Does Using Power Series Help Approach the Classical Limit in Physics?

1. What is a power series in physics?

A power series in physics is a series of terms that represent a mathematical function, where each term is a multiple of a variable raised to a power. In physics, power series are used to approximate and represent functions that are difficult to solve analytically.

2. How are power series used in physics?

Power series are used in physics to represent a variety of functions, such as position, velocity, and acceleration. They are also used to represent physical phenomena, such as electromagnetic fields and quantum mechanical wave functions.

3. What is the significance of the coefficients in a power series?

The coefficients in a power series have physical significance and can tell us about the behavior of a system. For example, in the Taylor series expansion of a function, the coefficients represent the derivatives of the function at a specific point.

4. Can power series be used to model complex systems?

Yes, power series can be used to model complex systems in physics. In some cases, the power series may need to be truncated to a finite number of terms in order to be useful, but this can still provide a good approximation of the behavior of the system.

5. Are there any limitations to using power series in physics?

While power series are a useful tool in physics, they do have some limitations. They may not accurately represent a function or system at every point, and they may diverge or become more complex for certain types of functions. In addition, the convergence of a power series may be limited by the range of values for the variable of interest.

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