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

[CAF123]

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 $e^{(hc/\lambda kT)}$ but I don't understand really why using a power series approximation makes something tend to the classical limit?

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

Many thanks.

 

[voko]

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.

 

[CAF123]

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?

 

[voko]

f(x) is function f of argument x.

 

[Nickie]

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.