Dimensional Analysis and Linear Correlation

Question

I was introduced to a silly way to derive the 'de broglie formulae':

1. Both $\left(\frac{p}{k}\right)$ and $\left(\frac{E}{\omega}\right)$ have the same dimension, and it has to be $\left(\frac{M \cdot L^2}{T}\right)$.

2. Both $\left(\frac{p}{k}\right)$ and $\left(\frac{E}{\omega}\right)$ have to be equal to some constants whose physical dimension is $\left(\frac{M \cdot L^2}{T}\right)$.

3. Without further investigation, these constants will be the same, and they are the reduced planck constant.

The problems are the following:

1. Why these constants has to be the same for both $\left(\frac{p}{k}\right)$ and $\left(\frac{E}{\omega}\right)$?

2. With those assumptions, it is clear that there is a linear correlation between $\left(p, \, k\right)$ and $\left(E, \, \omega\right)$.

3. How that type of correlation enter in this derivation?

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DISCLAIMER: i already know all the physic behind the 'de broglie hypothesis' as well as more sophisticate ways to derive the formulae above. I just want to know in which way that physics enter into a derivation based ONLY on dimensional analysis.

Answer 1

From special relativity both $(\frac{E}{c},\vec{p})$ and $(\frac{\omega}{c}, \vec{k})$ are known to be 4-vectors (see Four-momentum and Four-wavevector). The Planck/Einstein relation $$E=\hbar\omega$$ was already known before de Broglie. Therefore it is quite natural to expect (like de Broglie did) $$\vec{p}=\hbar\vec{k}$$ with the same constant $\hbar$, because then both equations can consistently be summarized to one 4-vector equation. $$\left(\frac{E}{c},\vec{p}\right)=\hbar\left(\frac{\omega}{c},\vec{k}\right)$$

Answer 2

Your argument is slightly incorrect. If two quantities have the same dimensions, they could differ by a dimensionless ratio which need not be one. In fact they could differ by any function of one or more dimensionless variables.

In your specific case, $(p/k)$ and $(E/\omega)$ could differ by - say - a dimensionless factor like $4\pi$. They could also differ by a function of - say - the fine structure constant $\alpha$; to give an outrageous example, the ratios could differ by $\cos^2(\alpha)$.

Dimensional analysis does not provide any immediate insight into the dimensionless quantities that appear in problems. These quantities must be determined by other means, either from experiment or using another type of theoretical argument. In the specific case of $(E/\omega)$ the ratio is $\hbar$ and for $(p/k)$ it is part of the deBroglie hypothesis that this ratio is also $\hbar$. There’s no deep reason for this other than it has been verified experimentally.