Central to de Broglie's equation is Planck’s constant, denoted as \(h\) Planck’s constant is a fundamental constant of nature, representing what is beleived to be the smallest discrete unit of energy in quantum physics. Its value being \( 6.626\ 070\ 1499\ × 10 ^{-34} \text{Js}\). Planck’s constant relates the momentum of a particle to its corresponding wavelength, bridging the gap between classical and quantum physics.

$$\lambda = \frac{h}{p}$$

By assuming that \(h\) is truly a fundamental constant, just a number, effectively some mysterious scalar value, de Broglie fails to recognize the main components of Planck's constant \(h\) then proceeding to deduce the following equation; $$\lambda = \frac{h}{mv}$$

There is nothing about the equation that is unusual, it is a perfectly valid derivation. The problem arises in the assumptions that de Broglie makes. Observing the mass in the denominator, deBroglie made a leap of faith in the assumption that all matter objects must therefore have a wavelength. The problem becomes apparent when the Planck constant \(h\) is replaced with its component values;

$$\lambda = \frac{a_0 m_e v_e}{m_e c} = \frac{a_0 v_e}{c} = 2.426\ 310\ 2387 × 10 ^{-12}\text{m}$$ The purported "rest mass" of the electron drops out of the equation resulting in the Compton wavelength for the electron. Lacking any reference to mass in the equation, the entire idea of matter waves disappears.

Wavelengths

Based upon de Broglie's hypothesis and the fundamental equation itself there is simply no reason to assume that this it is quantum mechanical in nature, all that is being calculated is the wavelength of the electron. $$\text{Compton Wavelength} = \lambda = \frac{a_0 v_e}{c} = 2.426\ 310\ 2387 × 10 ^{-12}\text{m}$$ Indeed the same principle could be applied to any rotating object to calculate its wavelength including macro scale objects.

As an example, the wavelength of the Earth itself can be calculated as it maintains an orbit around the Sun. Applying the values to de Broglies equation results in the following equation; $$\text{Wavelength of the Earth}=\frac{\text{Orbital radius}\cdot\text{Velocity}}{\text{Speed of Light}}$$

Substituting the values results in the following; $$\lambda_E=\frac{1.496 × 10^{11}\cdot 2.978 × 10^{4}}{2.998 × 10^8} = 1.486 × 10^7 \text{m}$$ The result is an extremely long wavelength which falls into the radio wave region of the electromagnetic spectrum. A wavelength of \(1.486 × 10^7\) meters corresponds to a frequency given by: $$ f=\frac{c}{\lambda} = 20.173\text{Hz}$$ Which places it in the ELF (Extremely Low Frequency) range for wavelengths longer than 100 km, which is used for specialized applications like submarine communication and geophysical studies. A frequency of 20.19 Hz falls right at the lower boundary of human hearing, it would be barely audible as a very deep rumble, possibly felt more as a vibration rather than a distinct tone.

Conclusion

One might wonder: where does this entire notion originate? Undoubtedly, it can be traced back to Einstein's controversial proposal that light behaves as a particle. However, the fact that we can perceive light does not necessarily imply it consists of particles with the traditional properties associated with them. Interestingly, there are no "particles" attributed to radio waves, microwaves, heat, or other forms of electromagnetic radiation—despite light being just a small segment of the vast electromagnetic spectrum that humans have evolved to detect with their eyes. The concept of "light particles" could arguably be one of the greatest misconceptions in physics, spawning not only numerous theories but also an entire industry rooted in this contentious premise. The de Broglie hypothesis while being accepted by the physics community is not based upon solid foundation lacking any physical evidence to support it. Rather it is based upon on a simple declaration made by de Broglie himself backed by one decidedly unconvincing assumption.