Quantum Mechanics

The de-Broglie wavelength ( ) is given by the formula:

Where: - is Planck's constant - is the mass of the particle - is the velocity of the particle

Since the speed of the electron ( ), proton ( ), and He²⁺ ion ( ) are the same, the de-Broglie wavelength is inversely proportional to the mass of the particle.

- The electron ( ) has the smallest mass, so it will have the largest de-Broglie wavelength. - The proton ( ) has a larger mass than the electron, so its de-Broglie wavelength will be smaller. - The He²⁺ ion ( ) is the heaviest, so it will have the smallest de-Broglie wavelength.

Thus, the order of de-Broglie wavelengths is:

e > p > α

Answer: e > p > α