Understanding Matter Waves Wavelength

According to the de Broglie hypothesis, every moving particle has a wave associated with it, called a matter wave. The wavelength of this matter wave, known as the de Broglie wavelength, is inversely proportional to the momentum of the particle.

The formula for the de Broglie wavelength ($\lambda$) is given by:

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

where $h$ is Planck's constant and $p$ is the momentum of the particle.

For a particle with mass $m$ moving with velocity $v$, the momentum $p$ is typically given by $p = mv$ (in the non-relativistic case). For fast-moving sub-atomic particles, relativistic effects might need to be considered, but the fundamental relationship between wavelength and momentum remains.

Factors Influencing Matter Waves Wavelength

Let's examine how each factor mentioned in the options affects the matter waves wavelength of a fast-moving sub-atomic particle:

• (i) Charge: The charge of a particle determines how it interacts with electric and magnetic fields. However, the de Broglie wavelength is related to the particle's momentum, which is primarily determined by its mass and velocity, not its charge. Therefore, charge does not directly affect the matter waves wavelength.
• (ii) Mass: As seen in the momentum formula $p = mv$, the mass of the particle is a direct factor in its momentum. If the mass changes (while velocity is constant), the momentum changes, and consequently, the matter waves wavelength changes ($\lambda \propto 1/m$ for constant $v$). Mass is a crucial factor.
• (iii) Velocity: Similarly, velocity is a direct factor in the momentum formula $p = mv$. If the velocity changes (while mass is constant), the momentum changes, leading to a change in the matter waves wavelength ($\lambda \propto 1/v$ for constant $m$). Velocity is also a crucial factor.
• (iv) Spin state: Spin is an intrinsic property related to a particle's angular momentum. While important in quantum mechanics, the spin state does not directly contribute to the linear momentum $p$ of the particle, which determines the de Broglie wavelength. Therefore, spin state does not affect the matter waves wavelength.
• (v) Momentum: The de Broglie formula $\lambda = h/p$ explicitly shows that the matter waves wavelength is inversely proportional to the momentum. Momentum is the fundamental quantity that determines the wavelength. Since momentum is defined as mass times velocity ($p=mv$), factors that affect momentum (like mass and velocity) will also affect the wavelength.

Concluding Factors for Matter Waves Wavelength

Based on the de Broglie hypothesis, the wavelength of matter waves associated with a fast moving sub-atomic particle is determined by its momentum ($p$). Momentum itself depends on the mass ($m$) and velocity ($v$) of the particle ($p = mv$). Therefore, the correct factors influencing the matter waves wavelength are mass, velocity, and momentum.

The properties charge and spin state do not directly influence the linear momentum or the resulting matter waves wavelength.

Thus, the factors upon which the matter waves wavelength depends are (ii) mass, (iii) velocity, and (v) momentum.