Proportionality of Discrete Particle Speed with Quantum Number

Based on the quantum mechanical description of a particle of mass $m$ confined within a one-dimensional region of width $a$ (an infinite potential well):

1. Quantization Condition:
   The standing wave condition for the de Broglie wavelength $\lambda_n$ of the particle in a box of width $a$ is given by:
   $n \frac{\lambda_n}{2} = a \implies \lambda_n = \frac{2a}{n}, \quad \text{for } n = 1, 2, 3, \ldots$

2. Momentum and Speed of the Particle:
   Using the de Broglie relation $p_n = \frac{h}{\lambda_n}$, the momentum of the particle in the $n$-th state is:
   $p_n = \frac{n h}{2a}$

   Since the magnitude of momentum is related to the speed $v_n$ of the particle by $p_n = m v_n$, we obtain:
   $m v_n = \frac{n h}{2a} \implies v_n = \frac{n h}{2 m a}$

3. Proportionality:
   Since $h$, $m$, and $a$ are constants:
   $v_n \propto n$

Therefore, the speed of the particle is proportional to $n$.

Correct Option: D