QJEEAdv09Ph8059: JEE Advanced 2009 Physics - Class 12 JEEAdv09Ph8059 Solution: The speed of the particle, that can take discrete values, is…

To find the relationship between the speed of the particle and the quantum number $n$ , we need to follow a few steps using the given information and fundamental quantum mechanical principles.

Firstly, for a particle moving in a one-dimensional box of length $a$ , the allowed wavelengths $\lambda$ of standing waves can be given by:

$\lambda = \frac{2a}{n}$

for $n = 1, 2, 3, \ldots$ .

The de Broglie relation links the wavelength $\lambda$ of a particle to its momentum $p$ :

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

From this, we can express the momentum $p$ in terms of the quantum number $n$ :

$\frac{h}{p} = \frac{2a}{n}$

Solving for $p$ , we get:

$p = \frac{nh}{2a}$

The kinetic energy $E$ of the particle is related to its momentum $p$ by the equation:

$E = \frac{p^2}{2m}$

Substituting $p = \frac{nh}{2a}$ into the energy expression, we get:

$E = \frac{(nh/2a)^2}{2m} = \frac{n^2 h^2}{8ma^2}$

The kinetic energy can also be written in terms of the speed $v$ of the particle:

$E = \frac{1}{2} m v^2$

Equating the two expressions for energy, we have:

$\frac{1}{2} m v^2 = \frac{n^2 h^2}{8ma^2}$

Solving for the speed $v$ , we get:

$v^2 = \frac{n^2 h^2}{4m^2 a^2}$

Therefore, the speed $v$ of the particle is:

$v = \frac{nh}{2ma}$

From the above expression, we see that the speed $v$ of the particle is directly proportional to $n$ :

$v \propto n$

Therefore, the speed of the particle is proportional to:

Option D: $n$