**Magnitude of Momentum of a Particle in a Box in its nth State**

In quantum mechanics, the momentum of a particle is related to its quantum state. In a particle in a box system, the particle is confined within a box of length L, and its momentum is quantized. The allowed momentum values are given by:

p_n = (n * h) / L

where p_n is the momentum of the particle in its nth state, n is the quantum number, h is the Planck's constant, and L is the length of the box.

To find the magnitude of the momentum, we take the absolute value of the momentum:

|p_n| = |(n * h) / L|

where |p_n| represents the magnitude of the momentum.

Therefore, the magnitude of the momentum of a particle in a box in its nth state is |(n * h) / L|.

**Minimum Change in Momentum and Uncertainty Principle**

The minimum change in the particle's momentum that a measurement can cause corresponds to a change of ±1 in the quantum number n. Let's consider the initial momentum as p_i and the final momentum as p_f. The difference between the initial and final momenta can be written as:

Δp = p_f - p_i

Since the minimum change in momentum corresponds to a change of ±1 in the quantum number n, we can write:

Δp = ±(h / L)

Now, let's consider the uncertainty in the position of the particle, denoted as Δx. In this case, Δx is equal to the length of the box, L.

According to the Heisenberg uncertainty principle, the product of the uncertainties in position (Δx) and momentum (Δp) must be greater than or equal to the reduced Planck's constant divided by 4π:

Δx * Δp ≥ h / (4π)

Substituting the values of Δx and Δp, we have:

L * (h / L) ≥ h / (4π)

Simplifying the expression, we get:

h ≥ h / (4π)

Since h is a positive value, this inequality is always true.

Therefore, we have shown that Δx * Δp ≥ h / (4π) for the particle in a box system, where Δx is the length of the box and Δp is the minimum change in momentum corresponding to a change of ±1 in the quantum number n.