Uncertainty Relation in a 1D Infinite Square Well Potential

Problem Statement

Consider a one-dimensional infinite square well potential of width $L$L. The potential $V(x)$V(x) is $V(x) = 0$V(x)=0 for $0 \le x \le L$0≤x≤L and $V(x) = \infty$V(x)=∞ elsewhere.

The stationary states of a particle of mass $m$m confined in this potential are described by the Schrödinger equation.

In the ground state of this particle, the square of the product of the position uncertainty $\Delta x$Δx and the momentum uncertainty $\Delta p$Δp is expressed in the following form:

$$(\Delta x \Delta p)^2 = \left( \frac{A\pi^2 + B}{C} \right) \hbar^2$$(ΔxΔp)2=(CAπ2+B​)ℏ2

Here, $\Delta x = \sqrt{\langle x^2 \rangle - \langle x \rangle^2}$Δx=⟨x2⟩−⟨x⟩2​, $\Delta p = \sqrt{\langle p^2 \rangle - \langle p \rangle^2}$Δp=⟨p2⟩−⟨p⟩2​, and $\langle \cdot \rangle$⟨⋅⟩ denotes the expectation value. $\hbar$ℏ is the Dirac constant.

Given that $A$A and $C$C are coprime positive integers and $B$B is a negative integer, find the value of the product $A \times |B| \times C$A×∣B∣×C.

Constraints

• $A$A and $C$C are coprime positive integers (greatest common divisor is 1)
• $B$B is a negative integer

Input Format

Give the result of $A \times |B| \times C$A×∣B∣×C as a positive integer.