Imagine a particle of mass m, constrained to move along the x-axis, subject to some specified force F(x, t). The program of classical mechanics is to deter- mine the position of the particle at any given time: x(t). Once we know that, we can figure out the velocity (\( v=\frac{dx}{dt}\) ), the momentum (p = mv), the kinetic energy ( \( T=\frac{1}{2}mv^2 \) ), or any other dynamical variable of interest. And how do we go about determining x(t)? We apply Newton's second law: F = ma. (For conservative systems the only kind we shall consider, and, fortunately, the only kind that occur at the microscopic level---the force can be expressed as the derivative of a potential energy function, \( F=-\frac{\partial V}{\partial x} \) , and Newton's law reads \( m\frac{d^2x}{dt^2}=-\frac{\partial V}{\partial x} \) .) This, together with appropriate initial conditions (typically the position and velocity at t 0), determines x(t). Quantum mechanics approaches this same problem quite differentl...
The Prophet Muhammad (peace be upon him) said, "Islam is built upon five pillars; bearing witness that there is no god but Allah, and Muhammad is the Messenger of Allah, establishing the prayer, giving zakat, fasting in Ramadan, and performing the pilgrimage to the House of Allah." The Pillars of Islam are the obligatory acts of worship for every Muslim. One's Islam is not considered complete unless they believe in the obligation of these pillars and practice them all, as Islam is built upon them. Therefore, they are referred to as the Pillars of Islam. The First Pillar: The testimony 'Lā ilāha illallāh' and 'Muḥammad rasūlullāh' Allah the Almighty has stated," "Therefore, know that there is no deity worthy of worship except Allah." [Muhammad/47: 19] Allah the Almighty also says, "Indeed, there has come to you a Messenger from among yourselves. Grievous to him is what you suffer; [he is] concerned over you and to the believers is kind a...