Math 421 Essay—Schrodinger’s Equation During this course, we have mostly discussed the wave equation, characterized by the following PDE: 1/c^2 (∂^2 u(x,t))/(∂t^2 )=(∂^2 u(x,t))/(∂x^2 ) We have also looked at the heat equation. Both of these partial differential equations we have looked at with a variety of boundary conditions and initial conditions. However, this essay will focus on a very different equation, giving a brief background on its idea, how it can be applied, and give a solution to it for one of its simpler cases. The equation in question is the time-dependent Schrodinger Equation of a single non-relativistic particle, which is given below: iℏ ∂Ψ/∂t=(-ℏ^2)/2m (∂^2 Ψ)/〖∂x〗^2 +V(x)Ψ(x,t) We will discuss the purpose and how to solve it with the given case throughout this essay. …show more content…
In classical physics, Newton’s second law (F = ma), is used to predict what an object will do at any time in the future given an initial condition. Schrodinger’s equation is the analogue of Newton’s second law for quantum mechanics, which explains the movement of a system (usually composed of atoms, subatomic particles, or molecules) given an initial condition and its wave function. The general form of Schrodinger’s equation is given