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This is the (non-relativistic) time-dependent Schrödinger wave equation for a particle subjected to a potential V(r,t). It is interesting and important to note that according to equations (6.3) and (6.4) we can make the following correspondence between the classical and quantum mechanical representations of energy and linear momentum ∂
E = p2 2m + U(x, t), E = p 2 2 m + U (x, t), where p is the momentum, m is the mass, and U is the potential energy of the particle. The wave equation that goes with it turns out to be a key equation in quantum mechanics, called Schrӧdinger’s time-dependent equation.
The Schrödinger equation is the heart of non-relativistic quantum me-chanics, in that virtually all the physics is derived from its solutions in var-ious systems. The origin of the equation is difficult to pin down, as every book on introductory quantum mechanics has its own way of introducing it.
For each of the five chapters in the text, I've recorded a series of video podcasts in which I explain the most significant principles, equations, and figures contained within that chapter. Supplemental material and references
Schrödinger Equation 6 In Quiz 2, you show that the free propagator satisfies the Schrödinger equation − for ℏ i ∂K(x B,t B;x A,t B) ∂t B = − ℏ2 2m ∂2K(x B,t B;x A,t A) ∂x2 B t B > t A and that the free particle wave function also satisfies the Schrödinger equation − ℏ i ∂ψ(x,t) ∂t = − ℏ2 2m ∂2ψ(x,t) ∂x2 ...
But what exactly is an operator, and what is the relation of any other observable quantity to an operator? Let us take this moment to flesh out some mathematical definitions. An operator is a rule for building one function from another.
This is a 2nd-order, ordinary differential equation (ODE), that yields complete knowledge of v(t) = ˙x(t) and x(t) given knowledge of any two positions, or any two velocities, or any combination of two, at any time.
