# (del^2Psi)/(delx^2) + (8pi^2m)/(h^2)(1/2mv^2)Psi = 0 Find xmv ?

## Find $x m v$ in terms of this differential equation? I hope the solution is the uncertainty principle

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3
Feb 9, 2018

It's not... the free-particle Schrodinger equation is:

(d^2psi)/(dx^2) + (2mE)/(ℏ^2) psi = 0

If for some miraculous reason the energy was $\frac{1}{2} m {v}^{2}$, then you've assumed implicitly that this is going to be a plane wave solution, which is well-known in physics and has nothing to do with Heisenberg uncertainty...

(d^2psi)/(dx^2) + (2m)/(ℏ^2) (1/2 mv^2)psi = 0

(d^2psi)/(dx^2) + (p/ℏ)^2psi = 0

A general solution is proposed, i.e. $\psi = {e}^{r x}$ such that

r^2e^(rx) + (p/ℏ)^2e^(rx) = 0

=> r = (ip)/ℏ

Thus,

color(blue)(psi(x) = c_1e^(ip x//ℏ) + c_2e^(-ip x//ℏ))

This is not normalizable, as allspace is literally all of the world.

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