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

Find #xmv# in terms of this differential equation?
I hope the solution is the uncertainty principle

1 Answer
Truong-Son N.
Feb 8, 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 #1/2mv^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^(rx)# 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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