Gaussian wavepacket and position-momentum uncertainty

[solas99]

We know that momentum is proportional to k so by adding more waves
to localise our particle we are adding more waves with independent
momentum values
Upon measurement, the gaussian wavepacket must collapse into one
eigenstate of momentum, but if we have a very localised packet there will be
a great many possible momentum vales for it to collapse into
Here we have a physical reason for the position-momentum uncertainty
principle

am i right with this?

thanks

 

[solas99]

does this explain the origin of position-momentum uncertainty, using gaussian wavepacket.

 

[Sam Wong]

what you are saying is correct, but it's not the origin of uncertainty principle.

the uncertainty principle can be purely a mathematical theorem, based on the mathematical formulation of quantum mechanics

you can read 《Modern Quantum Mechanics》by J.J. Sakurai, P34-36

or even wikipedia gives a proof of it

 

[The_Duck]

I think Sam Wong is too discouraging; this is indeed the origin of the position-momentum uncertainty principle (and similar reasoning establishes the time-energy uncertainty principle). The math just makes this more rigorous and shows how to generalize it to arbitrary noncommuting operators. Of course, you do need the mathematical formulation of quantum mechanics to establish that momentum is represented in position space by the operator -i d/dx, but once you have this the position-momentum uncertainty principle follows from your reasoning.

 

[Sam Wong]

I'm sorry if I made you think I was discouraging.

But indeed, we have to be serious about the origin of something.

I would say, the uncertainty of gaussian wave packet is a consequence of the uncertainty principle, not the origin.

Dirac, Schrödinger and those physicists developed the Quantum theory based on physical insights and experimental results (and some postulations). And the uncertainty principle is derived from the theory.

If you look at the history of uncertainty principle (from Heisenberg), their argument was from matrix wave mechanics.

gaussian wave packet is just a special case, not the origin