How Are Position and Momentum Connected by the Uncertainty Principle?

[CollectiveRocker]

How are position and momentum related?

 

[cepheid]

The momentum of an object depends on its velocity, which is the time derivative of its position. So if you have the position as a function of time, you know the velocity, and therefore the momentum of a particle:

However: as I've just learned elsewhere on the forum, a particle can begin its motion at a specific location with a specific velocity...these initial conditions are completely arbitrary (and independent of each other...at whatever time we are considering to be the initial time, the particle could start off anywhere with any velocity. ). Yet these initial conditions, or some boundary conditions, must be known in order to 'fix' a solution to the equations of motion. Only then is the relationship between position and velocity (and therefore momentum) is known. It is not known beforehand. I hope this helps.

 

[Andrew Mason]

How are position and momentum related?

First of all, position and momentum must be defined with respect to a frame of reference. You can only talk about an object's velocity, momentum and position relative to something else.

Velocity is the rate of change of position with respect to time. Momentum is defined as mass multiplied by its velocity. So position and momentum are related by mass and time.

AM

 

[CollectiveRocker]

The reason why I ask is that I'm working on a problem where the position and momentum are simultaneously discovered. I know the uncertainty in the position, yet was wondering how to find the uncertainty in the momentum. I'm not asking for you to do the problem for me; just to give me a smalll push in the right direction. Thanks.

 

[Andrew Mason]

The reason why I ask is that I'm working on a problem where the position and momentum are simultaneously discovered. I know the uncertainty in the position, yet was wondering how to find the uncertainty in the momentum. I'm not asking for you to do the problem for me; just to give me a smalll push in the right direction. Thanks.

Ok. Your question is not how position and momentum are related but how uncertainty of position is related to the uncertainty of momentum. That is the Heisenberg uncertainty principle:
$$\Delta p \Delta x = h$$ where h = Planck's constant and p refers to momentum and x to position

AM

 

[CollectiveRocker]

Are you positive that your equation is correct?

 

[CollectiveRocker]

How do I find the percentage of uncertainty in the particle's momentum?

 

[dextercioby]

Are you positive that your equation is correct?

Not really correct.Now I'm sure that Andrew knows the correct mathematical formulation of Heisenberg's principle,but he presented u with a form that could be easier to use in calculus,since it's an equality,while the real form in not.

I guess the push in the right direction has been given,since u have one equation with one unknown very simple to find.

 

[Andrew Mason]

Are you positive that your equation is correct?

I should have used $\Delta x \Delta p \approx h$. It is an uncertainty principle after all. It is really just an order of magnitude relationship which states that the uncertainty of position multiplied by the uncertainty of momentum is on the order of Planck's constant.

AM