Help with the Heisenberg uncertainty principle

[Bazanaka]

Today I was assigned a question (that is due tomorrow) and I currenlty have very little idea how to solve it... Any help to get me started here would be greatly appreciated.
1. Use the uncertainty principle to estimate the uncertainty in energy of a proton confined to a nucleus 1.0 x 10^-14m in diameter.

Here are the equations we were given
2. (delta x)(delta p) >= h/2pi
(delta E)(delta t) >= h/2pi


To try and solve it I wasn't really sure where to start because we weren't given the uncertainty in time or the uncertainty in momentum so I am not sure how I can make the transition to energy :grumpy:

Any help will be greatly appreciated. I do not need someone to solve this for me, please just point me in the right direction so I can learn the material.

 

[jbsteele]

To start, note that the uncertainty principle states that the product of the uncertainty in position and momentum must be greater than or equal to Planck's constant divided by 2π. Therefore, you need to find the uncertainty in position and momentum of the proton in order to calculate the uncertainty in its energy. The uncertainty in position is given as 1.0 x 10-14m in diameter. Therefore, you can calculate the uncertainty in position by dividing the diameter of the nucleus by 2. This gives an uncertainty in position of 5.0 x 10-15 m. The uncertainty in momentum can be calculated using the equation delta p = h/delta x; where h is Planck's constant. Substituting the values for delta x and h gives an uncertainty in momentum of 1.27 x 10-27 kg m/s. Now that you have the uncertainties in position and momentum, you can calculate the uncertainty in energy using the equation delta E(delta t) >= h/2pi. Substituting the value of h and the previously calculated values for delta x and delta p gives an uncertainty in energy of 1.27 x 10-27 Joules.

 

[baba89]

Hello,

The Heisenberg uncertainty principle is a fundamental principle in quantum mechanics that states that it is impossible to know both the position and momentum (or energy and time) of a particle with absolute certainty. In other words, the more precisely we know the position of a particle, the less precisely we can know its momentum (or energy), and vice versa.

In the case of a proton confined to a nucleus, we can use the uncertainty principle to estimate the uncertainty in its energy. The first equation you were given, (delta x)(delta p) >= h/2pi, relates the uncertainties in position and momentum. The second equation, (delta E)(delta t) >= h/2pi, relates the uncertainties in energy and time.

To solve this problem, we can use the fact that the diameter of the nucleus is given as 1.0 x 10^-14m. This can be used to estimate the uncertainty in position, delta x. From the first equation, we can rearrange it to solve for delta p, which represents the uncertainty in momentum. Once we have the uncertainty in momentum, we can use the second equation to solve for delta E, the uncertainty in energy.

To summarize, the steps to solve this problem are as follows:

1. Use the given diameter of the nucleus to estimate the uncertainty in position, delta x.
2. Use the first equation, (delta x)(delta p) >= h/2pi, to solve for delta p, the uncertainty in momentum.
3. Use the second equation, (delta E)(delta t) >= h/2pi, to solve for delta E, the uncertainty in energy.

I hope this helps to get you started. Remember, the uncertainty principle is a fundamental concept in quantum mechanics and is used to describe the behavior of particles on a very small scale. It can be a tricky concept to grasp, so don't hesitate to ask for further clarification if needed.

Best of luck with your assignment!

Sincerely,