How Does Particle in a Box Relate to Quantum Mechanics?

[Zeynaz]

Homework Statement

A ball of mass 3 g is trapped in a box with infinitely high walls. The width of the box is 8 cm.
a) Calculate the speed of the ball when it is in the ground state of the box
b)Estimate the quantum number n if the ball is moving with a constant speed of 2 m/s in the box

Relevant Equations

Ek= 1/2 (m)(v)^2
E=hf
E-n= (n^2*h^2)/ (8*m*L^2)

I am having difficulties understanding this concept. Particle in a box (as far as i understood) shows that the electrons or particles need a certain about of potential energy to escape the one-dimensional potential well. I think its pretty similar to the concept of the electons and ionising energies. (correct me if i am wrong).

so in question a) i tried to use the formula by just using the values i knew, L=0.08 m and m=3e-3 kg. I also took n=1.
As a result E=4.32e-30 J. When i equate this energy with KE. I get a small velocity of 3m/s. (The correct answer is 1.0e-30 m/s)
I am not sure how to find the right answer.
Also, because the ball is in a box with infinitely high walls, it should have an infinite potential energy thus inifinite KE. So how is it possible to find a value for this.

can someone explain/help?

thanks!

 

[PeroK]

As a result E=4.32e-30 J. When i equate this energy with KE. I get a small velocity of 3m/s. (The correct answer is 1.0e-30 m/s)
I am not sure how to find the right answer.

It's difficult to see how such a tiny energy could equate to $3 m/s$ for an object of $3g$.

 

[PeroK]

Also, because the ball is in a box with infinitely high walls, it should have an infinite potential energy

The potential is 0 inside the box.

 

[Zeynaz]

oh right i used m as the electron but i had to use the mass of the ball.
this gives me v= 5.37e-14 m/s which is still not the correct answer.
Also, if the potential is 0 inside the box, E=KE.

 

[PeroK]

oh right i used m as the electron but i had to use the mass of the ball.
this gives me v= 5.37e-14 m/s which is still not the correct answer.
Also, if the potential is 0 inside the box, E=KE.

You need to show your calculations. Although it looks like you've simply not squared Planck's constant.

 

[Zeynaz]

yes you were right i made a mistake with my calculations.
then in this case my only question would be;
would it be possible to say that E(n)= PE+KE= hf?

 

[PeroK]

yes you were right i made a mistake with my calculations.
then in this case my only question would be;
would it be possible to say that E(n)= PE+KE= hf?

Thers is no potential in the box. That's a basic assumption about the problem. The only potential is the "infinite" potential outside the box, which stops the particle getting out.

 

[Zeynaz]

alright, thanks!