Determine potential due to spring on particle is weak

[samee]

Homework Statement

A particle is in a box with length L. A spring attached to the particle is fixed at x=0. How can we determine the potential due to the spring is weak?

Homework Equations

H=p²/2m
Si(x)=sqrt(2/L)sin(npix/L)
E_n=n²pi²hbar²/2mL²
V(x)=kx²

The Attempt at a Solution

If I base my answer solely off of the quadratic equation, than that something must be x. If x is very small next to k, than wouldn't V(x) be small? I'm not sure if this is correct, however since there was so much information given in the problem... Help me please?

 

[nickjer]

Well, look at the units of 'k'. It is energy per unit length squared. Your problem has similar quantities, such as ground state energy and length of the box. So you will want your 'k' to be small compared to some form of those quantities that have the same units.

 

[samee]

So is it k<<V(x)/x² then? This can't be true... Am I comparing it to E_n/L²? k<<n²pi²hbar²/smL⁴ ?

 

[nickjer]

The last thing you said is better. Except you choose the ground state since it is a smaller energy. Also you can get rid of the pi and any other constants that have no units, since they are just numbers that don't contribute in this comparison. Not sure what the 's' is?

 

[samee]

meant 2 by the s.

Okay, so k<<n²hbar²/mL⁴ then?

 

[nickjer]

'n' has no units.

 

[samee]

awesome ^_^ thank you so much for your help.

I have k<<hbar²/mL⁴

 

[nickjer]

That is correct.