Determine potential due to spring on particle is weak

In summary, the conversation discussed determining the potential due to a weak spring in a particle box with length L, using the quadratic equation and comparing the units of 'k' to similar quantities such as the ground state energy and length of the box. It was determined that k<<hbar2/mL4 would result in a weak spring potential.
  • #1
samee
60
0

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)
En=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?
 
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  • #2
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.
 
  • #3
So is it k<<V(x)/x2 then? This can't be true... Am I comparing it to En/L2? k<<n2pi2hbar2/smL4 ?
 
  • #4
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?
 
  • #5
meant 2 by the s.

Okay, so k<<n2hbar2/mL4 then?
 
  • #6
'n' has no units.
 
  • #7
awesome ^_^ thank you so much for your help.

I have k<<hbar2/mL4
 
  • #8
That is correct.
 

1. What is the potential due to a spring on a particle?

The potential due to a spring on a particle is a measure of the energy stored in the spring when it is stretched or compressed. It is directly proportional to the displacement of the particle from its equilibrium position.

2. How do you determine the potential due to a spring on a particle?

The potential due to a spring on a particle can be determined using the formula V = 1/2 * k * x^2, where V is the potential, k is the spring constant, and x is the displacement of the particle from its equilibrium position.

3. What is the relationship between potential due to a spring and the spring constant?

The potential due to a spring is directly proportional to the spring constant. This means that as the spring constant increases, the potential also increases.

4. Can the potential due to a spring on a particle be negative?

Yes, the potential due to a spring on a particle can be negative. This occurs when the particle is at a displacement in the opposite direction of the spring's natural direction of compression or extension.

5. How does the displacement of the particle affect the potential due to a spring?

The potential due to a spring is directly proportional to the square of the displacement of the particle. This means that as the displacement increases, the potential also increases, but at a faster rate.

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