Solving the Van der Waals Force Problem - Talib

[Talib]

Hello,

Can anyone lead me or give me a hint on how to solve this problem?

Interaction between neutral atoms and molecules can be decomposed to
two different forces: an attractive force at large distances (the Van der
Waals force), and a repulsive force at short distances (due to overlap between electron wavefunctions). A simple mathematical description of this
interaction between two particles is given by the Lennard-Jones potential:


U (r) = 4*epsilon [(sigma/r)^12 - (sigma/r)^6]


where [sigma] and [epsilon] are empirical parameters (epsilon is the depth of the potential well and sigma is the (finite) distance at which the potential is zero), and r is the distance between the particles.

(a) Sketch this potential.

(b) Are there any forbidden regions? If so, what are they and why? If
not, why not?

(c) Obtain an estimate for the number of allowed energy levels corresponding
to bounded states.

(d) Calculate the energies of these (bounded) energy levels.

Thanks a lot! :D

Talib

 

[Pythagorean]

Hello,

Can anyone lead me or give me a hint on how to solve this problem?

Interaction between neutral atoms and molecules can be decomposed to
two different forces: an attractive force at large distances (the Van der
Waals force), and a repulsive force at short distances (due to overlap between electron wavefunctions). A simple mathematical description of this
interaction between two particles is given by the Lennard-Jones potential:


U (r) = 4*epsilon [(sigma/r)^12 - (sigma/r)^6]


where [sigma] and [epsilon] are empirical parameters (epsilon is the depth of the potential well and sigma is the (finite) distance at which the potential is zero), and r is the distance between the particles.

(a) Sketch this potential.

(b) Are there any forbidden regions? If so, what are they and why? If
not, why not?

(c) Obtain an estimate for the number of allowed energy levels corresponding
to bounded states.

(d) Calculate the energies of these (bounded) energy levels.

Thanks a lot! :D

Talib

have you even tried yet?

I'll get you started on a). You're basically just looking at U as a function of r, everything else is a constant. plot the points r at 0 and r at infnite to get a couple ideas for how the graph looks. You can also take the derivative and set it to 0 to find minimums and maximums.

 

[Talib]

aite thanks .. i figured the drawing .. but how can i obtain an estimate of the # of allowed energy levels??

 

[Pythagorean]

aite thanks .. i figured the drawing .. but how can i obtain an estimate of the # of allowed energy levels??

I'm taking solid-state right now, I haven't done quantum yet, so that's where I saw Leonard-Jones potentials, and we just solved for minimum energy.

I'm thinking it will have to do with your minimums on your graph, since minimums in potential energy usually represent a stable equilibrium.