The Cubic Potential : Consider a particle of mass m moving in one dimension under the influence of potential energy
u(x) = 
Here ꞷ, δ and a are real and positive.
A
Sketch typical plots of u(x) and identify extrema if any.
B
Consider the case where (in appropriate units) we have m = 1, ꞷ= 
, α = 1 and δ = 1/2. Sketch the potential energy u(x). If the total energy of the particle moving in this one-dimensional potential is E = 0 (in same units), discuss the motion of the particle in terms of allowed regions, boundedness and periodicity.
, α = 1 and δ = 1/2. Sketch the potential energy u(x). If the total energy of the particle moving in this one-dimensional potential is E = 0 (in same units), discuss the motion of the particle in terms of allowed regions, boundedness and periodicity.
Text Solution
Verified by ExpertsThe correct answer is:
CHECK THE SOLUTION.
Sol. U(x) = 
– δ x – 
U’(x) = – αx 2 + mꞷ 2 x – δ
if D < 0. m 2 ꞷ 4 – 4αs < 0.
D > 0, m 2 ꞷ 4 – 4αs > 0.
U(x) = x 2 – 
– 
U’(x) = – x 2 + 2x – 
D = 4 – 4× 
= 2
U = 0
x = 0, 
x = 0, 
between x = 0 and x = 
. U is (–ve). So, K.E. is +ve.
So, this the only allowed region.
The particle will oscillate between these two points periodically.
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