Concept of potential well | Oscillations and Waves

 Equilibrium and concept of potential well

• In all conservative field, potential energy U = U (x, y, z)
• Since force

                    

• If particle moves only along x-axis, then force F_x = - (∂U/∂x)
• Similarly F_y = - (∂U/∂y) and F_z = - (∂U/∂z)

• Force = slope of tangent at any point of the curve
• Since the tangent at P, Q, R and S are parallel to x-axis

• These positions are known as equilibrium positions.
• If a particle is slightly displaced from stable equilibrium position P, then it starts to oscillate between points A and B until it crosses point B.
• P is the position having minimum potential energy and is called stable equilibrium.
• The region of minimum potential energy bounded between points A and B is called  potential well .
• The difference between the maximum and minimum potential energy of a potential well is called the  binding energy of potential well.
• If the energy of particle is less than the B.E. of well, then it is always bound in the potential well and such state is called  bound state .
• Let a particle be slightly displaced from its mean position P = (x = x₀) then from Taylor sereis expansion, its potential energy at any point x

• For stable equilibrium position P

• If P lies at origin i.e., x₀ = 0 and U(x₀) = 0

• For small displacement x³ → 0, x⁴ → 0, ...

                    

                    

• In this position a curve between displacement and potential energy will be a  parabola and F ∝ x
Therefore the motion of particle in a parabolic potential well is always oscillatory and is simple harmonic .

To know more about potential well, click on the  https://youtu.be/9-hY4FPeQqg  or  https://youtu.be/GIG6_or5vV0