7. Suppose a particle is connected to a spring and undergoes one-dimensional mot

Question

7. Suppose a particle is connected to a spring and undergoes one-dimensional motion.

(a) Write an expression for the total (kinetic plus potential) energy of the particle in terms of its position x, its mass m, its momentum p, and the force constant ? of the spring.

(b) Now treat the particle as a wave. Assume that the product of the uncertainties in position and momentum is governed by an uncertainty relation ?x?p ? ½?.

Also assume that because x is, on average, zero, the uncertainty ?x is roughly equal to a typical value of |x|. Similarly, assume that ?p |p|. Eliminate p in favor of x in the energy expression.

(c) Find the minimum possible energy for the wave, and express it in terms of the vibrational frequency of a mass on a spring: f = 1/(2?) sqrt(?/m)

Explanation / Answer

a) Energy = Kinetic + Potential

Energy = 1/2*mv2 + 1/2kx2

Energy = 1/2*m*p2 / m2 + 1/2kx2 ( as p = mv)

Energy = p2 / 2m + 1/2kx2

b) E = (? / 2 ?x)2 / 2m + 1/2k ((?x)2

c) Differentiation (b) with respect to ?x , we get'

dE / d ?x = - ?2 / 4m ?x3 + k (?x)

solving for ?x, we get

?x =  (?2 / 4mk)1/4

so, E = 1/2*?*SQRT(k/m)

we know that ? = h /2*pi

so, E = 1/2*h/2*pi*SQRT(k/m)

as 1/(2?) sqrt(?/m) = f ( given in the question)

E = 1/2*h*f

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