Explanation Given info: The uncertainty product of harmonic oscillator for the ground level is Δ x Δ p x = ℏ 2 . Consider a Newtonian oscillator with amplitude A . When it reaches the amplitude the kinetic energy will become zero and total energy of the oscillator become equal to potential energy at that point. Write the expression for the total energy of a Newtonian oscillator. E = 1 2 k ' A 2 (1) Here, E is the total energy of the Newtonian oscillator k ' is the force constant A is the amplitude of the oscillator Write the expression for the energy of the n ’ th level of harmonic oscillator. E n = ( n + 1 2 ) ℏ ω (1) Here, n is the excited state quantum number ω is the angular frequency of the oscillator h is the planks constant ℏ is h 2 π Set total energy of the Newtonian energy equal to excited state energy of the quantum oscillator 1 2 k ' A 2 = ( n + 1 2 ) ℏ ω Solve the above equation to find the value of A . A = ( 2 n + 1 ) ℏ ω k ' (3) When the oscillator is moving at maximum speed v max then the total energy will becomes equal to kinetic energy at that point because the potential energy will be zero at that point. Write the expression for the total energy of the Newtonian oscillator at equilibrium point. E = 1 2 m v max 2 Here, m is the mass of the particle v max is the maximum speed of the oscillator Set this energy also equal to the energy of the excited state of the harmonic oscillator as in equation (2). 1 2 m v max 2 = ( n + 1 2 ) ℏ ω Solve the above equation to get v max . v max = ( 2 n + 1 ) ℏ ω m (4) Thus write the expression for maximum linear momentum. p max = m v max Here, Substitute ( 2 n + 1 ) ℏ ω m for v max in the above expression to find p max . p max = m ( ( 2 n + 1 ) ℏ ω m ) = ( 2 n + 1 ) m ℏ ω Write the expression for Δ x

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Author: YOUNG

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Chapter 40, Problem 40.39E

To determine

The uncertainty product ΔxΔpx for an excited level that has quantum number n .

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Chapter 40, Problem 40.38E

Chapter 40, Problem 40.40E

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Chapter 40 Solutions

MASTERINGPHYSICS W/ETEXT ACCESS CODE 6

Ch. 40.1 - Does a wave packet given by Eq. (40.19) represent...Ch. 40.2 - Prob. 40.2TYU Ch. 40.3 - Prob. 40.3TYU Ch. 40.4 - Prob. 40.4TYU Ch. 40.5 - Prob. 40.5TYU Ch. 40.6 - Prob. 40.6TYU Ch. 40 - Prob. 40.1DQ Ch. 40 - Prob. 40.2DQ Ch. 40 - Prob. 40.3DQ Ch. 40 - Prob. 40.4DQ

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The uncertainty product Δ x Δ p x for an excited level that has quantum number n .

Solution Summary: The author explains the uncertainty product of a Newtonian oscillator for an excited level that has quantum number n.

The uncertainty product ΔxΔpx for an excited level that has quantum number n .

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Chapter 40 Solutions