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Consider a charge q with mass m in a system with the Hamiltonian 2 H₁ = p² + 1 ma²x², Но 2m 2 i.e. a simple harmonic oscillator. The eigenfunctions are y,(x) with corresponding eigen-energies E. An external, constant, uniform electric field E is applied along x such that the charge is subjected to a force f = qE. (a) According to classical electrodynamics, this force f introduces an extra potential. What is then the new Hamiltonian H? (By now you should have learned the relationship between a conservative force and potential.) (b) Find the new eigen-energies E, of the Hamiltonian H. (c) Find the new eigenfunctions w", (x) of the Hamiltonian H. You can express w,,(x) in terms of the original y,(x). Hint: The classical counterpart of this problem is a mechanical spring with a spring constant k that is hung vertically to a fixed ceiling and attached at its bottom a mass m. Consider first the motion of the mass in a zero-gravity environment and observe what will change when you repeat the experiment on earth (with gravity). The gravitational force (mg) plays a role similar to the electric force (qE) in the current problem, and the gravitational potential (mgh) plays a similar role to the electric potential.

Consider a charge q with mass m in a system with the Hamiltonian 2 H₁ = p² + 1 ma²x², Но 2m 2 i.e. a simple harmonic oscillator. The eigenfunctions are y,(x) with corresponding eigen-energies E. An external, constant, uniform electric field E is applied along x such that the charge is subjected to a force f = qE. (a) According to classical electrodynamics, this force f introduces an extra potential. What is then the new Hamiltonian H? (By now you should have learned the relationship between a conservative force and potential.) (b) Find the new eigen-energies E, of the Hamiltonian H. (c) Find the new eigenfunctions w", (x) of the Hamiltonian H. You can express w,,(x) in terms of the original y,(x). Hint: The classical counterpart of this problem is a mechanical spring with a spring constant k that is hung vertically to a fixed ceiling and attached at its bottom a mass m. Consider first the motion of the mass in a zero-gravity environment and observe what will change when you repeat the experiment on earth (with gravity). The gravitational force (mg) plays a role similar to the electric force (qE) in the current problem, and the gravitational potential (mgh) plays a similar role to the electric potential.

University Physics Volume 3
University Physics Volume 3
17th Edition
ISBN:9781938168185
Author:William Moebs, Jeff Sanny
Publisher:William Moebs, Jeff Sanny
Chapter7: Quantum Mechanics
Section: Chapter Questions
Problem 41P: Show that (x,t)=Asin(kxt) and (x,t)=Acos(kxt) do not obey Schrödinger's time-dependent equation.
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Consider a charge q with mass m in a system with the Hamiltonian 2 H₁ = p² + 1 ma²x², Но 2m 2 i.e. a simple harmonic oscillator. The eigenfunctions are y,(x) with corresponding eigen-energies E. An external, constant, uniform electric field E is applied along x such that the charge is subjected to a force f = qE. (a) According to classical electrodynamics, this force f introduces an extra potential. What is then the new Hamiltonian H? (By now you should have learned the relationship between a conservative force and potential.) (b) Find the new eigen-energies E, of the Hamiltonian H. (c) Find the new eigenfunctions w", (x) of the Hamiltonian H. You can express w,,(x) in terms of the original y,(x). Hint: The classical counterpart of this problem is a mechanical spring with a spring constant k that is hung vertically to a fixed ceiling and attached at its bottom a mass m. Consider first the motion of the mass in a zero-gravity environment and observe what will change when you repeat the experiment on earth (with gravity). The gravitational force (mg) plays a role similar to the electric force (qE) in the current problem, and the gravitational potential (mgh) plays a similar role to the electric potential.
Transcribed Image Text:Consider a charge q with mass m in a system with the Hamiltonian 2 H₁ = p² + 1 ma²x², Но 2m 2 i.e. a simple harmonic oscillator. The eigenfunctions are y,(x) with corresponding eigen-energies E. An external, constant, uniform electric field E is applied along x such that the charge is subjected to a force f = qE. (a) According to classical electrodynamics, this force f introduces an extra potential. What is then the new Hamiltonian H? (By now you should have learned the relationship between a conservative force and potential.) (b) Find the new eigen-energies E, of the Hamiltonian H. (c) Find the new eigenfunctions w", (x) of the Hamiltonian H. You can express w,,(x) in terms of the original y,(x). Hint: The classical counterpart of this problem is a mechanical spring with a spring constant k that is hung vertically to a fixed ceiling and attached at its bottom a mass m. Consider first the motion of the mass in a zero-gravity environment and observe what will change when you repeat the experiment on earth (with gravity). The gravitational force (mg) plays a role similar to the electric force (qE) in the current problem, and the gravitational potential (mgh) plays a similar role to the electric potential.
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