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The most general form of the Schrodinger equation is: a(r, t) Ət ħ² -V²V(r, t) + V(r, t)(r, t), 2m where V(r, t) is the potential energy. We consider a free particle moving in the x-dimension. iħ 1) Show that the Hamiltonian operator can be written as

The most general form of the Schrodinger equation is: a(r, t) Ət ħ² -V²V(r, t) + V(r, t)(r, t), 2m where V(r, t) is the potential energy. We consider a free particle moving in the x-dimension. iħ 1) Show that the Hamiltonian operator can be written as

Modern Physics
Modern Physics
3rd Edition
ISBN:9781111794378
Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher:Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Chapter6: Quantum Mechanics In One Dimension
Section: Chapter Questions
Problem 35P
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The most general form of the Schrodinger equation is: a(r, t) Ət ħ² -V²v(r, t) + V(r, t)V(F, t), 2m where V(r, t) is the potential energy. We consider a free particle moving in the x-dimension. iħ 1) Show that the Hamiltonian operator can be written as ħ² d² 2m dx²¹ Ĥ = - H and verify that V₁(x) = eikx, V₂(x) = cos(kx) and V3(x) = sin(kx) are stationary solutions of the corresponding Schrodinger equation (write it ex- plicitely). 2) The momentum operator in one dimension is pa = -ih Calculate ÎxV₁(x), ÎxV₂(x) and pV3(x). Are V₁(x), V₂(x) or V3(x) eigenfunctions of the momentum operator? Why? What are the corresponding eigenvalues?
Transcribed Image Text:The most general form of the Schrodinger equation is: a(r, t) Ət ħ² -V²v(r, t) + V(r, t)V(F, t), 2m where V(r, t) is the potential energy. We consider a free particle moving in the x-dimension. iħ 1) Show that the Hamiltonian operator can be written as ħ² d² 2m dx²¹ Ĥ = - H and verify that V₁(x) = eikx, V₂(x) = cos(kx) and V3(x) = sin(kx) are stationary solutions of the corresponding Schrodinger equation (write it ex- plicitely). 2) The momentum operator in one dimension is pa = -ih Calculate ÎxV₁(x), ÎxV₂(x) and pV3(x). Are V₁(x), V₂(x) or V3(x) eigenfunctions of the momentum operator? Why? What are the corresponding eigenvalues?
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