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Consider an infinite square-well potential of width a, but shifted so that the infinite potential barriers lie at x=-a/2 and x = a/2 (see left diagram below). The eigen-wavefunctions of this Hamiltonian have the general form: Vn(x) = Ancos(nπx/α) + Вsin(nлx/a). V(x) V(x) x=-a/2 x=0 x = a/2 x= -a/2 x=0 x = a/2 (a) Write down the normalized wavefunctions for the ground state (n = 1) and the first-excited state (n = 2). You can do so by solving the Schrodinger equation with the appropriate boundary conditions or by otherwise using what you know about the infinite square-well. (b) Write down the general expression for the eigen-energies E,,. (c) Now the width of the well is reduced by half. The barriers now lie at x = 0 and x= a/2, as shown in the right diagram above. Write down the general expression for the eigen-energies E.

Consider an infinite square-well potential of width a, but shifted so that the infinite potential barriers lie at x=-a/2 and x = a/2 (see left diagram below). The eigen-wavefunctions of this Hamiltonian have the general form: Vn(x) = Ancos(nπx/α) + Вsin(nлx/a). V(x) V(x) x=-a/2 x=0 x = a/2 x= -a/2 x=0 x = a/2 (a) Write down the normalized wavefunctions for the ground state (n = 1) and the first-excited state (n = 2). You can do so by solving the Schrodinger equation with the appropriate boundary conditions or by otherwise using what you know about the infinite square-well. (b) Write down the general expression for the eigen-energies E,,. (c) Now the width of the well is reduced by half. The barriers now lie at x = 0 and x= a/2, as shown in the right diagram above. Write down the general expression for the eigen-energies E.

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 30P
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Consider an infinite square-well potential of width a, but shifted so that the infinite potential barriers lie at x=-a/2 and x = a/2 (see left diagram below). The eigen-wavefunctions of this Hamiltonian have the general form: Vn(x) = Ancos(nπx/α) + Вsin(nлx/a). V(x) V(x) x=-a/2 x=0 x = a/2 x= -a/2 x=0 x = a/2 (a) Write down the normalized wavefunctions for the ground state (n = 1) and the first-excited state (n = 2). You can do so by solving the Schrodinger equation with the appropriate boundary conditions or by otherwise using what you know about the infinite square-well. (b) Write down the general expression for the eigen-energies E,,. (c) Now the width of the well is reduced by half. The barriers now lie at x = 0 and x= a/2, as shown in the right diagram above. Write down the general expression for the eigen-energies E.
Transcribed Image Text:Consider an infinite square-well potential of width a, but shifted so that the infinite potential barriers lie at x=-a/2 and x = a/2 (see left diagram below). The eigen-wavefunctions of this Hamiltonian have the general form: Vn(x) = Ancos(nπx/α) + Вsin(nлx/a). V(x) V(x) x=-a/2 x=0 x = a/2 x= -a/2 x=0 x = a/2 (a) Write down the normalized wavefunctions for the ground state (n = 1) and the first-excited state (n = 2). You can do so by solving the Schrodinger equation with the appropriate boundary conditions or by otherwise using what you know about the infinite square-well. (b) Write down the general expression for the eigen-energies E,,. (c) Now the width of the well is reduced by half. The barriers now lie at x = 0 and x= a/2, as shown in the right diagram above. Write down the general expression for the eigen-energies E.
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