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3. Given the orthonormal basis functions X₁ = 000, X2 = 100, X3 = P010, X4 = P001 which are the eigenfunctions of the ground state and the first excited state of a three-dimensional isotropic harmonic oscillator in Cartesian coordinates, calculate the matrices for the position vector, r

3. Given the orthonormal basis functions X₁ = 000, X2 = 100, X3 = P010, X4 = P001 which are the eigenfunctions of the ground state and the first excited state of a three-dimensional isotropic harmonic oscillator in Cartesian coordinates, calculate the matrices for the position vector, r

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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3. Given the orthonormal basis functions X1 = Po00, X2 = $100, X3 = Po10, X4 = Poo1 which are the eigenfunctions of the ground state and the first excited state of a three-dimensional isotropic harmonic oscillator in Cartesian coordinates, calculate the matrices for the position vector, r %3D %3D
Transcribed Image Text:3. Given the orthonormal basis functions X1 = Po00, X2 = $100, X3 = Po10, X4 = Poo1 which are the eigenfunctions of the ground state and the first excited state of a three-dimensional isotropic harmonic oscillator in Cartesian coordinates, calculate the matrices for the position vector, r %3D %3D
Expert Solution
Step 1

Consider the given orthonormal basis functions,

χ1=ϕ000, χ2=ϕ100, χ3=ϕ010, χ4=ϕ001

For an isotropic oscillator, energy eigenvalues can be determined by,

Enxnynz=nx+ny+nz+32hω, where nx, ny and nzrepresents set of quantum number.

For ground state energy eigenvalue E000=3hω2 which can be found by substituting nx=0, ny=0, and nz=0.

hence, χ1=ϕ000 is a ground state eigenfunction.

For first excited state energy eigenvalue E000=5hω2 which can be obtained for  χ2=ϕ100, χ3=ϕ010, χ4=ϕ001.

Hence,  χ2=ϕ100, χ3=ϕ010, χ4=ϕ001 are the eigenfunction of the first excited state in three fold degenerate.

 

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