Consider the dimensionless harmonic oscillator Hamiltonian, = A-+ with = -i d 2 dx (a) Show...

 

 

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Consider the dimensionless harmonic oscillator Hamiltonian, = A-+ with = -i d 2 dx (a) Show the two wavefunctions \(x) = e x/2 and y(x) = xex/2 are 3 " 2 eigenfunctions of with eigenvalues and respectively. 1 2 (b) Find the value of the coefficient a such that 2(x) = (1+ax)ex/2 is orthogonal to y,(x) and show that 1/2 (x) is an eigenfunction of with eigenvalue 52 Consider the dimensionless harmonic oscillator Hamiltonian, = A-+ with = -i d 2 dx (a) Show the two wavefunctions \(x) = e x/2 and y(x) = xex/2 are 3 " 2 eigenfunctions of with eigenvalues and respectively. 1 2 (b) Find the value of the coefficient a such that 2(x) = (1+ax)ex/2 is orthogonal to y,(x) and show that 1/2 (x) is an eigenfunction of with eigenvalue 52