Consider the dimensionless Hamiltonian = 2+, with P = -id/dx. (a) Show that the wave...

 

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Consider the dimensionless Hamiltonian ÂĤ = ¿2+, with P = -id/dx. (a) Show that the wave functions wo(x) =e=x*/2//J and w1(x) = /2//axe-/2 are eigenfunctions of Ĥ with eigenvalues 1/2 and 3/2, respectively. (b) Find the values of the coefficients a and B such that 1 w2(x) = Tava (ax²? – 1)e2 and w3(x) = and w3(x) = :(1+ px²)e¬*/2 are orthogonal to wo(x) and wi(x), respectively. Then show that w2(x) and w3(x) are eigen- functions of H with eigenvalues 5/2 and 7/2, respectively. Consider the dimensionless Hamiltonian ÂĤ = ¿2+, with P = -id/dx. (a) Show that the wave functions wo(x) =e=x*/2//J and w1(x) = /2//axe-/2 are eigenfunctions of Ĥ with eigenvalues 1/2 and 3/2, respectively. (b) Find the values of the coefficients a and B such that 1 w2(x) = Tava (ax²? – 1)e2 and w3(x) = and w3(x) = :(1+ px²)e¬*/2 are orthogonal to wo(x) and wi(x), respectively. Then show that w2(x) and w3(x) are eigen- functions of H with eigenvalues 5/2 and 7/2, respectively.