1. (a) Outline how the quantum numbers n, 1 and m originate as a consequence of the solution of the Schrödinger equation for the hydrogen atom and explain their physical significance. (b) Show that H2 1,0,0 = exp(-r/ao) παρ is an eigenfunction of the Hamiltonian for the hydrogen atom: -h² 1 a მ Ĥ 2 + 2m r² Or Ər 1 მ r² sin 0 ǝ0 (· მ 1 a² e2 sin 0. + ᎧᎾ r² sin² 0 ǝo² Απερτ h² 2ma me2 ≈0.529 × 10-10m and the remaining symbols with eigenvalue The quantity ao = have their usual meanings. (c) Find the expectation value of r for the 1,0,0 state and give its physical significance. (d) Given that the energy levels of the hydrogen atom scale as answer to (b) to find the wavelength of the transition from the to the first excited state. Hint: The following integral may be of use, xne-bx dx = لوم n! bn+1 use your 1,0,0 state

1. (a) Outline how the quantum numbers n, 1 and m originate as a consequence of the solution of the Schrödinger equation for the hydrogen atom and explain their physical significance. (b) Show that H2 1,0,0 = exp(-r/ao) παρ is an eigenfunction of the Hamiltonian for the hydrogen atom: -h² 1 a მ Ĥ 2 + 2m r² Or Ər 1 მ r² sin 0 ǝ0 (· მ 1 a² e2 sin 0. + ᎧᎾ r² sin² 0 ǝo² Απερτ h² 2ma me2 ≈0.529 × 10-10m and the remaining symbols with eigenvalue The quantity ao = have their usual meanings. (c) Find the expectation value of r for the 1,0,0 state and give its physical significance. (d) Given that the energy levels of the hydrogen atom scale as answer to (b) to find the wavelength of the transition from the to the first excited state. Hint: The following integral may be of use, xne-bx dx = لوم n! bn+1 use your 1,0,0 state