Question

1. Translational motion (Particle in a box) We want to design an experiment about energy quantization of a hydrogen atom via radiating a light. It is needed to predict which range of light do we have to radiate to excite an electron. Let’s assume the electron feels same potential(V=0) in certain distance(L) from the nucleus of hydrogen. 1. Let’s assume L=5.30x10-11m, me(electron mass)=9.11x10-31kg and ℏ=1.05x10-34m2 kgs-1 .

(a) Make a Hamiltonian for an electron in a hydrogen atom.

(b) Solve the Schrödinger equation using the Hamiltonian you made with applying boundary conditions.

(c) If the electron is in a ground state, which wavelength of light do we need to excite electron to 2nd state? What range the light belongs to (infrared, visible light etc.)?

(d) Which wavelength of light do we need to excite electron from nth state to n+1th state? Does wavelength of light get bigger when n increases?

2. Rotational motion

(a) Show that [?̂ ?,?̂ ?] = ?ℏ?̂ ? , [?̂ ?,?̂ ? ] = ?ℏ?̂ ?, [?̂ ? ,?̂ ?] = ?ℏ?̂ ?.

(b) Show that [?̂2 ,?̂ ? ] = 0, and then, without further calculations, justify the remark that [?̂2 ,?̂ ?] = 0 for all ? = ?, ?, and ?. What does this mean in terms of uncertainty principles?

Answer 1