Question

Part A

Calculate the average value of the kinetic energy for the H atom in its ground state.

Express your answer in terms of e, ε0, a0.

〈Ekinetic〉 =

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Part B

Calculate the average value of the potential energy for the H atom in its ground state.

Express your answer in terms of e, ε0, a0.

Answer 1

The electron energies in the hydrogen atom do nor depend on the quantum numbers m and l which characterize the dependence of the wave function on the angles ? and ?. The allowed energies are

E_n = -me⁴/(2?²n²) = -13.6 eV/n².

Here n is called the principle quantum number. The values E_n are the possible value for the total electron energy (kinetic and potential energy) in the hydrogen atom.

Part -A

The average kinetic energy is me⁴/(2?²n²).

Part -B

Theaverage potential energy is -me⁴/(?²n²)

The wave functions ?_nlm(r,?,?) = R_nl(r)Y_lm(?,?) are products of functions R_nl(r), which depend only on the coordinate r, and the spherical harmonics Y_lm(?,?), which depend only on the angular coordinates. They are characterized by three quantum numbers, n, l, and m.