55Q8 The statements in the following list all refer to the hydrogen atom. Each s

Question

55Q8

The statements in the following list all refer to the hydrogen atom. Each statement is prefaced by a label in square brackets which indicates whether the statement is to be judged according to the Coulomb model or according to more exact theories beyond the Coulomb model. Check the boxes of the THREE TRUE statements in the following list: [Coulomb model] All hydrogen atom energy eigenfunctions with n > 1 have at least one radial node. [Coulomb model] The fact that hydrogen atom energy eigenfunction are proportional to spherical harmonics is a consequence of the Coulomb potential energy function being independent of the the angular coordinates theta and phi, and has nothing to do with the precise 1/r form of the Coulomb potential energy function. [Coulomb model] Quantization of energy in a hydrogen atom occurs because the radial function cannot be allowed to diverge as r rightarrow infinity. [Beyond Coulomb model] In the fine structure of the hydrogen atom spectrum, a state with quantum numbers n, l and j = l + 1/2 has a slightly lower energy than a state with quantum numbers n, l and j = l = 1/2. [Beyond Coulomb model] Fine structure corrections lower the energy of the ground state of a hydrogen atom by a factor of order 0.001% rather than by 1%.

Explanation / Answer

1. False. By Coulomb model, there cannot be any radial nodes in the enrgy eigenfunctions of Hydrogen atom

2. True. As the coulomb potential is independent of any angular coordinates, the integral calculation with the wavefunction (Schrodinger's equation) over the whole volume will show angular isotropy and hence the energy eigenfunction are proportional to the spherical harmonics.

3. False. Quantization of energy occurs because otherwise the electron would just radiate energy because of the electromagnetic force and collapse into the nucleus

4. True. In the Fine structure of hydrogen atom spectrum, the enrgy is dependent on 1/(j+ 1/2) and therefore as j goes higher, the energy goes lower.

5. True. The fine structure constant has the value approximately 1/137. The energy correction to the ground state of hydrogen atom depends on the square of the fine structure constant, (1/137)^2, which gives the correction of a factor of order 0.00001 times or 0.001% that of the ground state energy.

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