Spectrum of 1-electron atoms

1. [Eisberg-Resnick 4.38] He⁺ emits radiation at roughly the same wavelength as the H_α line (the first Balmer line of hydrogen. Which are the n involved for He⁺? Which case emits the longer wavelength? Compute the wavelength difference.
   RESULT: λ_H = 656.4671 nm for n=3→n=2. λ_He+ = 656.2039 nm for n=6→n=4.
2. [Written test 2001-07-05] The frequencies of the spectral lines emitted by an hypothetical one-electron atom are given by

   ν = 10¹⁵ (1/n₁² - 1/n₂²) Hz

   where n_m is the principal quantum number.

   a) Construct the level scheme for this atom. Indicate the binding and excitation energies for the four lowest levels.

   b) Explain clearly what processes can take place when a large ensemble of such atoms in their ground state is bombarded by (1) electrons of kinetic energy 3.5 eV or (2) 5.0 eV; (3) photons of wavelength 3373 Å and (4) 2480 Å.
   RESULT: b)
   (1) elastic scattering and excitation n=1→n=2;
   (2) elastic scattering, excitation n=1→any n, ionization;
   (3) elastic (Rayleigh) scattering, excitation n=1→n=3;
   (4) elastic (Rayleigh) scattering, photoemission (ionization).

3. [Inspired to Written test 1998-07-10] A 4.00 eV electron is captured by a bare He nucleus. If the process involves emission of a photon of wavelength λ=1234.15 Å determine the quantum number n of the bound state into which this electron has been captured. If n>1, which is the energy of the photon emitted in the transition to n-1?
   RESULT: n=3.   7.5577 eV
4. Beyond what nuclear charge Z, the nonrelativistic mechanics of a single-electron atom becomes entirely meaningless due to relativistic effects?
   RESULT: Z_max=1/α ≃ 137
5. a) Give an expression for the level density as a function of energy for the bound states of the Coulomb potential of H.
   b) Same thing for the density of states (the degeneracy of a level n is n²)
   c) Use this approximate expression to estimate the number of states between -10^-5Ry and -10^-6Ry.
   RESULT: b) dn_states/dE_n = (1 Ry)^3/2/(2 E_n)^5/2. c) 3×10⁸ states.

As the energy spectrum following Bohr's model coincides with the one of nonrelativistic Schrödinger's equation, these exercise can be solved equally well either using the Bohr model or the current nonrelativistic quantum-mechanical formalism.