Schrödinger's wave description of 1-electron atoms

The following exercises focus on the 1-electron atom or ion. Their solution assumes the quantum-mechanical approach, based on Schrödinger's wave equation and its solutions.

The potential energy is assumed to be exactly Coulomb -Ze²/r (no corrections due to the finite size of the nuclear charge distribution). The nuclear charge is assumed to be +Z q_e. a is the modified Bohr radius 4 πε₀ ℏ²/(µ q_e²) = a₀ µ/m_e, where µ = m_eM_n/(m_e+M_n) is the reduced mass of the 2-body electron-nucleus system and m_e is the electron mass. Fine-structure corrections (e.g. spin-orbit coupling) are to be neglected except where explicitly mentioned.

Radial part of the wavefunction:

1. Normalize an exponentially-decaying radial wavefunction R(r)=N exp(-r/r₀).
   RESULT: N=2 (r₀)^-3/2 [optionally multiplied by an arbitrary phase factor exp(iφ)]
   [Use has been made of the following integral: I_k= ∫₀^∞ dx x^k·exp(-x) = k! ]
2. 
   Find the best variational spherically-symmetric exponentially-decaying wavefunction for the 1-electron atom.
   In other words, take the normalized wavefunction of the previous exercise, and compute the expectation value of the H-like atom Hamiltonian on this state, assuming no angular dependence (so that the contributions of all the angular derivatives in the Laplacian vanish). This "mean value of the energy" depends on the parameter r₀: determine for which r₀ this mean energy is minimum.
   RESULT: One must find the exact ground-state wavefunction, i.e. r₀=a/Z
   NOTE: this is an exceptional variational exercise where among the (infinitely) many "variational" wavefunctions one tries, there happens to be the exact ground state, which obviously turns out to be the lowest in energy. In "real life" this almost never occurs: the best variational wavefunction is usually only an approximate ground state.
3. Compute the average (also called expectation value) of rⁿ over the 1-electron atom ground state wavefunction, for integer n.
   RESULT: For n≥-2: <rⁿ>=(a/2 Z)ⁿ·(2+n)!/2
4. In a one-el. atom in its ground state, what is the probability to find the electron at a distance from the nucleus greater or equal to a given d? Evaluate this probability for d=1 m and d=1 µm.
   RESULT: Indicating with u=2 Z d/a, P(outside d)=exp(-u)(1+u+u²/2).
5. [Written special test 2000-12-06] Indicate for which values of Z and of the quantum numbers (n,l,m) the radial distribution in a 1-electron atom is:

   P(r)=N r⁴ exp(-3 r/a)

   where N is the normalization constant. Compute N, <r>, <r^-1> and the radius r_M for which the radial distribution is maximum.
   The general structure of the radial wavefunction is:

   R_n,l(r)=N_n,l · (k r)^l L_n+l^2l+1(k r) exp(-k r),       with k=Z/(n a)

   where L_p^q(x) is a (p-q)-degree polynomial in x with nonzero x⁰ term, and N_n,l is a constant.
   RESULT: l=1, n=2, Z=3, r_M=4a/3

Angular part of the wavefunction:

1. Show that the parity of the one-electron wavefunction ψ_3,2,1 is (+1).
2. Evaluate the probability to find the electron in an equatorial "conical" band of ±10° around the xy plane θ=π/2, assuming the atom is in the ψ_3,2,2 orbital state.
   RESULT: P = [150 sin(10°) + 25 sin(30°) + 3 sin(50°)]/128 = 0.319104

Transition rates

1. Suppose you want to build a 1 W fluorescent bulb based on the 2p -> 1s transition in atomic H. Assume you would be able to place the H atoms well separated from each other so that the light emitted from one atom is unaffected by that emitted from the other atoms. Neglect all the fine and hyper-fine structure of H. How many 2p-excited atoms are needed at any given time?
   RESULTs: (In the numerics, we neglect the reduced-mass correction).
   Radial integration yields 128/81 sqrt(2/3) a/Z.
   Angular integration gives for the (1,m) -> (0,0) amplitude: 3^-1/2.
   Total single-atom transition rate = (2/3)⁸ Z⁴ α³ E_Ha/ℏ = 6.2683×10⁸ Z⁴ s^-1.
   Number of atoms = 2187/32 ℏ Power/(α³ E_Ha² Z⁶) = 9.75791×10⁸ atoms [at Power=1 W, Z=1].
2. Compute the the spontaneous-emission rates of the np -> 1s transitions in 1-electron atoms, for n=2 (see ex. above), 3, 4 and 5.
   RESULT: In units of Z⁴ α³ E_Ha/ℏ = m_eq_e¹⁰Z⁴ /[c³ (4πε₀)⁵ ℏ⁶] = 1.6065×10¹⁰ Z⁴ s^-1 one gets for n=2 to 5: 1/6561, 1/24576, 162/9765625, 40/4782969.
3. Compute the ratio between the spontaneous-emission transition rates of the 4p -> 3d and the 4p -> 3s decays in H.
   RESULT: 4096/36125

Relativistic corrections

1. [Written special test 2000-12-06] In an atomic H gas, let the electron have a significant probability to occupy the levels n=1,2,3. Accounting for the fine-structure corrections: a) draw a level scheme; b) compute the splittings in eV between levels within the same-n multiplets; c) indicate, in the dipole approximation the number of emission lines present in the Balmer and Lyman series, and compute the maximum wavelength in both these series.
   [reminder: the 1-electron atom levels are expressed by

   E_n,j = E_n [1 + (Z α)² / n (1/(j+1/2) - 3/(4n)) ]

   where E_n is the energy as given by the simple Z/r potential].
   RESULT: c) 121.566964 nm and 656.47268 nm