The hydrogen atom consists of a proton of mass m_p=1.7´10^-27kg and charge q_e=1.6´10^-19C and an electron of mass m_e=9.1´10^-31kg and charge -q_e.  The dominant part of the interaction between the two particles is the electrostatic interaction.  The electrostatic potential energy is

in SI units.

If we confine ourselves to studying the relative motion of the two particles, and if we neglect any external forces, and if we treat the particles as spinless particles, then the Hamiltonian of the system is

.

The reduced mass m of the system is nearly the same as the electron mass m_e, and the center of mass of the system is nearly in the same place as the proton. We therefore often call the relative particle "the electron" and the center of mass "the proton".

The eigenvalues and eigenfunctions of H(r,p)

.

H is the Hamiltonian of a fictitious particle moving in a central potential.  We already know that the eigenfunctions of H are of the form

,

where u_kl(r) satisfies the radial equation

with the condition u_kl(0)=0.

We define the effective potential .

In terms of the effective potential the radial equation becomes

.

As r approaches infinity, V_eff(r) goes to zero.  For every E_kl>0 there therefore exists a solution u_kl(r) with an oscillatory behavior at infinity.  These solutions represent unbound states.

For E_kl<0 u_kl(r) will be proportional to  as r approaches infinity.  The real positive exponential solution is physically unacceptable.  As r goes to zero, u_kl(r) becomes proportional to r^l+1.  Only for certain discrete values of E_kl is it possible to properly match these two asymptotic solutions.

In order to simplify the radial equation, we introduce the dimensionless quantity  , where .  The radial equation then becomes

.

We want to solve this equation for E_kl<0 to find the energies of the bound states of the hydrogen atom.

Defining

we have

.

We therefore try a solution of the form

.

c_kl satisfies the equation

.

We have

.

Therefore

.

We may write this as .

The coefficients a_q for each power q of r must vanish for this to hold for all r.  For the power q+l-1 we therefore must have .   .

We have found a recurrence relation for the coefficients c_q.  Fixing c₀ allows us to calculate all c_q.

We have

.

As .

This implies that as c_kl(r) becomes proportional to , since

.

The asymptotic behavior of u_kl(r) is then dominated by which is physically unacceptable.

In order to keep the wavefunction finite at infinity the power series defining must terminate.  For some integer q=k we need c_k=0.  We then have c_k(q>k)=0. For c_k=0 we need .

Since we require c₀ to be non zero, the possible values for k are k=1, 2, 3, × × ×.  For a given l the possible values of E_kl then are .

c_kl(r) is a polynomial whose lowest order term is r^l+1and whose highest order term is r^k+l.

In terms of c₀ the various coefficients c_q are calculated using the recurrence relations

.

.

.

The associate Laguerre polynomials are defined as

.

We therefore have

.

c_kl(r) is proportional to a polynomial of order k-1 and therefore has k-1 radial nodes.

For u_kl(r) we have

.

.  Normalization requires .

.

We therefore have

.

.

Energy levels

For each l there exists an infinite number of possible energies, corresponding to values of k=1, 2, 3, × × ×.  Each of them is at least 2l+1 fold degenerate.  This is the essential degeneracy, since E_kl does not depend on m.  Accidental degeneracies also exist.  For the hydrogen atom we find that E_kl is only a function of k+l, and we define k+l=n for the hydrogen atom.

The possible eigenenergies therefore are .  Here n is called the principal quantum number, n fixes the energy of the eigenstate.  Given n, l can take on n possible values  . n characterizes an electron shell, which contains n subshells characterized by l.  Each subshell contains 2l+1 distinct states.

The total degeneracy of the energy level E_n is for a hydrogen atom made from spinless particles.

• The wavefunctions of the hydrogen atom
• Probability densities (1)
• Probability Densities (2)
• The hydrogen atom

Definitions and notations

The ground state energy of the hydrogen atom is -E_I.

where

.

a is the fine structure constant.

.

We also have

.

R_H is the Rydberg constant.

.

We denote the energy eigenstates of the hydrogen atom by { |n,l,m> }, where n=k+l.  The corresponding eigenfunctions are usually written as .

, where k=n-l, and u_kl(r) is given above. The number of radial nodes of R_nl(r) is n-l-1.

Spectroscopic notation

Letters of the alphabet are associated with various values of l.

Continue in alphabetic order.

The subshells are denoted by:

etc.

The principal shells are denoted by:

Continue in alphabetic order.

Problem:

In the interstellar medium electrons may recombine with protons to form hydrogen atoms with high principal quantum numbers.  A transition between successive values of n gives rise to a recombination line.
(a)  A radio recombination line occurs at 5.425978 * 10¹⁰ Hz for a n = 50 to n = 49 transition.  Calculate the Rydberg constant for H.
(b)  Compute the frequency of the H recombination line corresponding to the transition n = 100 to n = 99.
(c)  Assume the mean speed in part (b) is 10⁶ m/s.  At what frequency or frequencies would the recombination line be observable?
(d)  Consider that radio recombination lines may be observed at either of two facilities, the 11 meter telescope at Kitt Peak near Tuscon, Arizona, and the 1.2 meter radio telescope at Columbia University in New York.  Relative larger blocks of time are available on the smaller telescope, but its intrinsic noise is moderately high.  Where would you choose to map recombination radiation emanating from an external galaxy.  Discuss both technical and non technical aspects of your choice.

Solution:

• Concepts:
  The hydrogen atom, transitions between energy levels, the Doppler shift
• Reasoning:
  The wavelengths of the light emitted by excited hydrogen atoms can be found using the formula
  1/λ = R_H(1/n² - 1/n'²).  The observed wavelengths depend on the relative speed of the source and the observer.

• Details of the calculation:
  (a)  E_n' - E_n = hν = hcR_H(1/n² - 1/n'²).
  5.425978 * 10¹⁰ = 2.99792458 * 10⁸R_H(1/49² - 1/50²),  R_H = 1.097373 * 10⁷ m^-1.
  (b)  ν = cR_H(1/99² - 1/100²) = 6.679710 * 10⁹ s^-1.
  (c)  The line is Doppler shifted.
  ν' = ν[(1 + v/c)/(1 - v/c)]^1/2 ~ ν(1 + v/c) if v/c << 1.
  Here v is the relative velocity of source and observer; v is positive if the source and the observer approach each other, and v is negative if the source and the receiver recede from each other.
  -10⁶/(3*10⁸) < v/c < 10⁶/(3*10⁸),   6.657 * 10⁹ s^-1 < ν' < 6.702 * 10⁹ s^-1.
  (d)  Here are some factors worth considering:
  The amount of radiation gathered by a telescope is proportional to the square of its diameter D.
  The smallest angle that can be resolved is approximately θ = λ/D.
  (For each photon we have ΔxΔp_x ~ h, Δx ~ D,  ΔP_x ~ h/D, θ = Δp_x/p = hc/(Dhν) = λ/D.)
  If you assume that the radiation has a frequency of ~ 10^-10 s^-1, then  λ = c/ν = 3 cm.  For the small telescope we therefore have θ = 0.03/1.2  while for the large telescope we have θ = 0.03/11.  It takes (11/1.2)² = 84 times as long to gather the same amount of radiation with the small telescope as with the large telescope.