CHAP. 12: BASIC TERMS OF CHEMICAL PHYSICS [CONTENTS] 426
Forn= 1 the Schr ̈odinger equation has only one possible solution corresponding to the
minimum energy of an atom. It is theground stateof the electron at equilibrium. The
wavefunction is spherically symmetrical with the most probable distance between the electron
and the nucleus at
r=ε 0 h^2
π μ e^2 Z. (12.37)
If we substituteμ=me= 9. 109 × 1031 kg andZ= 1 into this equation, we obtain a quantity
denoteda 0 ,
a 0 =ε 0 h^2
π mee^2= 5. 292 × 10 −^11 m. (12.38)It is called theBohr radiusand is used as the atomic unit of length.
If the atom absorbs energy, the electron may transit from the state of equilibrium to a
state with a higher energy (n >1), called theexcited state. For these states there are always
several wavefunctions of the same energy but differing in the values of theangular momentum
quantum number^6 (l) and themagnetic quantum number(m). Their energies are thus
degenerate.
In the presence of a magnetic field the energy of atoms with quantum numbersm > 0
slightly changes. The change can be interpreted as an interaction of the magnetic fieldBwith
a moving electron creating a current loop around the nucleus. It holds
∆Eel=e h
4 π mem B=mBm B , (12.39)wheremB = 9. 274 × 10 −^24 A m^2 is the magnetic moment of a free electron called theBohr
magneton^7. Since the magnetic quantum numbermhas both positive and negative values,
the energy of the atom may both increase and decrease against (12.36).
During interaction with the magnetic field, the electron and the nucleus display another
property calledspin. The magnitude of the electron spin is s= 1/2 while the nucleus spin
depends on the number of nucleons and acquires values which are integer multiples of one half.
An energy change due to the interaction of the magnetic field with the electron spin is given by
∆Eel=gemBmsB , (12.40)(^6) also calledazimuthalquantum number
(^7) The magnetic moment of a free electron is actuallyge/2 = 1.00116 times larger, see (12.40).