ONE-ELECTRON ATOMS

The asymptotic condition as ( ) 0→rR ∞→r eliminates the divergent exponential solutions and indicates the acceptable asymptotic form as ( ) ,rer λξ −∼ provided we set

2 ⎟⎟⎠ ⎞

⎜⎜⎝ ⎛= =

ελ em (32.3) Therefore a trial solution to Eq.(32.2) is

( ) ( )rfer rλξ −= (32.4) and this leads to

( ) ( ) ( ) ( ) 0122 22 2 0

=⎥⎥⎦ ⎤

⎢⎢⎣ ⎡ +−+− rf

r ll

r Zem

dr rdf

dr rfd e

=λ (32.5) Assuming a power series solution

(32.6) ( ) ∑∞ =

= 0k

a procedure already used for the Hermite (30.20) and Legendre (31.32) equations, yields

m Zek k l l a k a rλ −+ ⎧ ⎫⎛ ⎞⎪ ⎪⎡ ⎤+ − + − − =⎜ ⎟⎨ ⎬⎣ ⎦ ⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭∑ = 0

and hence, we obtain

( ) ( ) ( ) kek allkk

Zemk a

11 /2 220

1 +−+ −=+ =λ (32.7)

For large k, Eq.(32.7) becomes identical to the recurrence relation for the terms in the series expansion ( )

ka a

r k

e k

k r λλλ 2or

! 2 1

However, this form is not acceptable for ( )rf because the solution (32.4) will diverge like as r tends to infinity. Thus, the series must terminate after a finite number of terms, for instance so that from Eq.(32.7), where

,nk = ,0and0 2 =≠ +nn aa we obtain

2 0 2or ===

eZm n

mZem n eee == ελ (32.8)

This is a quantum condition for the energy levels of one-electron atoms, of the form

m e eZ Z ZRh an n n

ε = − = − = −= (32.9) where R is the Rydberg constant (28.21), and is the Bohr radius (28.17). This result was first derived in Eq.(28.18) from Bohr's model for one-electron atoms. The acceptable values of the principal quantum number n must obey

0a

1+≥ ln (32.10) as required by the recurrence relation (32.7). Otherwise, for1, nn l a += is infinite and so are the succeeding coefficients. From Eqs.(31.43) and (32.10) we may conclude that, for a one-electron atom, the three quantum numbers, required for the physically acceptable solutions of the time-independent Schrödinger equation, have the following allowed integer values

1, 2, 3,n = … (32.11) while

0,1, 2, ,l 1n= −… (32.12) and

lml ≤≤− The radial eigenfunctions are obtained from Eqs.(31.25), (32.4), (32.6) and (32.8) as

( ) ∑∑ =

nrZem nl raer

rae r

rR e 0

0 11 = (32.13)

where the polynomials are solutions to Eq.(32.5), which is called the associated Laguerre equation and is the Bohr radius, as defined by Eq.(28.17). The expression (32.13) can be then combined with the appropriate spherical harmonics (31.44) to obtain the functions (31.17) in the form

( ) ( )0/ 0

1, , cos n

k r N e a r P e

r imϕθ ϕ θ− ±

⎛ ⎞Ψ = ⎜ ⎟⎝ ⎠∑ (32.14) where is a normalization constant. These are the complete time-independent state functions of the bound states of one-electron atoms. They will be used to describe the case of the hydrogen atom, which corresponds to

.1=Z

32.2. THE HYDROGEN ATOM We first consider the ground state of the hydrogen atom, given by the lowest values of the quantum numbers: 1, 0, 0.n l m= = = The eigenfunction (32.14) reads ( ) 0/00100 areNr −=Ψ (32.15) and shows a complete spherical symmetry, as it is not -θ nor -ϕ dependent. According to the Born interpretation, formulated in Eq.(29.1) for one dimension, the probability of finding the electron in a given volume element of space 2V sind r dr d dθ θ ϕ= about a point ( )ϕθ ,,r may be written as

( ) drddreNddrP ar ϕθθsinVV 2/22002100 0−=Ψ= (32.16) The constant is given by the normalization condition (29.2), which takes the form 00N

πϕθθ The integral has the form (IV.15), Appendix IV, so that

( )1/ 22 300 0 00 01 or 1/N a N aπ= = 3π (32.17) and the ground state function of the hydrogen atom becomes

( ) ( ) 0/2/130100 /1 arear −=Ψ π (32.18) In view of the spherical symmetry of the ground state, we may average the probability (32.16) over all the ,θ ϕ values to obtain

44 r a r aP r dr N r e dr r e d a

π − −= = r (32.19)

which gives the radial probability of finding the hydrogen electron in a spherical shell of thickness dr at a distance r from the nucleus, which is at the origin. The probability density plotted in Figure 32.1 as a function of r, is zero at the nucleus and in the asymptotic region, where

( ),rP ,∞→r so that a radial position of maximum probability must

exist, as given by

10 or 2 2 0 i.e.,r a dP r

e r r r dr a

− ⎛ ⎞= − =⎜ ⎟⎝ ⎠ a=

Therefore, the maximum occurs at a distance from the origin equal to the Bohr

radius However, we must emphasize that although the energy and the most probable radial position of the electron correspond to Bohr's results, a spherical distribution of the ground state probability is obtained, as illustrated by the angular probability density, given on a polar diagram in Figure 32.2, instead of a planar orbit predicted by Bohr's picture. The expectation radius of this spherical distribution is given by Eq.(29.8) as

0.a

4V r ar r d r e a

∞∗ −= Ψ Ψ =∫ ∫ dr (32.20) where we may use Eq.(IV.15), Appendix IV, to obtain

02 3 ar = (32.21)

We may further obtain from Eqs.(29.8) and (IV.15) that

2 34 0 adrer a

r ar == ∫ ∞

− (32.22)

which allows us to estimate of the standard deviation for the expectation electron radial position as

2 3 arrr r =><−><==∆ σ (32.23)

The expectation potential energy of the ground state can be calculated, starting from

1141 0

a drer

rar ar == ∫

which, if substituted into Eq.(32.1), gives

eZm rU e (32.24)

where the energy 1ε was inserted according to Eq.(32.9). Since we have

( )rUTH +== ˆˆ1ε the relation between the expectation values of the kinetic and potential energy operators, in the ground state, given by

( ) ( )rUrUT 2 1ˆ

11 −=−=−= εε obeys the classical virial theorem, Eq.(26.13), provided that the expectation values are interpreted as the average values of observables in the ground state.