Central potential. The hydrogen atom
6.2 The hydrogen atom
Before the advent of quantum physics people imagined atoms as miniature solar systems with the positively charged heavy nucleus in the center and negatively charged electrons orbiting around it, bound by the electric Coulomb field of the nucleus. The simplest atom in the Universe is the hydrogen atom. It only contains a positively charged proton as a nucleus and a single negatively charged electron. The proton is 1836 times as heavy as the electron, so according to the classical model the proton does not move perceptibly while the electron is orbiting it3. Because an accelerating charge loses energy by radiation this is not a classically consistent model. For a little generalization let us consider hydrogen like ions, i.e. atoms with Z protons in their nucleus ionized until only a single electron remains.
Example 6.3. After the discovery of de Broglie that electrons may have wave like prop-erties the danish physicist Niels Bohr proposed a simple model for the hydrogen atoms which we now call the Bohr model. He proposed that the electron will not emit radia-tion if it moves on such (classical) circular orbits where the circumference is an integer multiple of the wavelength of the electron (c.f. 3.1):
2r π =n λe =nh
p, where n= 1,2, ...
2Associated Legendre polinomials are the solution of the differential equation (1−x2)y00−2xy0+ l(l+ 1)− m2
1−x2
y= 0;
The first few associated Legendre polynomials are
P00(x) = 1, P10(x) =x, P11(x) =−p 1−x2,
3We could get rid of this assumptions by using the reduced mass of this 2-body problem instead of the electron massme:
m= memN
me+mN,
where mN is the mass of the nucleus. For hydrogenm= 0.99945me.
This is the same as the statement that the orbital angular momentum of the electron is quantized, as with simple rearrangement
r p= h
2πn =n~ (6.2.1)
and in our case the momentum vector is perpendicular to the radius vector, therefore
|L| = |r×p| = r p = r mev. This semiclassical result is different from (6.1.6) (not known to Bohr at his time).
The electron is held in orbit by the Coulomb force, which provides the necessary centripetal force:
This gives the total energy of the hydrogen atom to be half of its potential energy Etot = 1
i.e. the electron can orbit only at discreet radii rn. rn = (4π o)~2
meZ e2 n2 (6.2.5)
The radius of the first orbit r1 in a hydrogen atom (Z = 1) is denoted by ao
ao = 4π o~2
mee2 ≈0.0529nm (6.2.6)
and is called the Bohr radius4. For a hydrogen like atom:
rn=aon2 Z
4Sometimesao is written as
ao= ~ mec α,
wherecis the speed of light in vacuum andαis called thefine structure constant, introduced by Arnold Sommerfeld in 1916, which is the coupling constant characterizing the strength of the electromagnetic interaction. Being a dimensionless quantity, it has the same numerical value in all systems of units.
The current recommended value ofαis 7.2973525698(24)·10−3= 1/137.035999074(44)
The total energy then
where RE ≈ 13.6eV is called the Rydberg constant or the Rydberg unit of energy and the index n denotes the n-th electron orbit.
In quantum mechanics to get the possible energy levels and wave functions of the electron in the H atom we must solve the 3 dimensional time independent Schr¨odinger equation (3.5.4) for the centrally symmetric Coulomb potential of the proton:
H ϕ(x, y, z) =ˆ Eϕ(x, y, z) Because this potential is spherically symmetric we will be better off if we write this equation in spherical coordinates (see (5.1.19), or (22.9.6)):
H ϕ(r, θ, φ) =ˆ Eϕ(r, θ, φ) and remember, the eigenfunctions and eigenvalues of ˆL2 we already know. Therefore we can separate the radial and angular dependence of ϕ. Let ϕ(r, θ, φ) = R(r)Y`m(θ, φ), where Y`m(θ, φ) is a spherical harmonics, the eigenfunction of ˆL2, with the eigenvalue
`(`+ 1)~2. With this
Now multiply both sides withr2 and divide them withϕ=R Y`m then reorder the terms containing only the radial and only the angular part on the opposite sides of the equal sign:
therefore the equation for the radial part of the wave function is
− ~2 Note that this is “only” an ordinary second order differential equation, which nevertheless contains the unknown eigenvalue E, i.e this is also an eigenvalue equation. A mathemat-ical trick leads to a much simpler form. Let us introduce a new function u(r) with the formula: R(r) = u(r)
r . After some easy mathematical steps we get:
− ~2 which is a 1D Schr¨odinger equation with the effective potential
Vef f(r) = V(r) + ~2`(`+ 1)
The non-Coulomb part of this is sometimes called thecentrifugal potential This potential is repulsive. The resulting effective potential have a minimum where ∂ Vef f
∂ r = 0.
rmin = 4π o~2`(`+ 1) meZ e2 = 1
Z ao`(`+ 1)) for l >0 (6.2.15) (where ao is defined in (6.2.6)).
Even after such simplifications this equation is hard to solve and we will not attempt to do it here. For our purposes it is sufficient to show the eigenvalues and eigenfunctions.
The eigenfunctions can be characterized by 2 quantum numbers: a positive integer n = 1,2, ..., which determines the energy and`which describes what spherical harmonics
Figure 6.2: Radial and angular parts of the full wave functions and the total wave function for the first three energy level in a hydrogen atom (Z=1).
belong to this function5. In Fig. 6.2 we summarized the different wave functions for the first three energy levels (n = 1,2 and 3) in a hydrogen atom.
5The form of the radial part is
Rn,`(r) = s
2Z nao
3
(n−`−1)!
2n(n+`)! e−Zr/nao 2Zr
nao
`
L2`+1n−`−1 2Zr
nao
, (6.2.16)
where L(k)n are the generalized (or associated) Laguerre polynomials. These can be defined as:
L(k)n =exx−k n!
dn
d xn e−xxn+k
(6.2.17) The first few generalized Laguerre polynomials are
L(k)0 (x) = 1 (6.2.18)
L(k)1 (x) =−x+k+ 1 (6.2.19)
L(k)2 (x) =x2
2 −(k+ 2)x+(k+ 2)(k+ 1)
2 (6.2.20)
L(k)3 (x) =−x3
6 +(k+ 3)x2
2 −(k+ 2)(k+ 3)x
2 +(k+ 1)(k+ 2)(k+ 3)
6 (6.2.21)
In quantum mechanics the classical notion of a trajectory or orbit does not apply6. What quantum mechanics have instead is called anatomic orbital in an atom and molec-ular orbital in a molecule.
Important 6.2.1. An atomic or molecular orbital is a one electron wave function in an atom or a molecule respectively.
If an atom has more than one electron than both the Hamiltonian and its eigenfunctions contain the coordinates of all electrons. In this case the eigenfunctions are usually written as a linear combination of atomic orbitals.
Fig. 6.3 shows the radial part of the wave function for n= 1,2 and 3.
Using the correct quantum mechanical calculation for a hydrogen like ion gives exactly the same energy levels as the Bohr model:
En=−
(Z e2)2me 2 (4π o)2~2
1
n2 (6.2.22)
But, in contrast with the Bohr model, in the ground state the angular momentum of the electron is 0, the electron is not orbiting the nucleus! Similar states exist for at every n, because ` can be 0. And even in cases where the total angular momentum is not zero the electron does not move on classical orbits.
Important 6.2.2. The reason electrons in an atom or molecule do not emit electromag-netic radiation (except when exited from one stationary state to an other one) is that they do not move around the nucleus on classical orbits, therefore they do not accelerate in their stationary states.
The solutions of the original 3D Schr¨odinger equation of the hydrogen atom are characterized by 3 quantum numbers:
• the principal quantum number n= 1,2, ..., that determines the energy level
• the (orbital)azimuthal (or angular momentum) quantum number `= 0,1,2, ..., n−
1, which corresponds to the length of the angular momentum (sub-level orsubshell), and
• themagnetic quantum numberm =±1,±2, ...,±`, which determines the z-components of the angular momentum7.
6When chemists talk about the “orbit” of an electron in atoms or molecules they usually refer to the range in space where the electron can be found with an (arbitraryly chosen) 90% probability.
7There is a 4th quantum number, calledspin which we will discuss later.
Figure 6.3: Radial part Rn,` of the hydrogen wave function for the first three principal quantum numbers
Because all possible orbitals with the same principal quantum number have the same energy most of the energy levels are degenerate.
For historical reasons the states with different ` values have single letter names8 as well, as summarized in the following table.
` name
0 s
1 p
2 d
3 f
... ...
Table 6.1: Names for the different angular momentum states
The expressions 1s22s12p5, ... etc denotes the orbitals or electron shells where the number before the letter is the value of n, the letter determines the subshell (the value of
`) and the “exponent” is the number of electrons occupying that subshell in the ground state of the atom. The electron structure of an atom is then written as a series of these expressions. E.g. hydrogen has an electron structure of 1s1, helium with 2 electrons is 1s2 and argon with 10 electrons is 1s22s22p6.
As you can see the radial part of the ground state (the lowest energy) wave function R1,0 = 2
Z ao
3/2
e−Z r/ao (6.2.23)
neither have a minimum nor a maximum around ao/Z. The corresponding Y00(θ, φ) = 1
2√
π is constant. So not only R1,0 does not have a minimum, but neither have ϕ1,0,0 = R1,0Y0(0). So what does the Bohr-radiusao correspond to?
The probability that the electron is found in a dr range around the distance r from the nucleus is
Pn`m(r)d r = Z
angular part
Pn`m(r)dV
8Letters are abbreviations for“sharp”, “principal”,”diffuse”and“fundamental”. These are historical names used for the spectroscopic lines.
where dV ≡d3r=r2 sinθ dφ dθ dr: since the integral equals to 1
4π 4π = 1. This, however has a maximum. For the ground state of a hydrogen like ion, substituting P1,0,0:
P1,0,0 =r2 The probability may have an extremum where dP1,0,0/d r= 0
dP1,0,0 which can only be zero if the expression in the square brackets are zero.
2r−2Z r2
ao = 0 ⇒ 1− Z
ao r= 0 ⇒ r = ao Z,
i.e. the maximum of the radial probability for an electron in the ground state of a hydrogen like ion is at a distance rmax probab=ao/Z. For hydrogen rmax probab =ao.
In Fig. 6.4 we depicted the spatial probability as a function of the distance for the 5 lowest orbitals.
Figure 6.4: Radial probability functions in a hydrogen atom.