Download Quantum Mechanics: The Hydrogen Atom

Quantum Mechanics: The Hydrogen Atom
12th April 2008
I. The Hydrogen Atom
In this next section, we will tie together the elements of the last several
sections to arrive at a complete description of the hydrogen atom. This will
culminate in the definition of the hydrogen-atom orbitals and associated
energies. From these functions, taken as a complete basis, we will be able to
construct approximations to more complex wave functions for more complex
molecules. Thus, the work of the last few lectures has fundamentally been
amied at establishing a foundation for more complex problems in terms of
exact solutions for smaller, model problems.
II. The Radial Function
We will start by reiterating the Schrodinger equation in 3D spherical coordinates as (refer to any standard text to get the transformation from Cartesian
to spherical coordinate reference systems). Here, we have not placed the constraint of a constant distance separting the masses of the rigid rotor (refer
to last lecture); furthermore, we will keep in the formulation the potential
V (r, θ, φ) for generality.
Thus, in spherical polar coordinates, Ĥ(r, θ, φ)ψ(r, θ, φ) = Eψ(r, θ, φ)
becomes:
"
−h̄2
2µ
1 ∂
∂
r2
2
r ∂r
∂r
∂
1
∂
+ 2
sinθ
r sinθ ∂θ
∂θ
∂2
1
+ 2 2
r sin θ ∂φ2
!
#
+ V (r, θ, φ) ψ(r, θ, φ) = Eψ(r, θ, φ)
Now, for the hydrogen atom, with one electron found in ”orbits” (note the
quotes!) around the nucleus of charge +1, we can include an electrostatic
potential which is essentially the Coulomb potential between a positive and
negative charge:
V (|r|) = V (r) =
−Ze2
4πo r
It is important to note that the Coulomb potential as we have
written it here is simply a function of the magnitude of the position vector between the 2 masses (i.e, between the electron and
1
nucleus). There is no angular dependence!.
Recall:
2
L̂ = −h̄
2
1
∂
∂
sinθ
2
r sinθ ∂θ
∂θ
∂2
1
+ 2
r sinθ ∂φ2
!
is the total angular momentum squared operator (function of θ and φ only!).
Thus, we can rewrite the Schrodinger equation as:
−h̄
2
∂
∂
r2
∂r
∂r
2
+ 2µr [V (r) − E] ψ(r, θ, φ) + L̂2 ψ(r, θ, φ) = 0
This demonstrates that the Hamiltonian is separable since the terms in
brackets are functions of r only, and the angular momentum operator is
only a function of θ and φ. Thus, the wavefunction can be written in a
form that lends to separation of variables. Recalling that the spherical
harmonics are eigenfunctions of the angular momentum operator:
ψ(r, θ, φ) = R(r)Ylm (θ, φ)
Separation of V ariables
L̂2 Ylm (θ, φ) = h̄2 l(l + 1)Ylm (θ, φ)
l = 0, 1, 2, ...
Accounting for separation of variables and the angular momentum resuls,
the Schrodinger equation is transformed into the Radial equation for the
Hydrogen atom:
−h̄2
2µr 2
dR(r)
d
r2
dr
dr
"
#
h̄2 l(l + 1)
+
V (r) − E R(r) = 0
2µr 2
The solutions of the radial equation are the Hydrogen atom radial wavefunctions, R(r).
II. Solutions and Energies
The general solutions of the radial equation are products of an exponential
and a polynomial. The eigenvalues (energies) are:
E=
−Zµe4
−Z 2 e2
=
8πo ao n2
82o h2 n2
n = 1, 2, 3, ..
The constant ao is known as the Bohr Radius:
2
2o h2
πµe2
ao =
The Radial eigenfunctions are:
"
(n − l − 1)!
Rnl (r) = −
2n [(n + l)!]3
#1 2
2Z
nao
l+ 3
2
l
re
−Zr
nao
L2l+1
n+l
2Zr
nao
2Zr
The L2l+1
n+l
nao are the associated Laguerre functions. Those for n = 1
and n = 2 are shown:
n = 1;
n = 2;
l = 0;
n = 2;
l = 0;
L11 = −1
L12 = −2! 2 −
l = 1;
Zr
ao
L33 = −3!
How to normalize:
Spherical Harmonics:
Z
2π
dφ
0
Z
π
0
dθ sinθ Ylm∗ (θ, φ)Ylm (θ, φ) = 1
Radial Wavefunctions:
Z
∞
0
∗
dr r 2 Rnl
(r)Rnl (r) = 1
The total hydrogen atom wavefunctions are:
ψ(r, θ, φ) = Rnl Ylm (θ, φ)
Table 1. Nomenclature and Ranges of H-Atom quantum numbers
Name
Symbol
Allowed Values
principle quantum number
n
1,2,3,....
angular momentum quantum number
l
0,1,2,...,n-1
magnetic quantum number
m
0,±1, ±2, ±3, ..., ±l
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ENERGIES: DEPEND on ”n”
E=
−Z 2 e2
8πo ao n2
ANGULAR MOMENTUM: DEPEND on ”l”
Lz−component : DEP EN Don”m”
.
p
|L| = h̄ l(l + 1)
Lz = mh̄
Total H atom wavefunctions are normalized and orthogonal:
Z
2π
dφ
0
Z
π
dθ sinθ
0
Z
∞
0
∗
drr 2 ψnlm
(r, θ, φ)ψn∗ 0 l0 m0 (r, θ, φ) = δnn0 δll0 δmm0
Lowest total Hydrogen atom wavefunctions: n=1 and n=2 ( define
σ ≡ Zr
ao )
Table 2. Hydrogen Atom Wavefunctions
n
l
m
1
0
0
2
0
0
1
0
1
±1
Orbital Name
ψnlm
√1
32π
ψ200 =
ψ210 =
ψ21±1 =
√1
32π
√1
64π
√1
π
ψ100 =
Z
ao
3
3
2
Z
ao
Z
ao
Z
ao
2
2
e−σ
(2 − σ)e
3
3
2
σe
σe
−σ
2
−σ
2
ψ1s
−σ
2
ψ2s
cosθ
ψ2pz
sinθe±iφ
Transforming to real functions via normalized linear combinations
1
±1
ψ2px =
√1
32π
Z
ao
3
σe
1
±1
ψ2py =
√1
32π
Z
ao
3
σe
2
2
−σ
2
−σ
2
sinθcosφ
ψ2px =
√1
2
(ψ21+1 + ψ21−1 )
sinθcosφ
ψ2py =
√1
2
(ψ21+1 − ψ21−1 )
Thus, we have come to the point where we can connect what we
already know from previous analysis:
4
Table 3. Quantum numbers
n = 1, 2, 3, ...
l = 0, 1, 2, ..., n − 1
m = 0, ±1, ±2, ..., ±l
Orbital
1
0
0
1s
2
0
0
2s
2
1
0
2pz
2
1
+
2px
2
1
-
2py
What are the degeneracies of the Hydrogen atom energy levels?
Recall they are dependent on the principle quantum number only.
III. Spectroscopy of the Hydrogen Atom
Transitions between the energy states (levels) of individual
atoms give rise to characteristic atomic spectra. These spectra
can be used as analytical tools to assess composition of matter.
For instance, our knowledge of the atomic composition of the sun
was in part aided by considering the spectra of the radiation from
the sun. For the Hydrogen atom, early scientists observed that
the emission spectra (generated by exciting hydrogen atoms from
the ground to excited states), gave rise to specific lines; the spectra were NOT continuous. The understanding of the quantum
mechanical nature of the hydrogen atom helps us understand how
these lines arise.
Series of lines in the hydrogen spectrum, named after the scientists who observed and characterized them, can be related to the
energies associated with transsitions from the various energy levels of the hydrogen atom. The relation, simple enough as it is,
turns out to accurately predict the spectral lines. The equation
relating the wavelength (and thus energy via E = hν) associated
with a transition from a state n1 to another tate n is given by:
1
= RRydberg
λ
1
1
− 2
2
n1 n
n1 = 2; Balmer Series
n1 = 1; Lyman Series
n1 = 3; P aschen Series
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