Physics 249 Lecture 18, Oct 17th 2012
Reading: Chapter 7.1, 7.2, 7.3
HW due Friday Oct 19th. Available on web site.
1) Bound states of an electrostatic potential: The Hydrogen atom
The potential energy:
Vr=-ke2r
The time independent Schrodinger equation in spherical coordinates is:
Eψ=-ℏ22μ1r2∂∂rr2∂ψ∂r-1r21sinθ∂∂θsinθ∂ψ∂θ+1sin2θ∂2ψ∂ϕ2+V(r)ψ
Separating variables:
ψr,θ,ϕ=Rrfθg(ϕ)
gϕ=eimϕ
where Cϕ=m2, m=0,±1,±2…
Lets consider m=0. The simplest solution for fθ is:
fθ=C
Then
Cr=0
1R(r)∂∂rr2∂R∂r+2μℏ2r2E-Vr=0
1R(r)∂∂rr2∂R∂r+2μℏ2r2E+ke2r=0
∂∂rr2∂R∂r+2μℏ2r2E+ke2rRr=0
Again not an easy differential equation to solve: Let’s try a negative exponential. Negative so that the probably goes down with large r.
Rr=Ce-r/a0
∂∂rr2-1a0R(r)+2μℏ2r2E+ke2rRr=0
-2r1a0+r21a02Rr+2μℏ2r2E+ke2rRr=0
-2a0r+1a02r2Rr+2μℏ2Er2+2μke2ℏ2rRr=0
a0=ℏ2μke2
E1=-ℏ22μa02=-μ2ke2ℏ2=13.6eV
ψr,θ,ϕ=Ce-r/a0
where the constant will come from normalizing the wave function.
ψr,θ,ϕ=14π2a03e-r/a0
The first wave function is spherically symmetric with a distribution that falls off exponentially characterized by a radius a0, which is identical to the Bohr radius!
Looking at the wave functions above it is clear that the full set of wave functions can be more complex than spherically symmetric though they will have phi symmetry. The solutions are called orbitals where the distributions are analogs of orbits with a characteristic vector z about which there is phi orbital symmetry but otherwise complex functional dependences in theta.
The radial wave function, though independent of phi and theta will have complex functional dependences as well.
2) Full electrostatic potential (Hydrogen atom) solution:
We explored the simplest solution to the problem, which corresponds to the ground state of all three quantum numbers. Both the theta and radial wave equations clearly have more complex solutions. Also, since each separate wave equation is related by constants, which are different depending on the quantum numbers there will be relationships between the three quantum numbers of the system.
ψnlmr,θ,ϕ=CnlmRnlrflmθgm(ϕ)
gmϕ=eimϕ
flmθ=sinθm2ll!ddcosθl+mcos2θ-1l
m=-l…0…+l
Where the f functions are known as the Legendre functions
The combination of f and g functions are often expressed as spherical harmonics
Ylmθ,ϕ=Clmflmθgmϕ
l=0, m=0
Y00θ,ϕ=14π
l=1, m=1,0,-1
Y11θ,ϕ=-38πsinθeiϕ
Y10θ,ϕ=34πcosθ
Y1-1θ,ϕ=38πsinθe-iϕ
Where we have normalized the wave functions over the angular space. The l=0, m=0 state is the one we already solved.
Cr=ll+1 connecting the radial and angular equations and quantum numbers.
Note that this is a constant factor with no dependence on the angular or radial coordinates. However, it is a different constant for each value of l each of which is a separate solution the angular portion of the Schrodinger equation. For each value of l, and thus different constants, the solutions to the radial equation will be different.
Rnlr=Cnle-r/na0rlLnlr/a0
n= 1,2,3 …
l= 0, …, n-1
Where the L functions are known as the Laguerre polynomials.
n=1, l=0
R10r=2a03e-r/a0
n=2, l=0,1
R20r=12a031-r2a0e-r/2a0
R21r=126a03ra0e-r/2a0
and
En=-ℏ22μn2a02=-μ2n2ke2ℏ2=-E1n2
3) Exploring this solutions
There are three quantum numbers.
n: principle quantum number. The energy is set by this quantum number. For a given n all the states of l and m will be degenerate in energy. As expected we find an energy degeneracy.
l: the orbital quantum numbers: We will find that this quantum number is associated with the angular momentum of the orbit.
m: the magnetic quantum number: We will find that the projection of the angular momentum on the z axis is associated with this quantum number.
Radial distributions.
The radial distributions are easiest to understand in case where l=m=0. In those cases the probability distribution is spherically symmetric. These cases are the so-called s orbitals.
Though since the angular portion is separated we can also look at the radial distributions in the cases where l>0. The probability distributions as a function of the radial coordinate are an exponential or an exponential times a polynomial distribution. A distance scale set by the Bohr radius characterizes them. They can be understood by graphing the probability distribution and quantified by calculating the maxima, minima and expectation values. For higher n these distributions will be pushed out to higher values.