Introduction to Mathematical Models and Quantum
Mechanics
Goals:
• become familiar with how to work with expressions involving
functions, derivatives, and integrals in a symbolic math program
such as MathCad
• discover the relationship between a physical picture (observed
energy levels of hydrogen) and a mathematical model (radial
functions that are the only allowed solutions to an equation).
Reference:
Your textbook presents some background on pages 103-104.
In the pre-lab discussion, we saw how the hydrogen atom is understood
to have only certain allowed radial orbits for the electron. Each orbit
corresponds to a different energy level. Transitions between these
particular energy levels and no others have been observed, thus
demonstrating the hydrogen atom is a quantum mechanical entity.
We wish to explore a mathematical model (an equation if you will) that
will have these allowed energies and orbits (and no others) to emerge
naturally. We will replace the circular orbits of Bohr theory with the
radial wavefunctions, R(r). R is the function that depends on r, the
distance between the nucleus and the electron. These R(r) will be
solutions to a differential equation called the Hamiltonian.
)()(
ˆrRErRH ⋅=⋅
We read this as “ H
ˆworks on )(rR to yield Etimes )(rR .” E is the energy.
In other words, the Hamiltonian works on the radial function to give the
function back except for a factor out in front (the energy).
The Hamiltonian for the radial solutions to the hydrogen atom can be
expressed most easily in atomic units:
)()(
1
)(
2
)1()(
2
1
2
2
2rRErR
r
rR
r
dr
rdR
r
dr
d
r⋅=⋅−⋅
+
+
⎟
⎠
⎞
⎜
⎝
⎛
−ll
There are only three variables in this equation:
• r is the distance between the nucleus and the electron
• lis an integer with possible values 0,1,2,… which corresponds to
s,p,d,… that you learned in general chemistry!
• E is the energy
Docsity.com
