EEE434/591: Quantum Mechanics
1. The potential energy of an electron in a hydrogen atom (in MKS units) is
V r e
r
( ) = −
2
0
4πε ,
where
e
is the electron charge and
r
is the distance of the electron from the
proton. The hydrogen atom is, of course, a three-dimensional system, and
r
is
the radial coordinate of a spherical coordinate system with the origin at the proton.
However, we can model this system by a simple, continuous one-dimensional
potential energy whose TISE we can solve. For this purpose, we can assume that
the potential energy term in the 1D TISE is of the form:
V x
e
xx
x
( ) ,
,
=− >
∞ ≤
R
S
|
T
|
2
0
40
0
πε
The bound state energies of an electron in this potential can be determined using
the method of power series that was explained on the example of a simple
harmonic oscillator (SHO). Let's find the solution step-by-step:
(a)
Simplify the TISE for the total energy
E
by changing the variable of
differentiation from
x
to one that is dimensionless. Define this new
variable
ξ
in terms of the constant:
a
me
0
0
2
2
4
=πε
ℏ
,
which represents the Bohr radius of the hydrogen atom. Introduce a
dimensionless energy parameter
ε
defined by:
ε
−
= −
20
2
2
2mEa
ℏ
,
and the dimensionless length variable
ξ
ξε
=2
0
x
a .
(b)
The equation that you have obtained can not be immediately solved by, for
example, inserting into a power series expansion. You need to calculate
the asymptotic limit, i.e. the solution
g( )
ξ
for
ξ → ∞
.
(c)
Once you have calculated the asymptotic solution
g( )
ξ
, express the
wavefunction as
ψ ξ ξ ξ
( ) ( ) ( )
=
Ag f
, and find the differential equation for
f( )
ξ
.
(d)
Solve the differential equation for
f( )
ξ
using a power series expansion,
i.e.
