Download Stationary Schrödinger equation (1.5 LP) Vibrational states of a HCl

Potsdam University, Department of Physics
Computational Physics
WS 2013/2014
A. Pikovsky
Stationary Schrödinger equation (1.5 LP)
The one dimensional stationary Schroedinger equation belongs to the class of elliptic differential
equations.
ĤΨ(x) = En Ψ(x)
where Ĥ is the Hamilton operator. This equation defines an Eigenvalue problem for the energy
levels En of a physical system. The Hamilton operator is represented by
Ĥ = −
h̄2
∂x + V (x)
2m
where V(x) is the potential of a single particle of mass m is exposed.
Vibrational states of a HCl molecule
Given a single HCl molecule, in harmonic approximation the potential of the system can be
described by:
mω 2 x2
V (x) =
2
where ω = 2πν and the vibrational frequency ν = 8.66 · 1013 s−1 .
The analytical solution of the problem of the harmonic quantum mechanical oscillator is well
known from textbooks. A common method to solve the stationary Schroedinger equation numerically is the shooting method:
• Start two solutions ψ+ , ψ− at large distances from center (X+ , X− )
• Solve ordinary differential equation toward central a point (eg x = 0)
′
′
• Calculate at this point the Wronskian w(ψ+ , ψ− ) = ψ−
ψ+ − ψ+
ψ−
• Values of energy E for which the Wronskian vanishes are the Eigenvalues (why?)
• So the problem reduces to finding zeros of a function w(E) of one variable
Problem 1: Solve the stationary Schroedinger equation of the one dimensional harmonic oscillator for a HCl molecule by using its reduced mass µ defined as
1
1
1
=
+
,
µ
MH
MCl
( where MH is the mass of the hydrogen ion and MCl the mass of the chloride ion) for the first
five Eigenvalues and Eigenfunctions of the harmonic oscillator. Compare the numerical values
for En and the computed wave function Ψ(x) with the analytical solution. Use a common and
freely available plotting program for visualization (i. e. Gnuplot).
(1) How the accuracy depends on the positions (X− , X+ )?
(2) How the accuracy depends on the initial conditions at these points (use WKB-based initial
conditions, and some other ones)?
(3) How the accuracy depends on the numerical method for solving ODE (use Runge-Kutta
4th order, Euler, and Numerov methods)?
Problem 2: The molecular potential can be represented in a more elaborated way when using
a Lennard-Jones potential:
V (r) = ǫ
r 12
m
r
−2
r 6 m
r
For HCl rm = 1.27 · 10−10 m. Determine the value of ǫ by using a Taylor series decomposition
of V(x) up to second order.
Solve the stationary Schroedinger equation numerically for the first five Eigenvalues and Eigenfunctions again by using the Lennard-Jones potential.
Problem 3: Potential U(x) = −U0 / cosh2 αx possesses a finite number of eigenvalues and
eigenfunctions (see Landau and Lifshitz, Quantum Mechanics, $ 23, problem 5). Find the
eigenvalues and compare them to the theoretical formula.