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WOLFRAM NOTEBOOK
Quasi-Exact Solutions of Schrödinger Equation: Sextic Anharmonic Oscillator
Quasi-Exact Solutions of Schrödinger Equation: Sextic Anharmonic Oscillator
Quasi-exact solutions to a Schrödinger equation pertain to limited regions of a spectrum of eigenstates for which closed-form eigenfunctions and eigenvalues can be derived, whereas the remainder of the spectrum can only be approximated. These occur for certain potentials with parameters in some limited range. The sextic anharmonic oscillator is the only one-dimensional polynomial potential that can be quasi-exactly solved if its parameters are appropriately chosen. Depending on the parameters, the system can be a single-, double- or triple-well potential.
Consider solutions of the time-independent one-dimensional Schrödinger equation, in atomic units:
-ψ''(x)+V(x)ψ(x)=Eψ(x)
1
2
One trick for finding quasi-exact solutions is to assume some appropriately behaved function and to use the relation
ψ(x)
-=E-V(x)
1
2
ψ''(x)
ψ(x)
to identify a potential function . For example, the simplest case of a quasi-exact sextic anharmonic oscillator follows from
V(x)
ψ(x)=
-β4
4
x
e
α2
2
x
e
which gives
V(x)=-3β-αβ+
1
2
2
α
2
x
4
x
2
β
2
6
x
with
E=-
α
2
More generally, it can be shown that the potential can have the form
V(x)=-3β-2nβ-αβ+
1
2
2
α
2
x
4
x
2
β
2
6
x
n=0,1,2,…
For the cases , and , we show plots of the quasi-exact eigenfunctions and of the potential functions , with superposed eigenvalues shown in red.
n=0
1
2
V(x)