Computational Physics - Department of Physics

470 14 Quantum Monte Carlo Methods

−

̄h^2 r^2

2 m

(

∂

∂r(r

2 ∂R(r)

∂r )

)

−

ke^2

r R(r)+

̄h^2 l(l+ 1 )

2 mr^2 R(r) =E R(r).

Introducing the functionu(r) =rR(r), we can rewrite the last equation as

−

̄h^2

2 m

∂^2 u(r)

∂r^2 −

(

ke^2

r −

h ̄^2 l(l+ 1 )

2 mr^2

)

u(r) =E u(r), (14.15)

wheremis the mass of the electron,lits orbital momentum taking valuesl= 0 , 1 , 2 ,..., and
the termke^2 /ris the Coulomb potential. The first terms is the kinetic energy. The full wave
function will also depend on the other variablesθandφas well. The energy, with no external
magnetic field is however determined by the above equation. We can then think of the radial
Schrödinger equation to be equivalent to a one-dimensionalmovement conditioned by an
effective potential

Veff(r) =−

ke^2

r

+ ̄

h^2 l(l+ 1 )

2 mr^2

.

The radial equation yield closed form solutions resulting in the quantum numbern, in
addition tolml. The solutionRnlto the radial equation is given by the Laguerre polynomials
[93]. The closed-form solutions are given by

ψnlml(r,θ,φ) =ψnlml=Rnl(r)Ylml(θ,φ) =RnlYlml

The ground state is defined byn= 1 andl=ml= 0 and reads

ψ 100 =^1

a^30 /^2

√

π

exp(−r/a 0 ),

where we have defined the Bohr radiusa 0

a 0 = ̄

h^2

mke^2

,

with lengtha 0 = 0. 05 nm. The first excited state withl= 0 is

ψ 200 =^1

4 a^30 /^2

√

2 π

(

2 −r

a 0

)

exp(−r/ 2 a 0 ).

For states with withl= 1 andn= 2 , we can have the following combinations withml= 0

ψ 210 =

1

4 a^30 /^2

√

2 π

(

r

a 0

)

exp(−r/ 2 a 0 )cos(θ),

andml=± 1

ψ 21 ± 1 =

1

8 a^30 /^2

√

π

(

r

a 0

)

exp(−r/ 2 a 0 )sin(θ)exp(±iφ).

The exact energy is independent oflandml, since the potential is spherically symmetric.
The first few non-normalized radial solutions of equation are listed in Table 14.2.
When solving equations numerically, it is often convenientto rewrite the equation in terms
of dimensionless variables. This leads to an equation in dimensionless form which is easier
to code, sparing one for eventual errors. In order to do so, weintroduce first the dimension-
less variableρ=r/β, whereβis a constant we can choose. Schrödinger’s equation is then
rewritten as

−

1

2

∂^2 u(ρ)

∂ ρ^2

−

mke^2 β

̄h^2 ρ

u(ρ)+

l(l+ 1 )

2 ρ^2

u(ρ) =

mβ^2

̄h^2

E u(ρ). (14.16)