Numerically Solving the Schrödinger Equation

Contributed by Andrei Tokmakoff
Professor (Chemistry) at University of Chicago

Often the bound potentials that we encounter are complex, and the time-independent Schrödinger equation will need to be evaluated numerically. There are two common numerical methods for solving for the eigenvalues and eigenfunctions of a potential. Both methods require truncating and discretizing a region of space that is normally spanned by an infinite dimensional Hilbert space. The Numerov method is a finite difference method that calculates the shape of the wavefunction by integrating step-by-step across along a grid. The DVR method makes use of a transformation between a finite discrete basis and the finite grid that spans the region of interest.

: Selection and discretization of a space bounding the region for which the TISE will be solved numerically. A space of length

L

is discretized into

N

points separated by a spacing

δ x

over which the potential varies slowly.

The Numerov Method

A one-dimensional Schrödinger equation for a particle in a potential can be numerically solved on a grid that discretizes the position variable using a finite difference method. The TISE is

[T + V (x)] ψ (x) = E ψ (x) (1.5.1)

with

T = - ℏ 2 2 m \partial 2 \partial x 2, (1.5.2)

which we can write as

ψ'' (x) = - k 2 (x) ψ (x) (1.5.3)

where

k 2 (x) = 2 m ℏ 2 [E - V (x)] . (1.5.4)

If we discretize the variable $x$

, choosing a grid spacing

δ x

over which

V

varies slowly, we can use a three point finite difference to approximate the second derivative:

f'' i \approx 1 δ x 2 (f (x i + 1) - 2 f (x i) + f (x i - 1)) (1.5.5)

The discretized Schrödinger equation can then be written in the form

ψ (x i + 1) - 2 ψ (x i) + ψ (x i - 1) = - k 2 (x i) ψ (x i) (1.5.6)

Using the equation for $ψ (x_{i + 1})$

, one can iteratively solve for the eigenfunction. In practice, you discretize over a range of space such that the highest and lowest values lie in a region where the potential is very high or forbidden. Splitting the space into N points, chose the first two values

ψ (x_{1}) = 0

and

ψ (x_{2})

x to be a small positive or negative number, guess

E

, and propagate iteratively to

ψ (x_{N})

. A comparison of the wavefunctions obtained by propagating from

x_{1}

x_{N}

with that obtained propagating from

x_{N}

x_{1}

tells you how good your guess of

E

was.

The Numerov Method improves on Equation $1.5.6$

by taking account for the fourth derivative of the wavefunction

Ψ^{(4)}

, leading to errors on the order

O (δ x^{6})

. Equation

1.5.5

becomes

f'' i \approx 1 δ x 2 (f (x i + 1) - 2 f (x i) + f (x i - 1)) - δ x 2 12 f (4) i (1.5.7)

By differentiating Equation $1.5.3$

we know

ψ (4) (x) = - (k 2 (x) ψ (x))'' (1.5.8)

and the discretized Schrödinger equation becomes

ψ'' (x i) = 1 δ x 2 (ψ (x i + 1) - 2 ψ (x i) + ψ (x i - 1)) + 1 12 (k 2 (x i + 1) ψ (x i + 1) - 2 k 2 (x i + 1) ψ (x i) + k 2 (x i + 1) ψ (x i - 1))

This equation leads to the iterative solution for the wavefunction

ψ (x i + 1) = ψ ( x i ) ( 2 + 10 δ x 2 12 k 2 ( x i ) ) - ψ ( x i - 1 ) ( 1 - δ x 2 12 k 2 ( x i - 1 ) ) 1 - δ x 2 12 k 2 ( x i + 1 ) (1.5.9)

Discrete Variable Representation (DVR)

Numerical solutions to the wavefunctions of a bound potential in the position representation require truncating and discretizing a region of space that is normally spanned by an infinite dimensional Hilbert space. The DVR approach uses a real space basis set whose eigenstates $φ_{i} (x)$

we know and that span the space of interest—for instance harmonic oscillator wavefunctions—to express the eigenstates of a Hamiltonian in a grid basis (

θ_{j}

) that is meant to approximate the real space continuous basis

δ (x)

. The two basis sets, which we term the eigenbasis (

φ

) and grid basis (

θ

), will be connected through a unitary transformation

$Φ^{†} φ (x) = θ (x)$

Φ θ (x) = φ (x)

For $N$

discrete points in the grid basis, there will be

N

eigenvectors in the eigenbasis, allowing the properties of projection and completeness will hold in both bases. Wavefunctions can be obtained by constructing the Hamiltonian in the eigenbasis,

$H = T (\hat{p}) + V (\hat{x}),$

transforming to the DVR basis,

H^{D V R} = Φ H Φ,

and then diagonalizing.

Here we will discuss a version of DVR in which the grid basis is set up to mirror the continuous $| X ⟩$

eigenbasis. We begin by choosing the range of

x

that contain the bound states of interest and discretizing these into

N

points (

x_{i}

) equally spaced by

δ x

. We assume that the DVR basis functions

θ_{j} (x_{i})

resemble the infinite dimensional position basis

θ j (x i) = Δ x - - - \sqrt δ i j (1.5.10)

Our truncation is enabled using a projection operator in the reduced space

P N = \sum i = 1 N | θ i ⟩ ⟨ θ i | \approx 1 (1.5.11)

which is valid for appropriately high $N$

. The complete Hamiltonian can be expressed in the DVR basis DVR

H D V R = T D V R + V D V R . (1.5.12)

For the potential energy, since ${θ_{i}}$

is localized with

⟨ θ_{i} | θ_{j} ⟩ = δ_{i j}

, we make the DVR approximation, which casts

V^{D V R}

into a diagonal form that is equal to the potential energy evaluated at the grid point:

V D V R i j = ⟨ θ i | V (x^) | θ j ⟩ \approx V (x i) δ i j (1.5.13)

This comes from approximating the transformation as $Φ V (\hat{x}) Φ^{†} \approx V (Φ \hat{x} Φ^{†}) .$

For the kinetic energy matrix elements $⟨ θ_{i} | T (\hat{p}) | θ_{j} ⟩$

, we need to evaluate second derivatives between different grid points. Fortunately, Colbert and Miller have simplified this process by finding an analytical form for the

T^{D V R}

matrix for a uniformly gridded box with a grid spacing of

∆ x

T D V R i j = ℏ 2 ( - 1 ) i - j 2 m Δ x 2 {π 2 / 3 2 / (i - j) 2 i = j i \neq j} (1.5.14)

This comes from a Fourier expansion in a uniformly gridded box. Naturally this looks oscillatory in $x$

at period of

δ x

. Expression becomes exact in the limit of

N \to \infty

Δ x \to 0

. The numerical routine becomes simple and efficient. We construct a Hamiltonian filling with matrix elements whose potential and kinetic energy contributions are given by Equations

1.5.13

and

1.5.14

. Then we diagonalize

H^{D V R}

, from which we obtain

N

eigenvalues and the

N

corresponding eigenfunctions.

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/01%3A_Overview_of_Time-Independent_Quantum_Mechanics/1.05%3A_Numerically_Solving_the_Schrodinger_Equation

Wednesday, September 15, 2021

Numerically Solving the Schrödinger Equation

The Numerov Method

Discrete Variable Representation (DVR)

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