Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.1

Lecture 8

Numerical Techniques : Differential

Equations - Eigenvalue Problems

Wave Motion Along a Spring, Hydrogen Atom

IKC-MH.55 Scientiﬁc Computing with Python at December 15,

2023

Dr. Cem Özdo

gan

Engineering Sciences Department

Izmir Kâtip Çelebi University

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.2

Contents

1 Differential Equations - Eigenvalue Problems

Eigenvalue Problems

Standing Waves on a String

Numerical Solutions of Schrödinger Equation

Hydrogen Atom

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.3

Differential Equations

1 Initial Va lue Problems.

2 Boundary Value Problems.

3 Eigenvalue (characteristic-value) Problems:

Even if the boundary conditions of the differential equation are

available, the solutions can only exist for some speciﬁc values

of a parameter in the system. For example,

−

+ V (x)ψ(x ) = Eψ(x )

•

In Schrödinger equation, there are solutions that ψ(x)

goes to zero at inﬁnity for c ertain values of the energy E.

•

These E

values satisfying this condition are the

eigenvalues of the differential equation.

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.4

Boundary/Eigen Value Problems

•

See the following 2nd degree differential equation:

′′

= f (x , y , y

′

: λ)

•

Again, let the boundary conditions of this equation to be

given at both ends.

•

If these conditions can only be satisﬁ ed for certain λ

values, we call it the eigenvalue problem.

•

e.g.: Vibrations o f a wire with both ends ﬁxed give stable

solution o nly for certain waveleng ths.

•

e.g.: Solutions of the Schrödi nger e quation that are zero at

inﬁnity exist only for certain energy eigenvalues.

•

Some boundary value problems in physics/engineering

have a solution based on an eigenvalue.

•

In terms of numerical solution, boundar y value and

eigenvalue problems are solved by the same method.

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.5

Eigenvalue Problems I

For example, standing waves are the solutions of the

following differential equation for a string ﬁxed at both

ends:

y(x)

+ k

y(x) = 0

Here k = 2π/λ represents the wavenumber.

•

If the two ends of the L-length string are ﬁxed, then

corresponding boundary values are:

y(0) = y(L) = 0

•

Although there are sufﬁcient boundary conditions, there

are only solutions for certain k values.

•

It is impossible to satisfy these boundary conditions for

other k values.

•

The standing wave solution in the string:

y(x) = Asinkx + Bcoskx

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.6

Eigenvalue Problems II

•

Boundary conditions to ﬁnd the coefﬁcients A, B;

y(0) = 0 → A ∗ 0 + B ∗ 1 = 0 →B = 0

y(L) = 0 → AsinkL = 0 →kL = nπ (n = 1, 2, . . .)

•

According to these results, there is only one set of solution

as k = π/L, 2π/L, . . . values. (k

= nπ/L → eigenvalue(s).)

2 Another example is the Schrödinger equation in quantum

mechanics. If we write it in one-dimensional space,

−

ψ(x)

+ V (x)ψ(x) = Eψ(x)

Here V (x) is the potential energy function of the particle,

and E is its total energy.

•

Boundary conditions for the wave function are given for

x = ±∞:

ψ(±∞) → 0

•

Again, this differential equation has solutions s atisfying the

boundary conditions only for certain E eigenvalues.

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.7

Eigenvalue Problems III

For eigenvalue problems, the trial-and-error (shooting) method

is also used.

•

However, an estimated value is given to the eigenvalue

instead of giving to the derivatives at the boundary .

•

Then, a trial solution is obtained.

•

By comparing this trial solution with the value at the

boundary condition, the eigenvalue is readjusted and

another trial is perfor med.

•

Finally, the solution is to be found when the true

eigenvalue is approached within a certain margin of error.

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.8

Standing Waves on a String I

•

The wave equation and boundary conditions in a string of

length L = 1 m with both ﬁxed ends are as follows:

y(x)

+ k

y(x) = 0 & y(0) = y(1) = 0

Firstly, transform this quadratic equation into

a system of linear (ﬁrst degree) equations:

y → y

→ y

with these values (y

and y

), the sy stem of

equations to be solved: (Now, we h ave a

set of equations.)

= y

= −k

and the boundary conditions are:

y(0) = 0

y(1) = 0

•

Here, the trial-and-error approach differs from the

previously discussed boundary value problem.

•

Different estimates for y

(0) values do not make it zero at

the other boundary.

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.9

Standing Waves on a String II

•

Instead, an estimated value for the k eigenvalue is

taken and a solution search is initiated.

•

Searching continues by increasing the value of k until the

boundary condition (here y(1) = 0) at the other end is

satisﬁed.

•

For example, the solution at the other boundary is y

(1)

for a given value of k.

•

Accordingly, the next step is ﬁnd the root of the following

equation:

F (k) = y

(1) − y(1) = 0

When we encountered an eigenvalue k, then F (k) will

change sign as indicating the root.

•

(Example py-ﬁle: The program to ﬁnd the 5 smallest of

the k eigenvalues in a string:

standingwawes.py)

•

The program can ﬁnd the eigenvalues

k = nπ/L = π, 2π, 3π, . . . on a str ing of length L=1 m.

•

However, the error margin is to be increased by increasing

eigenvalues (see k

/π values).

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.10

Standing Waves on a String III

0.0 0.2 0.4 0.6 0.8 1.0

−0.01

0.00

0.01

0.02

0.03

Approximate

and Exact Solution for EVP ODE:

(

)

(

) = 0

n=1

n=2

n=3

n=4

n=5

SciPy n=5

Figure: S olution for the Eigenvalue Problem for the ODE:

= −k

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.11

Numerical Solutions of Schrödinger Equation I

•

In quantum mechanics, the Schrödinger equation is used

to ﬁnd the eigenvalues and eigenfunctions of the particle

moving in one dimension at the potential V (x):

−

ψ(x)

+ V (x)ψ(x) = Eψ(x)

•

An analytical solution to this equation is available for only

very few potential functions such as harmonic oscillator,

inﬁnite well, hydrogen atom, ....

•

Therefore, numerical solutions of the Schrödinger

equation is an indispensable tool in physics research.

•

The numerical solution of the Sc hrödinger equation is

complicated for general solutions of the problem.

•

However, if we assume the potential function to be as

symmetric, the problem can be solved in a much easier

way.

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.12

Numerical Solutions of Schrödinger Equation II

•

For a symmetric potential,

V (−x) = V (x)

•

Therefore, the solutions of the Schrödinger equation also

fall into two groups:

Symmetrical wave functions: ψ(−x) = ψ(x)

Antisymmetric wave functions: ψ(−x) = −ψ(x)

•

This property allows us to determine exactly the initial

conditions necessary to start the eigenvalue problem.

Symmetric (even) wave functions: ψ(0) = 1 & ψ

′

(0) = 0

Antisymmetric ( odd) wave functions: ψ(0) = 0 & ψ

′

(0) = 1

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.13

Hydrogen Atom I

•

In quantum mechanics, the hydrogen atom is considered

as a system of electrons with a charge of −e around a

proton with a charge of +e.

•

If the electrostatic potential energy between the

electron-proton is substituted in the Schrödinger equation

as V (r ) = −e

/r,

−

∇

ψ(

r) −

ψ(

r) = Eψ(

r )

Here m

= m

/(m

+ m

) is the reduced mass of the

electron-proton system.

•

Since the potential energy depends only on the distance r ,

the solution is deﬁned with the spherical coordinates

(r, θ, φ) in three-dimensional space:

ψ(r, θ, φ) = R(r)Y (θ, φ)

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.14

Hydrogen Atom II

•

The solutions of the equation provided by the angular

variables (θ, φ) are independent of the V (r) potential and

consist of functions called spher ic al harmonics Y (θ, φ).

•

The equation provided by the R(r ) function is called radial

Schrödinger equation.

−







l(l + 1)~

−



R(r) = −|E|R(r)

•

Attempt to ﬁnd the bound energy (eigen)values and wave

(eigen)functions of the radial Schrödinger equation

numerically :

•

First, it is necessary to make the radial equation

dimensionless bu deﬁning a new wavefunction:

u(r) = rR(r )

•

The radial equation in terms of this new function u(r )

becomes simpler:

−



l(l + 1)

−



u(r) = −

|E|

u(r)

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.15

Hydrogen Atom III

•

Now, a variable change is made as the following:

k =

√

|E|

, ρ = 2kr , λ

ℜ

|E|

Here, the Rydberg constant ℜ and the Bohr radius a

are

deﬁned as:

, ℜ =

•

As a result of these changes, the dimensionless radial

equation becomes:

dρ

−

l(l + 1)

u +



−



u = 0

(1)

where l is orbital quantum number and λ is principal

quantum number.

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.16

Hydrogen Atom IV

Numerical Solution. Firstly, transform this

quadratic Equation 1 into a system of linear

(ﬁrst degree) equations:

u → y

dρ

→ y

with these values

and y

), the system of

equations to be solved:

(Now, we have a set of

equations.)

dρ

= y

dρ



l(l + 1)

−



−



•

For the initial conditions:

•

u(0) ∼ 0

•

Since the absolute magnitude of the wave function has no

physical meaning, the arbitrary va lue u

′

(0) = 1 can b e

taken.

•

Then;

(0) = 0 and y

(0) = 1

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.17

Hydrogen Atom V

(Example py-ﬁle: Program that solves the radial Schrödinger

equation for the hydrogen atom:

hydrogenatom.py)

•

Program does not graph the u(r) functions, but the |u|

probability densities, which is physically meaningful.

•

It calculates according to the l quantum number which is

supplied by the user.

•

The error margin is to be increased by increasing n

quantum number.

Figure: S olution for the Eigenvalue Problem for the ODE:

dρ

l(l+1)

−



−

i

Numerical Techniques:

Differential Equations -

Eigenvalue Problems

Dr. Cem Özdo

gan

LOGIK

Differential Equations -

Eigenvalue Problems

Standing Waves on a

String

Numerical Solutions of

Schrödinger Equation

Hydrogen Atom

8.18

Hydrogen Atom VI

0 1 2 3 4 5 6

0.000

0.001

0.002

0.003

0.004

0.005

(

Solution for EVP ODE:

dρ

−

(

+ 1)

(

−

)

= 0

Figure: S olution for the Eigenvalue Problem for the ODE:

dρ

l(l+1)

−



−

i