A Thesis entitled

Relativistic Treatment of Confined Hydrogen Atoms via Numerical Approximations

by Jacob M. Noon

Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Masters of Science Degree in Applied Mathematics

Dr. Ivie Stein, Committee Chair

Dr. Robert Deck, Committee Member

Dr. Geoffrey Martin, Committee Member

Dr. Amanda C. Bryant-Friedrich, Dean College of Graduate Studies

The University of Toledo May 2018 Copyright 2018, Jacob M. Noon

This document is copyrighted material. Under copyright law, no parts of this document may be reproduced without the expressed permission of the author. An Abstract of Relativistic Treatment of Confined Hydrogen Atoms via Numerical Approximations by Jacob M. Noon

Submitted to the Graduate Faculty as partial fulfillment of the requirements for the Masters of Science Degree in Applied Mathematics The University of Toledo May 2018

The study of particles and atoms confined to spherically symmetric regions have been used to illustrate the differences between classical and quantum systems since

Erwin Schrodinger’s famous equation was published in 1926. Paul Dirac later added his own equation, the Dirac Equation (1928), which integrated the quantum principles that had been developing in the decades prior to Schrodinger’s equation with the principles of Einstein’s Special Theory of Relativity.

The literature on Hydrogen atoms confined to spherically symmetric regions (using the Schrodinger equation) is abundant, with consensus on the results. To the best of our knowledge, no relativistic treatment of this problem exists (i.e., using the

Dirac equation). Some reasons for this are the complexity of the Dirac equation, its solutions, and problems that arise when trying to satisfy the boundary conditions.

In this paper, I will present solutions to the given problem, as well as limitations that are mathematical and physical in nature. The methods used to obtain solutions involve solving systems of first order linear ordinary differential equations analytically, and computing the roots of ratios of Kummer functions via two different numerical methods.

iii For my wife Kirstin and the rest of my family Acknowledgments

I would like to express my deepest gratitude towards both of my advisors, Professor

Ivie Stein and Professor Robert Deck. Without their guidance, I would not have been able to complete this thesis.

I would also like to thank my wife Kirstin and my entire family for all of their support throughout my two years in this program.

Lastly, I would like to thank all of my friends I have made these last two years and my professors who have helped me build a more solid mathematical foundation that I did not have when I entered this program.

v Contents

Abstract iii

Acknowledgments v

Contents vi

1 Introduction 1

1.1 Particle In A Box ...... 3

1.2 Extensions ...... 5

2 Relativistic Hydrogen 6

2.1 Dirac Equation ...... 7

2.2 Radial Equations ...... 7

2.2.1 (0 < r < R)...... 8

2.2.2 (R < r < ∞) ...... 12

2.2.3 (r = R) ...... 14

3 Numerical Results 16

3.1 Bisection Method ...... 17

3.2 Brent’s Method ...... 18

3.3 Results ...... 18

4 Conclusion 24

4.1 Overview ...... 24

vi 4.2 Limitations and Future Work ...... 25

References 27

A Code and Graphics 29

vii Chapter 1

Introduction

Since Erwin Schrodinger published his famous formula (also known as the Schrodinger equation) in 1926, physicists have been able to study properties of matter when con-

fined to a small region on the quantum level. These so called Potential Well problems aided in explaining certain types of phenomenon such as the discreteness of the hydro- gen energy spectrum, and predicted new properties of matter like tunneling. Mathe- matically speaking, these problems usually involve solving differential equations that must satisfy certain boundary conditions.

To mathematically model an atom or a single particle confined in a spherically symmetric finite potential well, the standard procedure is to use the Schrodinger equation. The problem of a single particle confined in a potential well is found in most introductory quantum mechanics textbooks. This involves the potential energy term in changing depending on the region of interest. Solving for the energy in any potential well problem comes from satisfying the boundary conditions. In the case of a finite potential well, the formula for energy in most cases cannot be solved for analytically. Therefore, numerical approximations must be used in order to obtain an energy value. An extension to this problem is the hydrogen atom (which includes the proton in the nucleus and the electron in orbit) and confining it in a spherically symmetric finite potential well. The potential energy term inside the spherical region

1 is now the coulomb potential. Energy will again need to be solved for via numerical approximations. Ley-Koo and Rubinstein [3] provided energy values (ground state only) for this problem while varying the radius of confinement and potential well height.

In most situations, the Schrodinger equation works quite well. As long as the region of confinement is not too small, the Heisenberg Uncertainty Principle allows the energy of the confined particle or atom to be small enough so that relativistic equations for momentum and energy will not be needed. However, as the region of confinement decreases in size, the uncertainty principle shows us that the uncertainty in energy increases. At a certain point, depending on the particle or atom being studied, non-relativistic formulas for kinetic energy and momentum may not be suffi- ciently accurate in predicting the total energy of the particle or atom because of the effects of special relativity. This is a problem because the the Schrodinger equation obeys classical energy and momentum formulas, where the total energy is just the sum of the potential energy and kinetic energy.

This thesis will be dedicated to providing a relativistic treatment (using the Dirac equation) to the problem of a hydrogen atom confined in a spherically symmetric potential well in order to determine how the energy of the electron changes as a function the radius of confinement and the potential well height. To accomplish this, solving two systems of linear first order ordinary differential equations, satisfying the boundary conditions, and numerically computing the roots of a nonlinear equation that will contain confluent hypergeometric functions of the first kind will help answer this question. These roots will in fact be the energy of the ground state relativistic hydrogen atom. For reasons that will be stated later in this paper, the ground state will be the only state that is studied in detail. A direct comparison with [3] will also be provided in order to discern whether relativistic effects are important in determining the modified ground state energy of a confined hydrogen atom. Before the answer to

2 this question is provided, an example of a simple confinement problem (or potential

well problem) will be shown. This example will display the dependence of the energy

of the confined particle on the size of area of confinement.

1.1 Particle In A Box

Consider the time-independent Schrodinger equation (1) in one spatial dimension

with the potential energy term V (x)(2) subject to the following constraint:

−¯h2 00 (1) 2m ψ (x) + V (x)ψ(x) = Eψ(x)   0 for 0 ≤ x ≤ L (2) V (x) =  ∞ for x > L, x < 0

The only case to consider is where V (x) = 0 for (1), which implies the particle cannot exist outside the box. So the solution ψ(x) must go to zero at x = 0 and x = L.

A graphic that displays the region of interest can be found in the appendix [A1]. As stated in the introduction, the eventual area of interest will be a spherically symmetric region of confinement. In the non-relativistic case, this would involve taking the three dimensional Schrodinger equation in Cartesian coordinates and transforming it into spherical coordinates. Since the goal of this section was to provide a simple example of the dependence of energy on confinement, only a 1 dimensional example in Cartesian coordinates will be provided.

Putting V (x) = 0, (1) becomes

−¯h2 00 (3) 2m ψ (x) = Eψ(x)

The following solution will follow the format taken in [9]. Solving (3), the general solution is of the form:

q 2mE ψ(x) = Asin(kx) + Bcos(kx), k = ¯h2 3 The boundary conditions require that ψ(0) = ψ(L) = 0. Thus,

→ B = 0 and Asin(kL) = 0 → kL = nπ, n = 1, 2, ...

E can now be solved for using what is known about k.

¯h2π2n2 En = 2mL2 , n = 1, 2, ...

The formula above for En shows that the energy for the particle is ”quantized”, or discrete. The above formula for energy also shows the relationship with the size of the area of confinement L. As L shrinks, the energy increases for each n. The E1 energy is called the ”ground state” energy by physicists. This is the lowest energy state for a bounded particle or atom. Thus, as L decreases, the lowest energy of the particle in this configuration also increases. Later on in this paper, the concern will be with the ground state of a relativistic hydrogen atom. Based on this simple confinement model, we can predict that the ground state for the reltivistic hydrogen atom confined to a region should also increase as the region of confinement decreases.

The particular solution ψn(x) will not be solved for here because it was not the purpose of this section. The main goal was to show how the energy levels are discrete for a confined particle and how those discrete energy levels are dependent on the size of confinement.

Furthermore, the wave function ψ(x) should be normalizable, meaning:

R ∞ 2 −∞ |ψ(x)| dx = 1

The probability of finding a particle at position x is given by |ψ(x)|2. Therefore, summing over all probabilities has to equal one.

It should be noted that an infinite potential is not actually a physical reality. This infinite potential well problem is only a good approximation for a particle confined to a region with very low probability of the particle exisisting outside the region.

4 1.2 Extensions

This type of one dimensional problem can be extended to two or three dimensions

as well as different one (two, or three) dimensional coordinate systems. The potential

term V (x) can also be chosen to be finite, which will unveil another aspect of quantum

mechanics known as tunneling.

Eventually, physicists applied this framework to the hydrogen atom in spherical

coordinates for both finite and infinite potentials. Sommerfeld and Welker [10] found

the radius of confinement at which the electron in a hydrogen atom placed in a

spherically symmetric infinite potential well becomes unbounded (so that the binding

energy is positive). They did this for all energy states as well. The normal unconfined

−13.6eV hydrogen atom binding energy levels are En = n2 .

For the case when the confining potential V (r) = V0 is constant (for some r > R), the boundary conditions become much more difficult to satisfy in the case of the hydrogen atom. As stated in the introduction, the energy must be solved for numerically. This is what will need to be done in the next chapter as well.

5 Chapter 2

Relativistic Hydrogen

The simple one dimensional model in section 1.1 showed that the energy of the confined particle is dependent on the size of the confinement region. Even though that relation was shown using the non-relativistic wave equation (Schrodinger equa- tion), the uncertainty principle still applies to the Dirac equation, and says that the uncertainty in energy of the confined particle will increase with a decrease in location uncertainty (radius of confinement). Not only is a three dimensional relativistic model needed, but the finite confining potential must be spherically symmetric. Therefore, the Dirac equation in spherical coordinates will be used.

The main goal of this chapter is to derive an equation as a function of energy E that can be used to solve for E numerically. In order to achieve this goal, the Dirac radial equations will need to be solved with two different potential energy terms along with satisfying the boundary conditions. Satisfying the boundary conditions will then give an equation that can be used to numerically compute the energy values of the electron at a fixed well height and confinement radius (which will be computed in the next chapter).

Before jumping into the derivation of this equation, a slight overview of the Dirac equation will be given below.

6 2.1 Dirac Equation

The Dirac equation is a relativistic wave equation that describes all spin 1/2 par-

ticles such as electrons and protons, and is consistent with the principles of quantum

mechanics and special relativity. This means that it also obeys Einstein’s energy-

momentum relation. Shown below is the Dirac equation in Cartesian coordinates

along with some properties of the components as stated in [1].

∂ψ(x) 2 ih¯ ∂t = [cα · p + βm0c + V (x)]ψ(x) 2 x = x(t, x, y, z), H = [cα · p + βm0c + V (x)], p = −ih¯5 (5 is the del operator)   0 σ  i αi =   (i = 1, 2, 3) where σi are the 2x2 Pauli spin matrices σi 0   I 0   β =  , where I is the 2x2 identity matrix 0 −I

Dirac wanted an equation that contained a first order derivative of the temporal

component and a first order derivative of the spatial components for symmetry rea-

sons. In order to achieve this, he derived the type of algebraic structures that needed

to exist in the equation in order for the derivatives to be first order and the operators

∂ (E = ih¯ ∂t and H) to obey the relativistic energy-momentum relation when squared.

As the algebraic structures derived by Dirac show (αi and β), the solutions to this new relativistic wave equation are four component column vectors instead of complex valued scalar functions.

2.2 Radial Equations

Now a relativistic treatment to a hydrogen atom confined to a spherically sym- metric potential well will be shown. Converting the time-independent Dirac equation

7 into spherical coordinates, the solutions are again four component column vectors,

but now have the form as stated in [14]:

G (r)Ω (θ, φ) ! 1 J jlm ψ(r, θ, φ) = r iFJ (r)Ωjl0m(θ, φ)

Here FJ (r) and GJ (r) are scalar functions of r (0 < r < ∞), and Ωjlm(θ, φ), Ωjl0m(θ, φ)

0 1 0 1 are the spherical harmonics (where l = l+1, l = J − 2 and l = J + 2 ). After convert- ing the time-independent Dirac equation to spherical coordinates, the radial functions can be shown to satisfy the equations

2 0 κ E+m0c −V (r) (3) G (r) + r G(r) = ¯hc F (r) 2 0 κ E−m0c −V (r) (4) F (r) − r F (r) = − ¯hc G(r) 1 1 3 (5) κ = ±(J + 2 ),J = 2 , 2 , ...,

where the subscripts on the functions F and G are dropped for convenience. Because it will be possible to obtain energy from solving the system of differential equations above and there is no need for the full solution ψ(r, θ, φ), there will be no need to involve the spherical harmonics in the calculations. Also, since the only state that energy will be solved for is ground state of relativistic hydrogen, κ will eventually

1 equal −1 in (5)(J = 2 ). The potential energy term for (3) and (4) is  2  −Ze for 0 < r < R V (r) = r  V0 for R < r < ∞

When finding the energy solutions in the following subsections, only bound states

2 energies (E < m0c ) will be computed. The reason for this will be explained later on.

2.2.1 (0 < r < R)

For r < R the approach taken by Greiner [1] will follow after making some variable

substitutions. Let 8 √ 2 2 2 e2 (m0c ) −E α = ¯hc , λ = ¯hc , ρ = 2λr

Above, α is the fine structure constant (≈ 1/137), e is a unit of elementary charge,

2 c is the speed of light,h ¯ is Planck’s constant divided by 2π, m0c is the rest energy of an electron, and E is energy (in units of eV). After some algebra, a new system of

ordinary differential equations is obtained.

2 0 κ E+m0c Zα (6) G (ρ) = − ρ G(ρ) + ( 2λ¯hc + ρ )F (ρ) 2 0 E−m0c Zα κ (7) F (ρ) = −( 2λ¯hc + ρ )G(ρ) + ρ F (ρ)

Solutions to look for will be of the form:

√ ρ 2 − (8) F (ρ) = ( m0c − E)e 2 (φ1(ρ) − φ2(ρ))

√ ρ 2 − (9) G(ρ) = ( m0c + E)e 2 (φ1(ρ) + φ2(ρ))

After substituting F (ρ), G(ρ), F 0(ρ), and G0(ρ) into (6) and (7) a new system of

first order differential equations in terms of φ1(ρ) and φ2(ρ) is obtained. After adding

(6) to (7) and subtracting (7) from (6) (after they become the system of φ1(ρ) and

φ2(ρ)), the final system is

2 0 ZαE κ Zαm0c (10) φ1(ρ) = (1 − ¯hcλρ )φ1 − ( ρ + ¯hcλρ )φ2 2 0 κ Zαm0c ZαE (11) φ2(ρ) = (− ρ + ¯hcλρ )φ1 + ¯hcλρ φ2

To solve this system of ordinary differential equations, the Frobenius Method will be

applied. Let

P∞ n+γ (12) φ1(ρ) = n=0 anρ P∞ n+γ (13) φ2(ρ) = n=0 bnρ

After substituting (12), (13), into (10) and (11),a four equation recursion relation is derived as stated in [1]:

9 2 ZαE Zαm0c (14)(γ + ¯hcλ )a0 = −(κ + ¯hcλ )b0 2 ZαE Zαm0c (15)(γ − ¯hcλ )b0 = (−κ + ¯hcλ )a0 2 ZαE Zαm0c (16)(n + γ + ¯hcλ )an − an−1 + (κ + ¯hcλ )bn = 0 2 ZαE Zαm0c (17)(n + γ − ¯hcλ )bn − (−κ + ¯hcλ )an = 0

Solving for a0 in (14) and substituting into (15),

2 2 ZαE 2 Zαm0c 2 2 γ − ( ¯hcλ ) = −( ¯hcλ ) + κ 2 2 2 2 4 2 Z α (E −m0c ) 2 2 2 → γ = (¯hcλ)2 + κ = κ − (Zα) → γ = ±pκ2 − (Zα)2

Because F (ρ) and G(ρ) need to remain finite, and thus normalizable as ρ → 0 (which

means as r → 0), the negative solution for γ will be neglected.

Solving for bn in (17), the following relation between an and bn is:

2 Zαm0c −κ+ hcλ¯ (18) bn = ( ZαE )an n+γ− hcλ¯

Substituting this into (16) and solving for an in terms of an−1,

ZαE (n+γ− hcλ¯ ) (19) an = n(2γ+n) an−1 γ ZαE → (20) φ1(ρ) = a0ρ M(γ − ¯hcλ + 1, 2γ + 1, ρ)

where M(a, b, x) is the confluent hypergeometric function of the first kind (also known

as Kummer’s function). To solve for φ2(ρ), an from (19) can be substituted into (18) to obtain

2 Zαm0c (−κ+ hcλ¯ ) (21) bn = n(2γ+n) an−1

Since all bn terms can be put into terms of a0 by using (19), (21) implies

2 Zαm0c γ ZαE κ− hcλ¯ γ ZαE φ2(ρ) = b0ρ M(γ − ¯hcλ , 2γ + 1, ρ) = ( ZαE )a0ρ M(γ − ¯hcλ , 2γ + 1, ρ) hcλ¯ −γ

10 It should also be noted that by [6], so long as b 6= −n ,where n ∈ N, M(a, b, x)

converges for all x. In this case, b = 2γ + 1 is never a negative integer. At last, the solutions for F (ρ) and G(ρ) for 0 < r < R are:

√ ρ 2 κ−Zαm0c /¯hcλ 2 − 2 γ 0 0 (22) F (ρ) = ( m0c − E)e ρ a0(M(1−n , 2γ+1, ρ)−( n0 )M(−n , 2γ+ 1, ρ))

√ ρ 2 κ−Zαm0c /¯hcλ 2 − 2 γ 0 0 (23) G(ρ) = ( m0c + E)e ρ a0(M(1−n , 2γ+1, ρ)+( n0 )M(−n , 2γ+ 1, ρ))

0 ZαE n = ¯hcλ − γ This derivation can be found in more detail in [1].

11 2.2.2 (R < r < ∞)

With V (r) = V0 constant for R < r < ∞, the system of ordinary differential equations from (3) and (4) with the same substitutions at the top of section 2.2.1 become:

0 κ a (24) G (ρ) = − ρ G(ρ) + 2λ F (ρ) 0 κ b (25) F (ρ) = ρ F (ρ) + 2λ G(ρ) where

2 E+m0c −V0 a = ¯hc 2 E−m0c −V0 b = − ¯hc

Differentiating (24),

00 −κ 0 κ a 0 (26) G (ρ) = ρ G (ρ) + ρ2 G(ρ) + 2λ F (ρ)

Solving for F (ρ) in (24) and substituting into (25) results in an equation with F 0(ρ)

strictly in terms of G(ρ) and G0(ρ). After plugging this into (26) and a few cancella-

tions, (26) becomes

00 κ(κ+1) ab (27) G (ρ) = ρ2 G(ρ) + (2λ)2 G(ρ)

Since the goal is to determine energy for the ground state, κ = −1, the solution to

(27) is:

√ √ ρ ab − ρ ab G(ρ) = c1e 2λ + c2e 2λ

The condition that F (ρ) and G(ρ) → 0 as ρ (and thus r) → ∞ in order for the wave

function to remain normalizable requires that c1 = 0. Thus,

√ − ρ ab (28) G(ρ) = c2e 2λ √ √ − ab ρ ab 0 − 2λ (29) G (ρ) = 2λ c2e

12 F (ρ) is then obtained by inserting (28) and (29) into (24) to produce

√ ρ ab q b 2λ − 2λ (30) F (ρ) = −c2e ( a + aρ ) (F (ρ) → 0 as ρ → ∞)

13 2.2.3 (r = R)

In quantum mechanics, the wave function must be continuous everywhere in its

domain. Because κ = −1 in the previous section, κ must also be set to −1 in (22)

and (23). To satisfy the boundary condition at r = R, Fr

Fr>R(ρ) at r = R and GrR(ρ) at r = R. Furthermore, to remove the need for solving for the normalization constants (a0 for r < R and c2 for r > R), the ratio equality FrR(ρ) will cancel those constants while still GrR(ρ) satisfying the boundary conditions. This technique has was used in [11].

An equation determining the energy of the electron in the ground state can be

derived from the equality of ratios at r = R. The ratio and numerical computations

2 can be simplified by factoring out rest energy terms (m0c ). Recall that,

2 E+m0c −V0 a = ¯hc 2 b = − E−m0c −V0 √ ¯hc 2 4 2 m0c −E λ = ¯hc F (ρ) q b 2λ G(ρ) = − a − aρ (r > R)

E V0 If  = 2 , W = 2 , then m0c m0c

√ F (ρ) q 1−+W 2 1−2 (31) G(ρ) = − 1+−W − ρ 1+−W

2 Z Zαm0c For r < R, the solutions F (ρ) and G(ρ) contain the terms ¯hcλ and ¯hcλ . Using the same substitutions for  and W as above,

Z = √Zα ¯hcλ 1−2 2 Zαm0c = √Zα ¯hcλ 1−2 √ 2 Also recall that ρ = 2λr. Thus, when r = R, define ρ0 = 2λR = 2β 1 −  R, where

2 m0c (22) β = ¯hc . Using these new terms and variables, setting (23) = (31), and performing some minor algebraic manipulations, an equation is derived:

14 √ q 1− q 1−+W 2 1−2 Zα Zα (32)( + + )( √ − γ)M(γ − √ + 1, 2γ + 1, ρ0) + 1+ 1+−W ρ0 1+−W 1−2 1−2 √ q 1− q 1−+W 2 1−2 Zα Zα ( − − )(1 + √ )M(γ − √ , 2γ + 1, ρ0) = 0 1+ 1+−W ρ0 1+−W 1−2 1−2

Using (32), the value of  (and thus E) corresponding to two given fixed parameters

R and W can be approximated.

15 Chapter 3

Numerical Results

Looking back (32), it is clear that to compute the roots of this function by hand is nontrivial. Therefore, it was decided that a computer program would be needed to compute the  values that make this equation equal to zero for fixed R and W .

To compute the roots, a sufficient root finding algorithm will be required. The

main hurdles in choosing a root finding algorithm are the confluent hypergeometric

functions that are contained (32). Recall that the two hypergeometric functions are

of the form

(33) M(γ − √Zα + 1, 2γ + 1, ρ ) 1−2 0 (34) M(γ − √Zα , 2γ + 1, ρ ) 1−2 0 √ 2 ρ0 = 2βR 1 − 

For fixed W and R,(32) essentially turns into a function of . The independent variable  not only is in the argument of (33) and (34), but is also in the first parameter of both of these functions. That raises a problem if one wants to use

Newton’s method as their root finding algorithm because there is currently no closed form method to finding the derivatives of hypergeometric functions with respect to one of their parameters. There are numerical approximations to the derivatives of these functions with respect to their parameters as shown in [12], but this would make the end result (a value for the energy) less accurate because numerical approximations 16 are still required to find the root. Therefore, a root finding algorithm (or multiple

algorithms) that will find the roots of (32) without taking derivatives would be the

most efficient method of finding an energy value.

3.1 Bisection Method

The bisection method is a common root finding algorithm based on the interme-

diate value theorem that does not require computing derivatives. The only require-

ments the function ((32) in this specific case) needs to satisfy are to be continuous on a closed interval [a, b] with the function taking opposite values at the endpoints.

Given below is the algorithm for finding a root p using the bisection method found in [5].

For a function f(x) that is continuous on a closed interval [a, b] where f(a)f(b) < 0,

a1+b1 let a1 = a, b1 = b, and p1 = 2 .

• If f(p1) = 0, then p = p1 and the root is found.

• If f(p1) 6= 0, then f(p1) has either the same sign as f(a1) or f(b1).

• If f(p1) and f(a1) have the same sign, then p is in (p1, b1). Set a2 = p1 and

a2+b2 b2 = b1 and repeat the procedure with p2 = 2 .

• If f(p1) and f(b1) have the same sign, then p is in (a1, p1). Set a2 = a1 and

a2+b2 b2 = p1 with p2 = 2 and repeat.

In the results section, a calculation to determine the number of iterations needed

to find a solution in the interval chosen with a specific tolerance will be shown.

17 3.2 Brent’s Method

Brent’s method, developed by R.P. Brent [4] is a root finding algorithm that also

does not require computing derivatives. In the computer software that will be used

to find these roots, Brent’s method turns out to be the default root finding method.

Just as with the bisection method, a real valued function defined on a closed interval

[a, b] such that the function takes opposite signs at the endpoints is needed. However,

in Brent’s method, the function does not need to be continuous. This method is

built on top of Deckker’s Algorithm, as shown in [4], and uses a combination of

linear interpolation and inverse quadratic interpolation with bisection. Given certain

conditions, Brent shows that convergence is usually superlinear and is rarely slower

than bisection.

Because the details of the method would require multiple pages of explanations,

no further detail of Brent’s method will be discussed. The paper outlining the method

in detail by R.P. Brent can be found in the reference listed in the previous paragraph.

3.3 Results

Determining the closed interval [a, b] that is needed will be based off of the phys-

ical value of energy that is expected. Because the hydrogen atom in its ground

state is confined to a spherically symmetric potenial well, the perturbed ground state

energy value should be greater than or equal to the ground state energy of uncon-

fined hydrogen. Furthermore, because bound states are the energy region of inter-

2 E est (E < m0c ), this implies that  = 2 must be strictly less than one. The m0c ground state energy E for unconfined hydrogen is 510985.340407eV and the rest en-

2 ergy m0c = 510998.9461eV . Therefore, it will be appropriate to look for roots  in the closed interval [0.9999733739682344, 0.9999997337419158].

This was done for both Brent’s method and the Bisection method. It should 18 also be noted that (32) was assumed to be continuous on this interval, because if it was not, the program for bisection will not execute. It was already stated that

Brent’s method does not require the function to be continuous on the interval, so that algorithm would work anyway.

For three different values of W and five different values of R (which are in terms

−9m of the Bohr radius, A0 = .0529177 · 10 , m in meters), the roots of (32) were computed using both methods mentioned above using the mathematical software

SageMath [8]. SciPy packages for Brent’s method and bisection were imported from the Python libraries. SageMath already contains most Numpy and SciPy packages already, which means the syntax in sage is mostly Python. The code used for finding the roots can be found in the appendix [A2]. Below is a table of the resulting energy

2 values in eV (where W is units of m0c ).

Radius (W=0.5) E (Bisection) E (Brent’s)

1.8500 −0.501061486313120 −0.501061511051375

2.8500 −11.0278104042518 −11.0278104605968

3.8500 −13.0397465226124 −13.0397461217945

4.8500 −13.4837991572567 −13.4837991198292

5.8500 −13.5810291545349 −13.5810287899803

Radius (W=1) E (Bisection) E (Brent’s)

1.8500 −0.438878895074595 −0.438878876913805

2.8500 −11.0172005614731 −11.0172006346402

3.8500 −13.0374527411186 −13.0374528431566

4.8500 −13.4832893389394 −13.4832892202539

5.8500 −13.5809207679704 −13.5809208413120

19 Radius (W=1.9995) E (Bisection) E (Brent’s)

1.8500 −0.354762069531716 −0.354762214061338

2.8500 −11.0028614168987 −11.0028611254529

3.8500 −13.0343536875444 −13.0343540612375

4.8500 −13.4826004818315 −13.4826001463807

5.8500 −13.5807746467763 −13.5807749296655

Physically speaking, this is what was expected. As the radius of confinement shrinks,

the binding energy becomes less negative, which means the electron is less bound to

the proton in the nucleus. Comparing a fixed confinement radius at different confining

potentials W , it can also be seen that the binding energy becomes less negative as

the confining potential increases. This agrees with the trend shown in [3].

A quick calculation shows the number of iterations needed to find a root using the

bisection method. Let δ0 = b − a = 0.000026359773681372545 and let δ = tolerance, which is set by default to 10−15 in Sage. Then using the equation in [5], the number

of iterations n in the above calculations is

ln(δ0)−ln(δ) n = ln(2) ≈ 35

It should be noted here that the default maximum number of iterations for the

bisection method in SciPy is 100.

It is interesting to compare the modified ground state energies in the tables above

with the energy levels of a relativistic hydrogen atom that is not confined to any

potential. The table below contains those energies from [1]. Since J = 1/2 in (5)

(for the n = 1 ground state), the only unperturbed energy states that were compared with also had J = 1/2.

20 nS1/2 Binding Energy (unperturbed)

1 −13.6058741710731

2 −3.40147986373631

3 −1.51176379673416

4 −0.850365013000555

5 −0.544232648331672

6 −0.377938852703664

7 −0.277669090835843

8 −0.212590236100368

9 −0.167972430004738

10 −0.136057600495405

Comparing the modified binding energies at the smallest value of R with the table above, the modified ground state effectively achieves an unperturbed n = 6 excited state. This results from confining the atom to a smaller bounded region. As R increased in value, the new binding energy approached the unperturbed n = 1 binding energy value.

As stated in the Introduction, one key point of interest that would be focused on in this paper was to discover whether relativistic effects deserve attention when talking about the confinement of atoms to small spaces. [3] calculated the effects of confinement on the ground state of non-relativistic hydrogen. To find an answer to this question, it was decided that a comparison of relativistic energy values approximated by the model in this paper should be compared with the energy values computed in

[3] at the same confining radii and confining potential. The units for W and E are in terms of the Rydberg energy (1Ryd ≈ −13.605693eV ). The table below contains the comparison.

21 Radius (Bohr) E (Ley-Koo, Rubinstein)[3] E (Deck, Noon) ∆E

5.728340 −0.99940 −0.99777806 0.00162009302129373

5.139670 −0.99840 −0.99429488 0.00410511838556393

4.579720 −0.99600 −0.98630609 0.00969391381179463

4.149640 −0.99200 −0.97350420 0.0184958025510178

3.574570 −0.980300 −0.93667024 0.0436297610358837

3.138860 −0.961200 −0.87760301 0.0835969942837815

2.467660 −0.890000 −0.65447707 0.235522933240757

1.962810 −0.756100 −0.20553224 0.550567762611254

Above, E = −1 is roughly equivalent to the n = 1 energy in the previous table.

Likewise, E closer to 0 implies a less bound electron. A graphical comparison of the relativistic energy values vs the non-relativistic energy values in [3] are displayed below. In the paper by Ley-Koo and Rubinstein, energies are calculated for confine- ment radii smaller than the radii used in this relativistic model. The reason for this is because the root finding program was not able to compute energies  < 1 for radii

below R ≈ 1.845. This implies that the relativistic electron becomes unbound at

larger confining radii than in the non-relativistic model provided by [3].

22 E(Ryd)

1

0.5 Ley-Koo, Rubinstein •

Deck, Noon /

R(Bohr) 1 2 3 4 5 6

-0.5

-1

At relatively large confinement radii, there appears to be no relevant difference between relativistic and non-relativistic energy values. However, as R decreases, the difference in energy values ∆E grows, up to a factor of 4 for the last confinement radius value. This plot seems to show the at small confining radii, relativistic effects must be taken into account when calculating bound state energies of hydrogen.

23 Chapter 4

Conclusion

4.1 Overview

In chapter one, a simple model (a particle confined in an infinite potential well) was presented using the time-indepedent Schrodinger equation. The model shows how the energy levels are discrete as well as how these discrete energy levels change when the size of confinement changes.

In chapter two, this problem was extended to the confinement of a ground state relativistic hydrogen atom in a spherically symmetric finite potential well using the time-independent Dirac equation in spherical coordinates. After solving the Dirac radial equations (system of first order linear differential equations) and satisfying the boundary conditions, an equation was obtained (32) that contained a pair of confluent hypergeometric functions. With fixed potential well height and confinement radius, this equation could then be used to obtain the corresponding ground state relativistic energy values.

In chapter three, two numerical methods, Brent’s method and Bisection method, were used to compute the roots of (32) (energy values). The main reason for the specific choices in methods was that computing derivatives of hypergeometric func- tions with respect to their parameters (which contained the independent variable )

24 requires further numerical approximation, which will make the modified energy val-

ues less accurate. It was subsequently shown that, like the non-relativistic case, the

binding energy of the electron moves closer to zero (i.e. the electron gains energy)

as the well height increases or the radius of confinement decreases. Furthermore,

the comparison of relativistic ground state energies and non-relativistic ground state

energies of hydrogen at a fixed confinement radius and fixed well height showed that

accounting for relativistic effects becomes essential when calculating accurate energy

values.

4.2 Limitations and Future Work

These numerical results were strictly for the ground state of a relativistic hydrogen

atom. If this result was to be generalized to all states (→ κ = −1, ±2, ±3, ...), satisfying the boundary conditions at r = R would prove more difficult to achieve.

The general solution for G(ρ) in the case of r > R would yield a linear combination of spherical Bessel and spherical Neumann functions instead of the well behaved, easily handled exponential solutions obtained with κ = −1. Further work is needed obtain a generalized method.

As the first tables in the results section shows, only energy values at radii equal to or greater than R ≈ 1.85A0 were found. This indicates that below this radius, the

2 electron becomes unbound (E > m0c ). If the closed interval were to include  > 1 values, then (32) would contain complex valued functions, including the confluent hypergeometric functions. While the complex valued hypergeometric series will not be difficult to deal with, the solutions G(ρ) and F (ρ) for r > R would change. In

fact, a linear combination of sin and cos functions would be the solutions for F (ρ)

and G(ρ). The normalization constants in this scenario become more difficult to get

rid of by taking a ratio of these new solutions just like what was done previously.

25 Further work is also needed to rectify this problem.

q 1−+W One other limitation can be spotted in (32). The terms that contain 1+−W limit how big W can be. If W ≥ 2, then the only way those terms can be real is if  ≥ 1. This in turn would force other terms to become complex valued, and the solutions F (ρ) and G(ρ) would also need to be changed in the way stated in the

2 paragraph above. Therefore, it would seem that for bounded electrons (E < m0c ), W ≥ 2 cannot be allowed. This is the reason the final W value in the third table of the results section is 1.9995. One possible remedy for this problem points to a phenomenon known as the ”Klein Paradox” [13]. Without going into excessive detail,

2 2 2 it can be summed up as the following: for 2m0c < E + m0c < V0 (E > m0c ) with

V0 being the potential well height, the electron becomes a free particle. This means that it will tunnel through the barrier and escape the confining potential well. This

avenue is currently being explored.

26 References

1. Wachter, A. (2010). Relativistic quantum mechanics. Springer Science Busi-

ness Media.

2. Greiner, W. (1990). Relativistic quantum mechanics (Vol. 3). Berlin: Springer.

3. LeyKoo, E., Rubinstein, S. (1979). The hydrogen atom within spherical boxes

with penetrable walls. The Journal of Chemical Physics, 71(1), 351-357.

4. Brent, R. P. (1971). An algorithm with guaranteed convergence for finding a

zero of a function. The Computer Journal, 14(4), 422-425.

5. Burden, R., Faires, J. (2011). Numerical Analysis 9th edn (Boston: Brooks/-

Cole).

6. Arfken, G. B., Weber, H. J., Harris, F. E. (2011). Mathematical methods for

physicists: a comprehensive guide. Academic press.

7. Jones, E., Oliphant, T., Peterson, P. (2016). others. SciPy: Open source

scientific tools for Python. 2001. URL http://www. scipy. org.

8. SageMath, the Sage Mathematics Software System (Version 8.0.0), The Sage

Developers, 2017, http://www.sagemath.org.

9. Griffiths, D. J. (2016). Introduction to quantum mechanics. Cambridge Uni-

versity Press.

10. Sommerfeld, A., Welker, H. (1938). Knstliche Grenzbedingungen beim Kepler-

problem. Annalen der Physik, 424(12), 56-65. 27 11. Deck, R. T., Amar, J. G., Fralick, G. (2005). Nuclear size corrections to the

energy levels of single-electron and-muon atoms. Journal of Physics B: Atomic,

Molecular and Optical Physics, 38(13), 2173.

12. Ancarani, L. U., Gasaneo, G. (2008). Derivatives of any order of the confluent

hypergeometric function F 1 1 (a, b, z) with respect to the parameter a or b.

Journal of Mathematical Physics, 49(6), 063508.

13. Klein, O. (1929). Die Reflexion von Elektronen an einem Potentialsprung nach

der relativistischen Dynamik von Dirac. Zeitschrift fr Physik, 53(3-4), 157-165.

14. Bjorken, J. D., Drell, S. D. (1964). Relativistic quantum mechanics.

28 Appendix A

Code and Graphics

1. Below is an image of the simple model in Chapter 1.1

2. Below contains the computer code to find the roots:

29 #This numberical calculation solves for the new ground

state binding energy for a relativistic hydrogen atom

#in a box

E=var ( ’E ’ ) #Will serve as our epsilon

V=var ( ’V ’ )

P=var ( ’P ’ )

R=var ( ’R ’ )

#below is the fine structure constant alhpa from the math a=1/137.035999

s=s q r t (1−(a ) ∗∗2)

#below is beta from the math b=2589.605192

#below is the rest energy of the electron

rest=510998.9461

#below is the nonrelativistic binding energy for

unperturbed hydrogen

Ryd=−13.605693 from sage.numerical.optimize import f i n d r o o t from s c i p y import optimize

#P is rho from the math, contains R

#R is confinement radius

#V is confining potential W in terms of rest energy

#Below we will vary the parameters R and V to find

perturbed ground states

#Find binding energy in eV via Bisection, brent’s at V

30 =0.5, V=1, and V=1.9995 l i s t 1 =[] l i s t 2 =[] l i s t 3 =[]

for i in range ( 0 , 5 ) :

V=0.5

P=2∗b∗ s q r t (1−E∗∗2) ∗(1.85+ i ) ∗.0529177

f(E)=(sqrt((1−E)/(1+E))+sqrt((1−E+V) /(1+E−V))+(((2/P)

∗ s q r t (1−E∗∗2) ) /(1+E−V))) ∗ (((E∗a ) / s q r t (1−E∗∗2) )−s ) ∗

hypergeometric M ( s −((E∗a ) / s q r t (1−E∗∗2) ) +1 ,(2∗ s ) +1,

P)+(sqrt((1−E) /(1+E) )−s q r t ((1−E+V) /(1+E−V)) −(((2/P

) ∗ s q r t (1−E∗∗2) ) /(1+E−V))) ∗(1+(a/sqrt(1−E∗∗2) ) ) ∗

hypergeometric M ( s −((E∗a ) / s q r t (1−E∗∗2) ) ,(2∗ s ) +1,P)

New1=optimize . bisect(f(E)

,0.9999733739682344,0.9999997337419158)

New2=optimize . brentq(f(E)

,0.9999733739682344,0.9999997337419158)

Bind1=New1∗ rest −r e s t

Bind2=New2∗ rest −r e s t

list1 .append((1.85+i ,Bind1,Bind2))

for i in range ( 0 , 5 ) :

V=1

P=2∗b∗ s q r t (1−E∗∗2) ∗(1.85+ i ) ∗.0529177

31 f(E)=(sqrt((1−E)/(1+E))+sqrt((1−E+V) /(1+E−V))+(((2/P)

∗ s q r t (1−E∗∗2) ) /(1+E−V))) ∗ (((E∗a ) / s q r t (1−E∗∗2) )−s ) ∗

hypergeometric M ( s −((E∗a ) / s q r t (1−E∗∗2) ) +1 ,(2∗ s ) +1,

P)+(sqrt((1−E) /(1+E) )−s q r t ((1−E+V) /(1+E−V)) −(((2/P

) ∗ s q r t (1−E∗∗2) ) /(1+E−V))) ∗(1+(a/sqrt(1−E∗∗2) ) ) ∗

hypergeometric M ( s −((E∗a ) / s q r t (1−E∗∗2) ) ,(2∗ s ) +1,P)

New1=optimize . bisect(f(E)

,0.9999733739682344,0.9999997337419158)

New2=optimize . brentq(f(E)

,0.9999733739682344,0.9999997337419158)

Bind1=New1∗ rest −r e s t

Bind2=New2∗ rest −r e s t

list2 .append((1.85+i ,Bind1,Bind2))

for i in range ( 0 , 5 ) :

V=1.9995

P=2∗b∗ s q r t (1−E∗∗2) ∗(1.85+ i ) ∗.0529177

f(E)=(sqrt((1−E)/(1+E))+sqrt((1−E+V) /(1+E−V))+(((2/P)

∗ s q r t (1−E∗∗2) ) /(1+E−V))) ∗ (((E∗a ) / s q r t (1−E∗∗2) )−s ) ∗

hypergeometric M ( s −((E∗a ) / s q r t (1−E∗∗2) ) +1 ,(2∗ s ) +1,

P)+(sqrt((1−E) /(1+E) )−s q r t ((1−E+V) /(1+E−V)) −(((2/P

) ∗ s q r t (1−E∗∗2) ) /(1+E−V))) ∗(1+(a/sqrt(1−E∗∗2) ) ) ∗

hypergeometric M ( s −((E∗a ) / s q r t (1−E∗∗2) ) ,(2∗ s ) +1,P)

New1=optimize . bisect(f(E)

32 ,0.9999733739682344,0.9999997337419158)

New2=optimize . brentq(f(E)

,0.9999733739682344,0.9999997337419158)

Bind1=New1∗ rest −r e s t

Bind2=New2∗ rest −r e s t

list3 .append((1.85+i ,Bind1,Bind2))

#Now we will compare our energies in Rydberg units to Ley

−Koo and Rubinstein at V=1(Ryd)

R1 =

[5.72834,5.13967,4.57972,4.14964,3.57457,3.13886,2.46766,1.96281]

E1 =

[ −0.9994 , −0.9984 , −0.9960 , −0.9920 , −0.9803 , −0.9612 , −0.8900 , −0.7561]

#We can compare the numbers from Ley−Koo and Rubinstein

directly for V=1

compare1=[] ournums1=[]

for j in range ( 0 , 8 ) :

V=0.9999733743227773

P=2∗b∗ s q r t (1−E∗∗2) ∗(R1 [ j ] ) ∗.0529177

33 f(E)=(sqrt((1−E)/(1+E))+sqrt((1−E+V) /(1+E−V))+(((2/P)

∗ s q r t (1−E∗∗2) ) /(1+E−V))) ∗ (((E∗a ) / s q r t (1−E∗∗2) )−s ) ∗

hypergeometric M ( s −((E∗a ) / s q r t (1−E∗∗2) ) +1 ,(2∗ s ) +1,

P)+(sqrt((1−E) /(1+E) )−s q r t ((1−E+V) /(1+E−V)) −(((2/P

) ∗ s q r t (1−E∗∗2) ) /(1+E−V))) ∗(1+(a/sqrt(1−E∗∗2) ) ) ∗

hypergeometric M ( s −((E∗a ) / s q r t (1−E∗∗2) ) ,(2∗ s ) +1,P)

New=f i n d r o o t ( f (E)

,0.9999733599733552,0.9999997336019598)

NewBind=New∗ rest −r e s t

RydBind=(NewBind/Ryd) ∗(−1) changeE=abs (E1 [ j ]−RydBind ) compare1.append((R1[ j ] ,E1[ j ] ,RydBind,changeE)) ournums1.append((R1[ j ] ,RydBind))

34