investigating the hydrogen atom as a quantum system

7/27/2019 Investigating the Hydrogen Atom as a Quantum System

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Extended Essay

Investigating the Hydrogen Atom

as a Quantum System

Cole Coupland

Candidate #002424-0016

Mulgrave School

International Baccalaureate Programme

Extended Essay in Physics

May 2014

3871 Words

Author:

Cole Coupland

Supervisor:

Dr. Michael Frewin


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Abstract

The purpose of this paper is to provide an analytic treatment of the hydrogenatom in its ground energy state by considering the hydrogen atom as a quantumsystem rather than a classical system. This analysis involves the method andthe solution to the Schrodinger wave equation for the said hydrogen atom. Fur-thermore, a graphical analysis is conducted of the plots of the electron densityfunction and the radial distribution function of the theoretical hydrogen atom in

question. The research question that will be investigated is, What conclusionscan be made regarding the information obtained by applying the Schrodingerequation on the hydrogen atom and investigating its relevant wavefunctions?.

By examining the intrinsic properties of the hydrogen atom and the spher-ical coordinate system, the necessary information can be obtained to solve theSchrodinger wave equation for the hydrogen atom. The solutions to this equa-tion are two pieces of important information that could not be found by simplyanalyzing the properties of the hydrogen atom; the energy of the single electronand the most probable radius for an electron to be located at surrounding theatom or the Bohr radius. Two other important functions, the electron densityfunction and the radial distribution function, regarding the quantum nature ofthe hydrogen atom are then derived, plotted and analyzed.

The solution to the Schrodinger equation for the hydrogen atom was that theenergy of the electron was 13.6eV and that the Bohr radius was 5.29 1011m.There were also many significant consequences that were then reached fromanalyzing the plots of the two aforementioned functions such as the fact thatthere is a probability of the electron in the hydrogen atom existing anywhere inthe universe for when the radius is greater than zero.

Word Count: 292 words


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Contents

Abstract i

Introduction 2

Important Concepts 3

1.1 Wave-Particle Duality . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Quantum Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 The Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 The Schrodinger Wave Equation . . . . . . . . . . . . . . . . . . 4

Considerations for the Hydrogen Atom 6

2.1 Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 The Spherical Polar Coordinate System . . . . . . . . . . . . . . 7

2.2.1 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . 72.2.2 The Wavefunction in Spherical Coordinates . . . . . . . . 82.2.3 Laplacian of the Wavefunction in Spherical Coordinates . 8

Solving the Schrodinger Equation for the Hydrogen Atom 10

3.1 Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Determining a Functional Solution . . . . . . . . . . . . . . . . . 113.3 Rearranging the Equation . . . . . . . . . . . . . . . . . . . . . . 113.4 The Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.4.1 The Atomic Radius . . . . . . . . . . . . . . . . . . . . . 133.4.2 The Energy of the Electron . . . . . . . . . . . . . . . . . 13

Related Functions 15

4.1 The Electron Density Function . . . . . . . . . . . . . . . . . . . 154.2 The Radial Distribution Function . . . . . . . . . . . . . . . . . . 15

Conclusion 18

Bibliography 20

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Introduction

The model used to describe the atom has evolved from the times of Aristotle tothe model which we currently use and is constantly adapting to new discoveries.With the fall of the Bohr model came the rise of the quantum mechanical modelof the atom which described the position of electrons within the atom as aprobability function rather than a definite position within space and time.

The Schrodinger equation is a generalized description of a quantum me-chanical system. When Schrodingers equation is applied to a specific quantummechanical system such as an atom of a particular element, one can determinecertain characteristics of the system such as the energy of certain electrons andthe various atomic radii. The significance of this discovery is profound as itis confirmation for the basis of quantum mechanics and the startling implica-tion that nothing in our universe is certain. The existence of such orbitals andsubshells also explains atomic trends such as the ionization energies of eachatom.

The research question that will be investigated throughout the course of thispaper is What conclusions can be made regarding the information obtained by

applying the Schrodinger equation on the hydrogen atom and investigating itsrelevant wavefunctions?. This question has great significance in the world ofphysics because if the information obtained from such an investigation matchedthat of corresponding experimental results it would provide further evidence thatparticles can be expressed with wave like properties. There is also significance inthe fact that the probabilistic nature of an electrons position can be expressedin terms of the wavefunction and therefore if experimental results do matchthat of the information gained by applying the concept of the wavefunction, thefoundation of determinism would be undermined.

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Important Concepts

1.1 Wave-Particle Duality

Before the dawn of quantum mechanics, scientists stood puzzled by numerous

experiments involving particles and waves which would not behave as predictedby theory. Louis de Broglie suggested the groundbreaking idea that particlescould act as both a wave and a particle; the idea that became well-known asthe wave-particle duality. De Broglie also hypothesized that,

p =h

suggesting that particles were related by momentum to an associated wave-length (Tsokos). De Broglies idea was later confirmed by numerous experi-ments, one of which was the famous electron diffraction experiment. ClintonDavisson and Lester Germer observed that when electrons were passed throughnormally to a crystalline piece of nickel, the electrons would not pass straightthrough but would rather diffract away from the expected path. Diffraction is

what would be expected if an electromagnetic wave of wavelength was passedthrough an aperture of length comparable to . Davisson and Germer real-ized that this was direct evidence of electrons behaving as waves and confirmedthis by finding that the experimental wavelength of the electron matched thewavelength predicted by de Broglies relations (Galitski, Part III: PioneeringExperiments (contd); Penrose, 500-01).

The Schrodinger wave equation is written in terms of the properties of wavesand therefore establishing the wave-particle duality allows one to treat an elec-tron as a wave and hence solve the Schrodinger equation for the hydrogen atom.

1.2 Quantum Numbers

Quantum numbers are used to describe many of the important features of aquantum system. In this case the quantum system that will be analyzed isthe hydrogen atom. Four quantum numbers are typically used to describe theelectron configuration in a general atom, three of which are relevant to thehydrogen atom.

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The principal quantum number, n, has a range of values from 1 to n andindicates the energy level or electron shell the valence electrons are in. The

azimuthal quantum number, l, describes which subshell is being dealt with.The subshells s,p,d, and f have l values of 0, 1, 2, and 3 respectively. The rangeof l is 0 to n - 1 where n is the principal quantum number. The magneticquantum number of an atom, ml, specifies the orbital within the subshell thatcontains the electron under question and has a range ofl to l. For instance,an electron in the d subshell with l value of 2, would have ml values of either2,1, 0, 1, and 2 and hence confirms that there are five distinct orbitals withinthe d subshell. The fourth quantum number is ms, which indicates the spinof the electron within an orbital. This quantum number is not necessary todescribe the hydrogen atom to solve the Schrodinger equation as the solution isnot dependent on this quantum number (Hua).

1.3 The Wavefunction

The wavefunction, defined by the Greek letter , is a function of both positionand time. The wavefunction alone is meaningless as it only yields complexvalues but is given meaning when it is squared. The square of the wavefunction,||2, evaluates to a real value and it is interpreted as the probability of a free-particle being located at a certain point (x ,y ,z) at a particular time t. Thewave function can be described by the equation,

(x, t) = ei(kxt)

where i is the imaginary unit, x is the free-particles displacement, is itsangular velocity and t is time. The angular wave number k is equal to 2

where is the wavelength of the particle. Thus, all of the key informationconcerned with the particles movement and position are contained within thewavefunction, making it crucial to determining the probability of a particlebeing located at a certain position in space (Galitski, Part I: Meaning of theWavefunction).

1.4 The Schrodinger Wave Equation

For the purposes of this essay we need only be concerned with the time-independentSchrodinger Equation. This formula, similarly to the time-dependent form ofthe equation, uses the variable coordinates of a free-particle, defined as x, y andz but lacks the variable of time. The time-dependent Schrodinger equation is

much more complex and would lead the investigation astray from its intendedpurpose. The time-independent Schrodinger Equation takes the form,

E (x ,y ,z) = h2

2m2(x ,y ,z) + U (x ,y ,z) (1.1)

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The variable E represents the energy of the free-particle, (x ,y ,z) is thewavefunction of the particle, m is its mass and U is its potential energy. h is

the reduced Plancks constant which is Plancks constant divided by 2. TheLaplacian operator, 2, of a function, in this case the wavefunction, indicatesthe sum of the second derivatives of the wavefunction with respect to each ofthe independent variables (Galitski, **Part IV: Deriving the Schrodinger Eq.).That is,

2(x ,y ,z) = d2

dx2+

d2

dy2+

d2

dz2(1.2)

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Considerations for the

Hydrogen Atom

2.1 Potential Energy

To solve the Schrodinger equation for the hydrogen atom one must define thefree particle in the equation as the electron of a hydrogen atom. One stepthat must be taken to reach a solution is to express the potential energy, U,of the electron in terms of r. To do this one must consider that an electronspotential energy is caused by a single proton, as in the case of a hydrogen atom(Richmond). Potential energy can be written in the form,

U = mda (2.3)

where m is the mass of the electron, a is the acceleration the electron expe-riences and d is the distance between the electron and the reference point whichhappens to be the proton. Therefore r, the atomic radius, can be substituted

for d. To determine the acceleration of the electron one must apply Newtonssecond law of motion stating that,

a =F

m(2.4)

where F is the force acting on the electron and m is its mass. Interchangingthe variables d and r, and substituting equation 2.4 into 2.3 yields,

U = F r (2.5)

In order to determine the force acting upon an electron in a hydrogen atom,Coulombs law must be used, as the situation deals with charged particles.Coulombs law consists of three variables: q1 being the charge of the first parti-

cle, q2 being the charge of the second particle and r being the distance betweenthe charges. 0 is the permittivity of free space and is a physical constant(Tsokos).

Coulombs law states that,

F =q1q2

40r2

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The magnitude of the charge of an electron and a proton are both the ele-mentary charge, e, but the charges act in opposite directions. Therefore,

F = e2

40r2(2.6)

By substituting equation 2.6 into equation 2.5, the potential energy as afunction of the radius can be obtained.

U = ( e2

40r2)r

U(r) = e2

40r(2.7)

2.2 The Spherical Polar Coordinate System2.2.1 Spherical Coordinates

Another step that must be taken in solving the Schrodinger equation is to usespherical polar coordinates rather than Cartesian coordinates. The reason whywe do this will be explained shortly but first the system itself will be described.Consider a point mass moving around a central point of origin called O. Letthe line segment that connects O and the point mass be OP. The positionof a point mass in space can be described by r, the magnitude of OP, , theangle between the x-axis and OP, and , the angle between the z-axis and OP.A visual representation of the spherical coordinate system is presented in thefollowing figure,

Figure 2.1: The Spherical Polar Coordinate System

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In order to eliminate the need for some variables the variables x, y and zshould be converted to the spherical coordinate variables r, and . It is

intuitive to write the x, y and z variables in terms of the spherical coordinatevariables as the potential energy is a function of r, which is also a variable inthe spherical polar coordinate system. This effectively allows for two of thevariables to become one unified variable, r. To do so the wavefunction must beexpressed as a function of the spherical coordinates, r, and (Richmond). Inthe next subsection it will be shown why it will eliminate both of the angularvariables, , and .

2.2.2 The Wavefunction in Spherical Coordinates

The wavefunction, in terms of the spherical coordinates, can be represented asthe product of three functions,

(r,,) = R(r)()()

where R(r) is the radial wave function, and () and () are the twoangular wave functions (Hua).

The two angular wave functions can be thought of the components of a singleangular wavefunction, namely,

Y(, ) = ()()

The quantum numbers for hydrogen in its ground state are all 0 exceptfor the principal quantum number which is 1. This is because the specificelectron is in an s-orbital in the 1s subshell which only contains one orbital.Each component of the wavefunction is dependent on certain quantum numbersand these quantum numbers that they are dependent on are indicated in the

subscripts of each wavefunction. The order in which the relevant quantumnumbers are expressed in the subscript of each wavefunction is n, l and thenml (Hua). Therefore for the hydrogen atom in its ground energy state, thewavefunction can be expressed as,

1,0,0(r,,) = R1,0(r)Y0,0(, )

The corresponding angular wavefunction evaluates to a constant and there-fore the wavefunction is only dependent on the variable r. In order to show thatthe angular wavefunction is a constant, Legendre polynomials must be used butthis would deviate from the purpose and the focus of this essay (Kuntzleman).

2.2.3 Laplacian of the Wavefunction in Spherical Coordi-

nates

The Laplacian of the wavefunction in Cartesian coordinates is much simpler tofind than that of the Laplacian of the wavefunction in spherical polar coordinatesand can simply be written as the sum of the second derivatives of the x,y, andz variables as shown in equation 1.2.

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By using the identities,

x = r sin cos

y = r sin sin

z = r cos

r =

x2 + y2 + z2

one can determine the Laplacian of the wavefunction in spherical coordi-nates. This Laplacian has been proved to be,

2(r,,) = d2dr2

+ 2r

ddr

+ 1r2 sin2

d2d2

+ r2 cos sin

dd

+ 1r2

d2d2

(2.8)

but deriving this Laplacian is extensive and beyond the scope of this essayand therefore it is reasonable to simply use this established Laplacian of thewavefunction in spherical polar coordinates (Weisstein).

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Solving the Schrodinger

Equation for the Hydrogen

Atom

3.1 Simplification

By putting together all the terms that have just been derived and using keyinformation concerning the hydrogen atom it is now possible to simplify theSchrodinger equation to the point at which it is possible to describe a func-tional solution of the equation in terms of r (Richmond). First the generalSchrodinger equation (1.1) must be rewritten in terms of the wavefunction inspherical coordinates and the potential energy as a function of r,

E (r,,) = h2

2m2(r,,) + U(r)(r,,) (3.9)

Then the potential energy function (Equation 2.7) and the Laplacian of thewavefunction in spherical coordinates (Equation 2.8) must be substituted intoequation 3.9,

E = h2

2m

d2

dr2+

2

r

d

dr+

1

r2 sin2

d2

d2+

r2 cos

sin

d

d+

1

r2d2

d2

e

2

40r

The wave function for an electron in a hydrogen atom in its ground stateis not dependent on the electrons angular position as it is simply a constantindependent of the variables and . Therefore the terms involving the first andsecond derivatives of the wavefunction with respect to and can be removedfrom the equation (Richmond). The Schrodinger equation hence becomes,

E = h2

2m

d2

dr2+

2

r

d

dr

e

2

40r

E = h2

2m

d2

dr2 h

2

2m

2

r

d

dr e

2

40r (3.10)

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3.2 Determining a Functional Solution

To proceed with solving the Schrodinger equation for the hydrogen atom wemust be able to simplify equation 3.10. This equation involves the first and sec-ond derivatives of the wavefunction with respect to r and therefore a functionalsolution must be determined. As determined previously, the wavefunction isdependent solely on the radius and therefore = kR(r), where k is a posi-tive constant determined by the angular wavefunction. As the first and secondderivatives of the wavefunction are with respect to r, they can also be rewrit-ten in terms of the radial wavefunction. Hence, if a general form of the radialwavefunction could be determined, it could be used to find a general solution tothe Schrodinger equation (Winter). The radial wavefunction of hydrogen with aprincipal quantum number of 1, azimuthal quantum number of 0, and magneticquantum number 0 is,

R(r) = 2

1b

32 e

rb

and the corresponding angular wavefunction is,

Y(r) = k =14

b is a constant known as the Bohr radius or the radius at which an electronis most probable to be at around the nucleus (Finley). Since the wavefunctionis the product of the angular wavefunction and the radial wavefunction it canbe expressed as,

=14

(2

1

b 3

2 er

b ) =1

1

b 3

2 er

b

To make matters simpler, the variable can be assigned to the constantbefore the exponential term,

=1

1b

32

This makes the equation easier to algebraically manipulate and does notimpede upon reaching a solution. Therefore the general functional solution forthe simplified Schrodinger equation is,

= er

b

3.3 Rearranging the EquationEquation 3.10 can now be simplified further since a functional solution hasbeen obtained. Seeing that the equation deals with the differentiated forms of

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the functional solution, these differentiated forms should be determined (Rich-mond). The first and second derivatives of the wavefunction can be obtained

by using the differentiation rule,

d

dxAef(x) = Af(x)ef(x)

where A is some constant. Applying this rule on the wavefunction,

d

dr(e

r

b ) = b

er

b

d2

dr2(

be

r

b ) =

b2e

r

b

Substituting these first and second derivatives of the wavefunction with re-spect to r, and the wavefunction itself into equation 3.10 provides for a more

manageable form of the Schrodinger equation for the hydrogen atom (Rich-mond).

E(er

b ) = h2

2m(

b2e

r

b ) h2

2m

2

r(

be

r

b ) e2

40r(e

r

b )

Eer

b +e2

40re

r

b = h2

2mb2e

r

b +h2

mrbe

r

b

3.4 The Solution

As we know that the radial wavefunction for the ground state of hydrogen takesthe form,

R(r) = 21

b

32 e

r

b

it can be determined there are no radii r which makes the radial wavefunctionevaluate to zero (Finley). This is because the exponential term is greater thanzero for all r and the Bohr radius is a positive constant. Since the wavefunctionis the product of the angular and radial wavefunctions, if the radial wavefunctionwas zero, the wavefunctions value would also be zero. The probability of anelectron occuring at a certain point is the square of the absolute value of thewavefunction and therefore if the wavefunction were zero, the probability ofan electron existing at that position in space would also be zero. Using thisreasoning it can be deduced that since the radial wavefunction is positive for

allr

, there are no points in space where the probability of an electron beinglocated there is zero. The lack of these points, or nodes, for the ground state ofthe hydrogen atom also implies there be an infinite number of solutions to theSchrodinger equation specific to the hydrogen atom (Hua). For the simplifiedform of the Schrodinger Equation to have infinite solutions two types of terms inthe equation must be considered; the

rerb terms and the e

r

b terms. There

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are two of each type of term and for there to be infinite solutions each of thelike terms must be equal to one another. Therefore,

Eer

b = h2

2mb2e

r

b (3.11)

and,

e2

40re

r

b =h2

mrbe

r

b (3.12)

3.4.1 The Atomic Radius

The atomic radius for the ground energy level of hydrogen can be obtained bycontinuing to solve equation 3.12 for b (Richmond). The

re

r

b term can bedivided out on each side of the equation leaving,

e2

40=

h2

mb

Rearranging to isolate b yields,

b =40h

2

me2

but since h2 = h2

42this value can be simplified further,

b =0h

2

me2

By substituting the known values for the constants on the right hand side

of the equation, an approximate value for b can be reached,

b =(8.85 1012)(6.626 1034)2

(9.11 1031) (1.6 1019)2 5.29 1011m

3.4.2 The Energy of the Electron

To determine the energy of the single electron in the ground state of hydrogen,equation 3.11 must be solved for E (Richmond). Dividing by e

r

b reduces theequation to,

E = h2

2mb2

The right hand side of the equation is a constant value consisting of thereduced Plancks constant, the mass of an electron and the Bohr radius. As wehave found the value for the Bohr radius in the previous subsection, the energyof an electron in the ground energy state of hydrogen can also be determined.

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Using these known constants,

E = (6.63 1034)2

2 42 (9.11 1031) (5.29 1011)2 2.18 1018J

Since we are dealing with very small amounts of energy it is appropriate toconvert this value into electronvolts where one electronvolt is equal to 1.60 1019J. Applying this conversion,

E =2.18 10181.6 1019 13.6eV

This is only the energy that must must be applied to the electron to moveit from its position to infinity and therefore the actual energy of the electron ispositive and is thus 13.6eV.

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Related Functions

4.1 The Electron Density Function

The electron density function of an atom describes the electron density at an

arbitrary point surrounding the nucleus (Winter). The electron density is theprobability of an electron being present at that point and is also therefore equalto the square of the absolute value of the wavefunction. The electron densitycan also be thought of as the relative probability of an electron being locatedat a certain radius (Shusterman). The relationship between the electron densityfunction and the radial distribution function is,

||2 = 4r2P(r)where P(r) is the radial distribution function. Therefore the electron density

is the probability per unit surface area at the specific radius (Shusterman). Insection 3.2, it was determined that,

=

1

1

b 32

e

r

b

Therefore the electron density function for this particular hydrogen atom is,

||2 = 1b3

e2r

b

Given that we want the radius of the function to be measured in Bohr radiiwe can let b = 1,

||2 = 1

e2r

The graph of the electron density function can be seen in figure 5.2 on thefollowing page.

4.2 The Radial Distribution Function

The radial distribution function is another useful tool to investigate the distri-bution of electrons in an atom. It is a function of the radius of a certain electron

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Figure 4.2: The Electron Density Function

around an atom and will evaluate to the probability of an electron being locatedat that radius at an arbitrary time (Winter). The radial distribution functiondescribes the probability of an electron being located at a certain radius, whichis a set of points whereas the electron density function only describes the proba-bility of an electron existing at a single point around the atom. For the hydrogenatom in its ground energy state, the probability of an electron being located ata certain point is the same for all other points with the same radius as the an-gular wavefunction is simply a constant (Finley). Therefore for a given sphere

surrounding the nucleus, each point on the sphere will be equally probably ofhaving an electron exist at that point. Thus, by multiplying the electron densityfunction by the surface area of a sphere, 4r2, it will then be equivalent to theradial distribution function (Winter). Hence,

P(r) = 4r2||2 = 4r2

b3e

2r

b

By letting the Bohr radius equal 1, the units that the radius will be measuredin is the number of Bohr radii. As the definition of the Bohr radius is the mostprobable distance for an electron to be from the nucleus, the maximum of theradial distribution function should occur when r = 1. By substituting b = 1into the radial distribution function this new function can be determined,

P(r) = 4r2e2r

The graph of the radial distribution function may be seen on the followingpage in figure 4.3.

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Figure 4.3: The Radial Distribution Function

This graph must have a restricted domain as the distance between the nu-cleus and the electron must be greater than or equal to 0. Therefore the region

of the graph for r < 0 can be left from analysis.It can also be confirmed that the total probability of an electron existing inthe universe is 1 and this is shown by integrating the radial distribution functionwith respect to r from zero to infinity.

0

4r2e2rdr

To determine this integral, integration by parts must be used,

4r2e2rdr = 2r2e2r

4re2r

= 2r2e2r + 2re2r +

2e2r

= 2r2e2r + 2re2r e2r

0

4r2e2rdr = [2r2e2r + 2re2r e2r]0 = 1

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Conclusion

Throughout the course of this paper it should have become evident that treatingthe hydrogen atom in its ground state as a quantum system has allowed for agreat understanding of the intrinsic properties of atoms in general and a greaterunderstanding of the implications of quantum mechanics. The solutions to theSchrodinger wave equation for the hydrogen atom in its lowest energy statethat have been obtained match that of the corresponding experimental values.13.6eV is the value that has been accepted by the scientific community for theenergy of the electron in question which was the value obtained by solving theSchrodinger equation (Serway). The same applies for the Bohr radius which hasbeen accepted to be approximately 5.29 1011m which is also the value thatwas obtained through solving the equation (Serway). The fact that someone candetermine these values in their exact forms by treating the hydrogen atom as aquantum system through using the Schrodinger wave equation is confirmationfor the wave-particle duality. The Schrodinger wave equation uses informationthat would normally apply to waves, yet accurately describes similar informationfor an atom and thus shows that particles may act as waves and vice versa.

The graph of the radial distribution function for the said hydrogen atomcontains crucial information regarding the probabilistic nature of an electronsposition. Notice that,

limr

4r2e2r = 0

This can be seen visually from the fact that the graph asymptotically ap-proaches zero as the radius tends to infinity. Therefore the probability of anelectron existing at a certain radius never reaches zero. This conclusion is pro-found as it means that an electron may exist anywhere in the universe sur-rounding the nucleus, not just in the confines of the atom. The only positionin the universe where the probability of an electron existing is zero is whenr = 0, or the center of the nucleus of the hydrogen atom. The maximum of the

graph also indicates the most probable distance away from the nucleus for theelectron of the hydrogen atom to be found. In this case the maximum occurswhen r = 1 but since we defined the radius to be measured in the number ofBohr radii, this is simply the distance of the Bohr radius. When solving theSchrodinger equation this was determined to be 5.29 1011m. This maximumis confirmation for the fact that the Bohr radius is the most probable radius for

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the electron to be at. If the maximum were at an r value such that r = 1 itwould contradict the definition of the Bohr radius. The fact that the integral of

the function from zero to infinity is one shows that the total probability of anelectron existing in space is 1. If it were more or less than 1 it would indicatethat the radial distribution function was not a probability density function andwould cause a fundamental problem in the foundations of quantum theory.

The graph of the electron density function also contains important concep-tual information. Intuitively one may believe that the maximum of the electrondensity function occurs at the Bohr radius since this is the radius which hasthe maximum probability of an electron existing. This is not the case since theelectron density is the measure of the relative probability of an electron existingat a certain radius. Therefore, even though there is a very low probability of anelectron being found infinitesimally close to the center of the nucleus, the surfacearea as the radius approaches zero also tends to zero. It is therefore evident thatthe surface area approaching zero has a greater bearing on the electron densitythan the low probability of an electron being found at a very small radius. Also,because the graph never reached an electron density of zero, it can be concludedthat there would be a positive electron density in the universe even if only asingle hydrogen atom existed within such a universe.

In the future it would be interesting to investigate the application of theSchrodinger equation on atoms of different elements, energy states, subshellsand orbitals. I was very keen on making an essay similar to this one but with theSchrodinger equation being applied on an atom with an angular wavefunctionthat was not a constant. The three dimensional plots of such electron densityfunctions are truly spectacular when the probability is dependent on the angleand many such graphs involved interesting geometric shapes such as torusesand shapes that resembled dumbbells. The reason I chose to instead apply the

Schrodinger equation on the hydrogen atom was because I did not think it waspossible to effectively treat the more advanced topics within four thousand wordsand thought the essay would therefore be more suited to the hydrogen atom inits ground energy state. I have already begun investigating the consequences ofconsidering other atoms of various energies for the Schr odinger equation and Iwill continue to investigate this area further. Another area which I have yet tofully investigate is the time dependent form of the Schrodinger equation. It alsocould not be effectively treated within the word constraint as there is a greaterdegree of complexity involved due to the extra time dimension. Although itwas not suitable to treat in this essay it will provide for interesting furtherinvestigation.

In conclusion, considering the hydrogen atom solely as a particle imposesdrastic limitations on ones ability to understand the hydrogen atom and fur-

thermore the laws of the universe. The ramifications of considering the hydrogenatom as a quantum system is profound and the newly discovered probabilisticnature of the universe will be crucial to future discoveries in physics should itcontinue to uphold.

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Bibliography

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investigating the hydrogen atom as a quantum system

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