foundation of quantum mechanics

8/2/2019 Foundation of Quantum Mechanics

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The

FOUNDATIONof

QUANTUM MECHANICS

The Foundation of Quantum Mechanics 1 of 29


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INDEXPage

1.0 INTRODUCTION 4-52.0 THE HISTORY OF QUANTUM MECHANICS 6-10

2.1 Classical mechanics 6

2.1.1 Isaac Newton 62.1.2 Electro-magnetism 62.1.3 Albert Einstein's influence on mechanics 6-7

2.1.4 Atoms the 'complete' picture 72.2 Quantum mechanics 7

2.2.1 The evolution of the atomic model 7-8

2.2.2 Louis de Broglie 9

2.2.3 Erwin Schrdinger 9-102.2.4 Werner Heisenberg 10

3.0 DIFFERENCES BETWEEN CLASSICAL ANDQUANTUM MECHANICS

10-19

3.1 Determinism 10-11

3.1.1 Laplace's Demon 113.2 Indeterminism 11-123.2.1 The 'Copenhagen' interpretation 12

3.2.2 The Pauli exclusion principle 12-133.3 The differences 13

3.3.1 Feynman's thought experiments 13-163.3.2 Dirac's notation 16-18

3.3.3 Heisenberg's uncertainty principle 18-19

4.0 SCHRDINGER'S EQUATION 19-234.1 The equation 19

4.2 The 'infinite square well' 19-214.2.1 The quantization of energy (n = 3, 2, 1, 0) 21-23

5.0 CONCLUSION 24

6.0 RESUM 257.0 APPENDIX 26-288.0 BIBLIOGRAPHY 29-30

8.1 Books 29

8.2 Videos 29-308.3 Websites 30

The layout of this project was inspired by the layout of Niels Bohr's On the Constitution of

Atoms and Molecules. It seemed only fitting to make the layout of my project imitate and

thereby pay homage to the truly inspiring work done by Niels Bohr. Nonetheless, my

project does reach all requirements and does not overshoot the maximum allowance of

pages (if the margins were to be set to the normal borders provided by the standard text

editor programs).



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1.0 INTRODUCTION

Since early times, an understanding of the universe has been sought. An

understanding which could explain the workings of everything. This

understanding has changed greatly over many thousands of years, even

impacting other fields of thought such as philosophy. Aristotle (384 322

BC) had a philosophy which stated that the world consisted of four

elements, earth, air, fire, and water, and that objects acted to fulfill their

purpose1. Nave as it may have been, this was the start of a search for what

can be called a grand unified theory. Once Pythagoras (570 495 BC)

created the first mathematical proof of a physical phenomenon2 the stage

was set to describe the entire universe. When Newton (1643 1727 AD), an

avid mathematician, discovered the laws of gravity, a way of thoughtevolved, determinism, which went unchanged for four centuries3. So

ingrained and powerful was this way of thought that when a new form of

mechanics, quantum mechanics (from now on abbreviated as QM),

appeared, not even the greatest thinker of modern time, Albert Einstein

(1879 1955 AD), could be dissuaded. This was because QM was based on

probabilities rather than certainties, leading to indeterminism, to which

Einstein exclaimed, God doesn't play dice with the world4! The universe

is an enormously complicated system which no one, even today, has ever

fully understood. It is therefore no wonder that a theory, that seems to

explain a lot, is doubted. Moreover, it makes sense as QM is not completely

understood, yet.

What this project shall attempt to achieve is a fundamental

understanding of QM through a review of the experimental basis for our

view of the structure of atoms. This project will therefore be a short

historical guide to the persons involved in the establishment of QM and the

experiments which they performed. Moreover, this project will explain and

1 The Grand Design, p. 51.2 Fermats Store Stning, p. 37.

3 Fermats Store Stning, p. 38.

4 Einstein and the Poet, William Hermann.



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discuss the differences between the determinism of classical mechanics and

the indeterminism of QM. In order to give the reader a better understanding

of the intricacies of QM, I will later in the project explain and discuss parts

of some of the tools of QM, namely Schrdinger's equation and Dirac's

notation. I will also touch upon a fundamental aspect of QM, Heisenberg's

uncertainty principle.

The explanations for the Schrdinger equation and Dirac notation are

based on a compilation of lectures, videos and other assorted material. It

would be too time consuming to reference every part of the mathematics.

There will, therefore, be a complete list of the all the relevant references in

the bibliography to supplement the lack of individual references.



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2.0 THE HISTORY OF QUANTUM MECHANICS

Quantum mechanics is a culmination of the work done by various scientists,

sometimes by accident, sometimes by design. However, it all starts with the

advent of what we today call 'classical' mechanics.

2.1 Classical mechanicsClassical mechanics is the name given to the mathematically natured

science which gave us the law of gravity. This was thanks to the work

of the intuitive Isaac Newton who invented, independent of Gottfried

Leibniz, calculus, or rather the rules of differentials and integrals

which are still used today.

2.1.1 Isaac Newton

Thanks to the invention of calculus, Isaac Newton was able toaccurately calculate the movement of terrestrial and heavenly

bodies and the force of gravity, among many others5. This was an

incredible tour de force of science which allowed scientists to

accurately predict the course of an event. Scientists could now

understand the dynamics of a system and continued to apply

Newton's methods to other aspects of physics. This led to the

discovery of magnetism and electricity.

2.1.2 Electro-magnetism

At the beginning of the 19th century, scientists believed that

magnetism and electricity were two separate forces, H.C. rsted

did not. He believed that they had the same origin6. By passing

electricity through a conductor, H. C. rsted created a magnetic

field, thus proving his theory of electro-magnetism.

2.1.3 Albert Einstein's influence on mechanics

Albert Einstein, a genius of unforeseen magnitude, greatly

influenced the interpretation of classical mechanics because of his

discovery of the law of the photoelectric effect. Here Albert

Einstein postulated that electro-magnetic radiation consisted of

5 Taming the infinite. p. 141

6 Fysikkens Verden 3. p. 210-211



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light quanta(photons) moving at the speed of light7. With this he

proved that a photon's energy is:

Ephoton = h v (1)

Where h is Planck's constant and v is the electro-magnetic

radiation's frequency. This discovery eventually helped lead to the

conceptualization of the wave-particle duality which we shall

expand upon later.

2.1.4 Atoms the 'complete' picture

At this point in time, the idea of atoms had seemingly been

completed, that is to say, scientists believed atoms to be the

smallest parts of matter. An indivisible part. However, there

seemed to exist a great many atoms for such a unique and

fundamental building block8. This is why J. J. Thomson's

discovery of electrons may have been shocking yet welcomed.

2.2 Quantum mechanics

2.2.1 The evolution of the atomic model

After J. J. Thomson (1856 1940) discovered electrons,

mechanics and physics as a whole, took a

turn towards the infinitesimally small. No

longer was the atom the smallest functional

particle, now it consisted of electrons. J. J.

Thomson was led to conclude, based on the

evidence provided by his experiments, that

the electron is thousands of times less massive than the atom9.

This conclusion led to the 'plum-pudding' picture of the structure

of the atom. In the 'plum-pudding' atom, the negative electrons are

randomly dispersed over the positive body of the atom, see figure

7 Elementr kvantemekanik. p. 5

8 The ideas of particle physics. p. 2




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A. This atomic model, however, could not account for the results

of Hans Geiger and Ernest Marsden. Geiger and Marsden

discovered that when high-energy -particles collide with thin

metal-foil, a tiny fraction of the -particles are deflected up to

18010

. It was first explained when Ernest Rutherford (1871

1937) theorized that atoms must consist of tiny nuclei that make

up the majority of the mass of an atom, and an amount of

negatively charged electrons at rest around the nucleus that

manage to neutralize the positively charged nuclei. Classical

mechanics, however, dictated that if that were the case, the

electrons would gravitate inwards, towards the nucleus, as

opposites attract. Even if the electrons are not at rest but in anaccelerated orbit around the nucleus, they would emit electro-

magnetic waves (see (1)) thus losing energy and spiralling

towards the nucleus11. This is not what was found experimentally.

It was up to Niels Bohr (1885 1962) to create a modern atomic

model which could answer the question of the orbits of electrons.

In Bohr's theory, he postulates that electrons revolve around the

nucleus in stationary orbits that allow for the movement of the

electrons without their necessitated emission of radiation12.

Although this model is not the complete atomic model that is used

today, it is still the most stable and understandable model and is a

major influence on QM. We will not expand upon the history of

the atomic model further as this visualization allows us to

ascertain the approximate workings and purpose of an electron; to

orbit a nucleus, at a quantized distance with a variable but limited

speed(slower than or equal to the speed of light), at a variable

time, and neutralize the charge of the nucleus.

10 Elementr Kvantefysik. p. 18-19

11 Elementr Kvantefysik. p. 20

12 Elementr Kvantefysik. p. 21



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2.2.2 Louis de Broglie

Louis de Broglie (1892 1987) was the next physicist to advance

the concept of QM. In his doctoral thesis, de Broglie suggested

that electrons could act like waves in a manner where the

wavelength is inversely proportional to its momentum13

:

=h

p(2)

Where h is Planck's constant. The beauty of his theory consists of

the notion that this applies to all particles, not just electrons or

photons. One might say that if a human were to have a small

enough momentum, the human would also exhibit wavelike

properties. De Broglie's hypothesis managed to explain the nature

of Bohr's atomic model as the electronic orbits are only allowed if

they contain an integral number of de Broglie wavelengths 14.

However, de Broglie did not manage to create a resounding theory

of mechanics which could trump Newton's classical theory but his

work did set the stage for the development of one15.

2.2.3 Erwin Schrdinger

Erwin Schrdinger (1887 1961) took de Broglie's hypotheses on

particle waves and developed them into wave mechanics16.

Schrdinger's approach was to look at the waves' behaviour in

space and time. What he discovered was an equation which

describes the behaviour of matter through a description of a

particle by its wavefunction. This equation shows how a wave

might behave given a set of limits. Given such limits, the

Schrdinger equation is able to predict the theoretical interference

of particles. Practically, however, the measurement of such

13 The ideas of particle physics. p. 17-1814 The ideas of particle physics. p. 18





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behaviour is affected by the measurement itself.

2.2.4 Werner Heisenberg

Werner Heisenberg (1901 1976) essentially formulated that only

a system which is observed can be defined but that this

observation in itself is an interaction with the system. The

interaction with the system affects it denying perfect knowledge

of the system to the observer17. He formulated this principle

mathematically in what is today known as, the uncertainty

principle.

3.0 DIFFERENCES BETWEEN CLASSICAL AND QUANTUM

MECHANICSClassical mechanics had been the supreme way of thinking for hundreds of

years when QM appeared. This new method of understanding the universe

brought with it a new way of thinking which forcefully rejected the ways of

determinism. Now, scientist were led to believe in probabilities instead of

certainties. But why could the two mechanics not coexist? What made them

so different?

3.1 Determinism

Determinism states that, given the state of the universe at one time, a

complete set of laws fully determines both the future and the past18.

The metaphor that Newton grants us is of a clockwork universe19 in

which every action is followed by a definite reaction. As a practical

example, imagine an object moving in a vacuum. If that object is

moving with the speed of 10m/s then we can, through simple

arithmetic, figure out how long it takes the object to traverse a

distance. Newton's laws of motion even allow us to figure out the

starting acceleration of an object as long as we know the distance and

speed over which the acceleration, at observation, has acted over. This


18 The Grand Design. p. 30

19 The Number Mysteries. p. 230



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is a simple universe with no surprises, no miracles, except perhaps for

the creation of the universe.

3.1.1 Laplace's Demon

This scientific determinism was formulated into what is today

known as Laplace's Demon:

We may regard the present state of the universe as the effect of its

past and the cause of its future. An intellect which at a certain

moment would know all forces that set nature in motion, and all

positions of all items of which nature is composed, if this intellect

were also vast enough to submit these data to analysis, it would

embrace in a single formula the movements of the greatest bodiesof the universe and those of the tiniest atom; for such an intellect

nothing would be uncertain and the future just like the past would

be present before its eyes.20

In this analogy, Laplace basically states the view of determinism,

that all is determined. It is a neat philosophy in the sense that it

allows for the total knowledge and understanding of the entire

universe at any time as long as we know all data at a given point

in time. Though, if it were completely true, nothing would be left

to chance, not even free will21.

3.2 Indeterminism

Indeterminism, unlike determinism, certainly allows for free will. In

indeterminism, everything is governed by chance, or rather,

probabilities. At the macroscopic level indeterminism is diminished,

that is to say, the chance of all the particles in a human being suddenly

spreading out in their own unique direction, is incredibly small.

According to Bohr's Correspondence Principle, the macroscopic

effects of QM are actually rather deterministic because of the scale of

20 A Philosophical Essay on Probabilities. p.4

21 On Determinism, Sean Carroll.



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the macroscopic universe in comparison to Planck's constant 22,23. At the

microscopic level though, probabilities play a huge role.

3.2.1 The 'Copenhagen' interpretation.

The 'Copenhagen' interpretation of QM is an attempt to

understand this complex nature of QM. This interpretation, named

after the place where most progress within the field was made at

the time, was a summation of multiple points which, in essence,

describe QM. These points are, at their simplest24(see Appendix I

for full list):

1. A system is represented by its state in the form of a

wavefunction.

2. Nature is probabilistic and the square of the amplitude of awavefunction is the wavefunction's probability.

3. All values of a system cannot be known at the same time.

4. Matter exists in a wave-particle duality.

5. Measuring devices can only measure classical properties

as they themselves are classical in nature.

6. The description of a macroscopic system will be

approximately deterministic.

3.2.2 The Pauli exclusion principle

The Pauli exclusion principle, hypothesized by Wolfgang Pauli

(1900 1958), is a principle which explains the spreading of

electrons in an atom. According to Pauli, electrons cannot

simultaneously occupy precisely the same quantum state (i.e.

have identical values of momentum, charge and spin in the same

region of space)25. The principle explains the reason as to why

electrons will not huddle in the lowest quantum state but spread

out in additional orbits. In the innermost orbit, only two electrons

22 Kvantemekanik, Tom Andersen. p. 423 Copenhagen Interpretation of Quantum Mechanics. Jan Faye.

24 http://en.wikipedia.org/wiki/Copenhagen_interpretation




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can coexist as there are only two different kinds of spin. Outer

orbits allow for varying degrees of angular momentum and

therefore allow more electrons in the same orbit. This means that

if we were to find a system with isolated electrons, extending over

all of space or a definite size, then only two electrons, with

opposite spins, would be allowed the same momentum.

3.3 The differences

As has already been stated, classical and quantum mechanics are

represented by determinism and indeterminism, two polar opposites.

These differences do not just appear in the use of probabilities as

opposed to certainties but in fact appear in the very nature of QM.

3.3.1 Feynman's thought experimentsThe following is a short account of some experiments, used by

Richard Feynman in his lectures on physics26; used to develop an

understanding of the nature of probabilities in QM and the

mechanics in their most basic aspects.

Figure B. Interference experiment with bullets

27

Electrons and photons behave alike, therefore, what we find

applying to electrons will also apply to photons. Electrons behave

as waves and particles which we shall show but in order for us to

26 The Feynman lectures on physics Vol 3.

27 http://quantummechanics.ucsd.edu/ph130a/130_notes/node68.html



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do that, we must first consider the implications of them being

either particles or waves. Consider, for example, a gun which

sprays bullets in random directions along a plane. If we were to

design an experiment in which we weanted to find out where the

bullets landed, we would use a backstop which could stop these

bullets. In between the gun and the backstop we place a wall with

two holes. After firing the gun we could count the bullets that

passed through the holes in the wall and their distance from the

point on the backstop directly opposite the gun, x,using a detector

of sorts, see Figure B. This way we can measure the probabilistic

outcome of our experiment in relation to x, where probability is

measured along the first axis and the correspondingx value alongthe second axis. This way we find that the probability of the

bullets through the first hole (P1) plus the probability of the bullets

through the second hole (P2) equal the total probability

distribution of the bullets reaching the backstop (P1 2). We also

find that there is no interference in this experiment. What we have

discovered, already with this first experiment, is the use of

probabilities in order to explain and, in some respects, quantify a

sequence of measurements. Now let us consider a wave, such as

the kind one would expect in a pond. Suppose that an object in a

water trough is bobbed up and down creating a steady source of

waves in the same kind of configuration as the previous

experiment, see Figure C.



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Figure C. Interference experiment with water waves28

Here the detector measures the intensity of the wave. As we can

see in this example, the total intensity, I1 2 , is interestingly

different from the previous experiment. This is caused by

diffraction of the waves at the holes in the wall. Unlike in the first

experiment, I1 2 , is not the sum of the intensities of the individual

intensities but rather only happens when both holes are open. In

this example, the waves interfere, both constructively and

destructively, creating this pattern. If we were to try this

experiment with an 'electron gun', a device which fires electrons

towards a target in much the same way as our gun, we would find

that the result is a curve such as the one in figure C (see P 1 2 in

figure D).

28 http://dbanach.com/feyn1.htm



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Figure D. Interference experiment with electrons29

From this experiment we see that if we were to close off hole 1 we

would obtain P2 and if we close off hole 2 we would get P 1. As we

could guess, the sum of P1 and P2 should equal the same

probability curve as obtained in the bullet experiment. It does not.

However, the square of the sum of the complex numbers 1 and 2

does create such a probability curve and the squares of the

individual complex numbers also equal their respective, isolated,

probability curves. What we can conclude is that electrons, andtherefore also photons and other particles, sometimes behave as a

wave and sometimes as particles.

3.3.2 Dirac's notation

One of the things we discovered in the above, is that, the

probability that a particle will arrive at x, when let out at the

source s, can be represented quantitatively by the absolute square

of a complex number called a probability amplitude30. This is

also called the first general principle of QM. The second general

principle of QM simply states that, when a particle can reach a

given state by two possible routes, the total amplitude for the

29 http://quantummechanics.ucsd.edu/ph130a/130_notes/node68.html

30 The Feynman lectures on physics Vol 3. Chapter 3, page 2.



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process is thesum of the amplitudes for the two routes considered

separately31 which we also discovered in the above. In QM, a

common practice used to represent these principles is what is

called the Dirac notation. Dirac notation is also known as Bracket

notation as it consists of a left 'Bra', < |, giving the final condition

of a probability amplitude, and a 'Ket', giving the starting

condition of a probability amplitude. Using this Dirac notation we

are able to simplistically write up the general principles as the

following:

(3)

both holes open = through hole 1 + through hole 2 (4)

If we consider that the process with which the electron moves

through a hole is a two-part movement we can rewrite the notation

for the second general principle to include which hole the electron

has passed through:

both holes open = + (5)

Because of what we discovered for the probability curve P1 2 in

figure D, we can state the probability of the electron passing

through one of the holes and towards the screen as32:

P1 2 = |1 + 2|2 = | + |2 (6)

Though, it must be said, this only occurs if we do not observe the

electron itself. This is because of the interference of the wave

nature of the particles. If we were to observe the electron as a

particle then the interference pattern would cease and we would

31 The Feynman lectures on physics Vol 3. Chapter 3, page 3.

32 Kvantemekanik, Tom Andersen. p. 6



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not get the same type of probability curve as the one in figure D.

This is is a consequence of Heisenberg's uncertainty principle.

3.3.3 Heisenberg's uncertainty principle

These types of experiments, were one to attempt them, would also

be governed by another part of the uncertainty principle. As stated

earlier, Heisenberg formulated a principle expressing that an

observation on a system, impacts the system. This impact also

seeks to verify the existence of a system as a lack of observation

leads to an indefinite system which is therefore irrelevant33.

Imagine we wanted to observe an electron's position. We would

use a photon with a very high momentum as it gives the photon a

very short wavelength. This way, we can measure the position ofthe electron, at impact, but it's momentum will be highly uncertain

because of the momentum of the photon. This is formulated

mathematically as:

p x with =h

2(7)

Where h is again Planck's constant. This is a statement which says

that the product of the uncertainties in the two conjugate

parameters must always be greater than or equal to some small

measure of the effect of measurement34. It turns out that

Heisenberg's uncertainty principle also applies to the energy and

time of a system.

E t (8)

Heisenberg, through other work, also managed to represent

observations on a quantum system mathematically leading to





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results that were the same as the wave mechanics presented by

Erwin Schrdinger.

4.0 SCHRDINGER'S EQUATION

Schrdinger formulated an equation to describe the wave mechanics of

matter. Schrdinger used a wavefunction, , to describe how a particle's

wavefunction evolves in space and time under a specific set of

circumstances35.

4.1 The equation

Schrdinger's equation is a linear partial differential equation which is

the QM equivalent of Newton's laws. It incorporates the time and

space that an electron moves in and is actually a deterministic equationas we are able to predict its quantum state36. This is also a theory in

which the electron inhabits continual states. This, argued Bohr, is not

possible as the electron can only assume discrete energy values,

leading to discontinuity. We shall therefore solve the equation for a

one-dimensional box of sorts popularly called the, 'infinity square

well', in order to prove or disprove Bohr's notion.

4.2 The 'infinite square well'37

We must first introduce the equation as Schrdinger left it, the time-

dependent, three-dimensional, Schrdinger equation:

i t

(r , t)=2

2 m2(r , t)+V(r , t)(r , t) (9)

Where m is mass, V is the potential energy of the particle, 2 is the

Laplacean (a differential operator to take the place of the three-

dimensional coordinates), and of course is the wavefunction. For the

'infinite square well' example we will only need a simplified, one-


36 On Determinism, Sean Carroll.

37 Quantum mechanics: infinite square well (part I)



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dimensional and time-independent version of the Schrdinger

equation:

2

2 m

2

x2=V (10)

Before we begin, we will create a sketch of what we intend to find out,

see figure E.

In the sketch we have shown our two

limits, 0 and a. We have also shown that

the potential, V, has no upper limit at the

limits, therefore the particle cannot

penetrate the walls. The particle is,

however, free to move inside the well.

That is to say, the sum of the probabilities of finding the particle inside

the well is 1, definite, and outside the box 0.

Now, what we can choose to do is, substitute parts of (10) so that we

get a simple differential equation.

2

x2 =k2

when k

2

=

2mV

2 (11)

We can now solve this equation and get the following function for :

(x )=Acos kx+B sin kx (12)

WhereA and B are constants. We can now look back at figure E and

see that:

(0)=0 and (a )=0 (13)

If we try to use some of this information in (12) we find that:



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(0)=A cos0+Bsin 0 (14)

(0)=A(1)+B(0) (15)

A=0 (16)

We can now use (a )=0 and A=0 in (12):

(a )=B sin ka=0 (17)

Which means that eitherB = 0 or ka = n, where n is any whole

integer, as we know from the unit circle that any whole integer

multiplied by is equal to 0 when coupled with sinus. If we were to

take B = 0, then the differential equation would always be 0 which

means that the wavefunction would always be zero. As this would not

satisfy our condition that the probability inside the well should be

equal to 1, we will need to consider the alternative our only viable

option. We can now find the energy values of the system using the

formula which can be used to equate the different energy values for the

different values ofn:

As k2=

2 mV

2 then V=

2

k2

2 m

(18)

Which means that V=

2n

2

2

2 ma2 because we found that k=

n

a

(19)

We will not continue further as we have found what we need in order

to complete our picture of QM. However, it is possible to find a value

forB so that it is possible to find the complete wavefunction and

thereafter its probability.

4.2.1 The quantization of energy (n = 3, 2, 1, 0)

We can now attempt to use our findings to calculate actual values

for the potential energy, V, in an electron. We will only attempt it



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for the first four values of n, taken in a reverse order. First we

need to define our variables for an electron; it has a mass (m)

9.10953 10-31 kg 38 and we will set our limit to 3 10-10 meters for

a.

Ifn = 3 then:

V=

2n

2

2

2 ma2V(3)=

(1.054591034)2 32 2

2(9.109531031)(31010)2(20)

V(3)=6.024781018Joule (21)

Ifn = 2 then:

V=2n

2

2

2ma2V(2)=

(1.054591034)2222

2(9.109531031)(31010)2(22)

V(2)=2.677681018Joule (23)

Ifn = 1 then:

V=

2n

2

2

2 ma2 V(1)=

(1.054591034)212 2

2(9.109531031)(31010)2 (24)

V(1)=6.69421019Joule (25)

Ifn = 0 then:

V=2n

2

2

2ma2V(0)=

(1.054591034)2022

2 (9.109531031)(31010)2(26)

V(0)=0Joule (27)

What we have discovered here is that the lowest possible value for

n must be 1 as 0 cannot be divided. However, as we have said that

the potential at the limits is infinite we can also assume that the

38 Elementr Kvantefysik.



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possibility of the electron having infinite potential is possible.

This is also the explanation as to why the problem is called the

'infinite square well': the electron can never actually leave the

nucleus, as if it were in an infinite well. What we discovered

earlier is that n must be a whole integer which means that Bohr

was right, the electron can only assume discrete energy values,

leading to discontinuity. If we now choose to plot our values into

our 'infinite square well' we shall see the effect of our

calculations:

Figure F shows the approximate positions of the discrete energy

levels that an electron can be at for the different integers of n. Of

course, these values are purely theoretical as we can never

actually measure the energy levels of electrons with such accuracy

given Heisenberg's uncertainty principle. It has, however, helped

proved Bohr's theory of quantization.



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5.0 CONCLUSION

The invention of the modern approach to science, through rigorous

mathematics and experiments, led to classical mechanics which answered

many questions and gave a relatively enormous understanding of the

universe. Through the impertinence of scientists even more was achieved.

The discovery of electro-magnetism paved the road to the discovery of the

inner workings of the atom. This, however, led to more questions being

raised then answered. Yet again, there was a need for understanding. Only

through the hard work of theorists and experimentalists was this finally

achieved. Under way, the foundation of mechanics itself was challenged. No

longer was determinism the ultimate solution. New methods had to be

created in order for understanding to be restored. De Broglie set, in thisregard, a continuing trend in motion. From his ideas came a better

understanding which enabled Heisenberg to formulate his uncertainties,

Schrdinger to create his equation and Dirac to note the system. The quest

was not complete. Not even after the realization that a system was

quantized; as Bohr had predicted and elaborated. Nonetheless, what can be

concluded from our work is that Bohr was right. Electrons orbit the nucleus

in integral numbers of de Broglie's wavelength, or in other words, in

quantized energy states. This proves that the orbits of electrons are not

continuous. We also found that electrons cannot have zero potential but can,

on the other hand, have unlimited potential, never really being let go of their

orbit around the nucleus. With these realizations it is no surprise that as

Bohr defined the orbits as quantized states, the new mechanics came to be

known as Quantum Mechanics.



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6.0 RESUM

Classical mechanics had maintained a deterministic approach to

understanding the universe. Everything could be figured out, everything was

specifically designed, everything had a destiny. However, this deterministic

utopia of classical mechanics led to the evolution of Bohr's atomic model,

further expanding to the discovery of Quantum mechanics. Here was a new

way of understanding the universe where everything was indefinite and

probabilistic. The universe could no longer be described by classical

methods so new methods had to be invented, enter Heisenberg, Schrdinger

and Dirac. Through their work, they managed to give us a blueprint of the

workings of some of the smallest building blocks of the universe, the

electrons. These electrons, Bohr stated, operated in quantized states, as wehave shown in our project, which was so accurate that it even lent

inspiration to the name, 'Quantum Mechanics'.



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7.0 APPENDIX

I.

From: Faye, Jan, "Copenhagen Interpretation of Quantum

Mechanics", The Stanford Encyclopedia of Philosophy (Fall 2008

Edition), Edward N. Zalta (ed.),

http://plato.stanford.edu/archives/fall2008/entries/qm-copenhagen/

Bohr's more mature view () on complementarity and the

interpretation of quantum mechanics may be summarized in the

following points:

1. The interpretation of a physical theory has to rely on an

experimental practice.

2. The experimental practice presupposes a certain pre-scientific

practice of description, which establishes the norm for

experimental measurement apparatus, and consequently what

counts as scientific experience.

3. Our pre-scientific practice of understanding our environment is

an adaptation to the sense experience of separation, orientation,

identification and reidentification over time of physical objects.4. This pre-scientific experience is grasped in terms of common

categories like thing's position and change of position, duration

and change of duration, and the relation of cause and effect,

terms and principles that are now parts of our common

language.

5. These common categories yield the preconditions for objective

knowledge, and any description of nature has to use these

concepts to be objective.

6. The concepts of classical physics are merely exact

specifications of the above categories.

7. The classical conceptsand not classical physics itselfare

therefore necessary in any description of physical experience in



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order to understand what we are doing and to be able to

communicate our results to others, in particular in the

description of quantum phenomena as they present themselves

in experiments;

8. Planck's empirical discovery of the quantization of action

requires a revision of the foundation for the use of classical

concepts, because they are not all applicable at the same time.

Their use is well defined only if they apply to experimental

interactions in which the quantization of action can be regarded

as negligible.

9. In experimental cases where the quantization of action plays a

significant role, the application of a classical concept does notrefer to independent properties of the object; rather the

ascription of either kinematic or dynamic properties to the

object as it exists independently of a specific experimental

interaction is ill-defined.

10. The quantization of action demands a limitation of the use of

classical concepts so that these concepts apply only to a

phenomenon, which Bohr understood as the macroscopic

manifestation of a measurement on the object, i.e. the

uncontrollable interaction between the object and the

apparatus.

11. The quantum mechanical description of the object differs from

the classical description of the measuring apparatus, and this

requires that the object and the measuring device should be

separated in the description, but the line of separation is not the

one between macroscopic instruments and microscopic objects.

It has been argued in detail (Howard 1994) that Bohr pointed

out that parts of the measuring device may sometimes be

treated as parts of the object in the quantum mechanical

description.



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12. The quantum mechanical formalism does not provide

physicists with a pictorial representation: the -function does

not, as Schrdinger had hoped, represent a new kind of reality.

Instead, as Born suggested, the square of the absolute value of

the -function expresses a probability amplitude for the

outcome of a measurement. Due to the fact that the wave

equation involves an imaginary quantity this equation can have

only a symbolic character, but the formalism may be used to

predict the outcome of a measurement that establishes the

conditions under which concepts like position, momentum,

time and energy apply to the phenomena.

13. The ascription of these classical concepts to the phenomena ofmeasurements rely on the experimental context of the

phenomena, so that the entire setup provides us with the

defining conditions for the application of kinematic and

dynamic concepts in the domain of quantum physics.

14. Such phenomena are complementary in the sense that their

manifestations depend on mutually exclusive measurements,

but that the information gained through these various

experiments exhausts all possible objective knowledge of the

object.



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8.0 BIBLIOGRAPHY

8.1 Books

1. Hawking Stephen, and Leonard Mlodinow. The Grand Design.

New York: Random House, 2010.

2. Singh, Simon. Fermats Store Stning. Trans. Jan Teuber.

Copenhagen: Gyldendal, 1997.

3. Hermanns, William. Einstein and the Poet. N.p.: Branden

Books, 1983.

4. Stewart, Ian. Taming the Infinite: The Story of Mathematics.

London: Quercus, 2008.

5. Elvekjr, Finn, and Brge Degn Nielsen. Fysikkens Verden 3.Kbenhavn: Gjellerup & Gad, 1990.

6. Hansen, William W., and Henrik Parbo. Elementr

Kvantemekanik. Denmark: Systime, 1981.

7. Coughlan, G. D., and J. E. Dodd. The Ideas of Particle Physics.

Cambridge: Cambridge university press, 1991.

8. du Sautoy, Marcus. The Number Mysteries. United Kingdom:

Fourth Estate, 2010.

9. Laplace, Pierre Simon. A Philosophical Essay on Probabilities.

Trans. into English from the original French 6th ed. by

Truscott, F.W. and F. L. Emory. New York: Dover Publications

1951.

10. Andersen, Tom. Kvantemekanik.

11. Feynman, Richard P., Robert B. Leighton, and Matthew Sands.

The Feynman Lectures on Physics, Volume III. United States

of America: Cambridge Institute of Technology, 1966.

8.2 Videos

1. jrgoldma. Quantum Mechanics: Infinite Square Well (part I).

2. jrgoldma. Quantum Mechanics: Infinite Square Well (part 2).



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3. Binney, Professor J.J.. 001 Introduction to Quantum

Mechanics, Probability Amplitudes and Quantum States.

University of Oxford. 11 November 2009.

4. Binney, Professor J.J.. Dirac Notation and the Energy

Representation. University of Oxford. 12 November 2009.

8.3 Websites

1. Carroll, Sean. On Determinism. Discover Magazine. 5

December 2011.

2. Faye, Jan. "Copenhagen Interpretation of Quantum

Mechanics." The Stanford Encyclopedia of Philosophy (Fall

2008 Edition). Edward N. Zalta (ed.).

3. Wikipedia. Copenhagen Interpretation.

4. Branson. Feynman Lectures on Physics, Vol. III Chapter 1.

22 December 2012.

5. Banach, David. Quantum Behaviour: from Richard Feynman's

Lectures on Physics. David Banach's Course Server.

6. Cal Poly Ponoma. Bubble Chamber Picture.


foundation of quantum mechanics

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