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Electrons behave like particles in some experiments, and like waves in others. The electron's 'wave/particle duality' has no real analogy in the everyday world. The quantum theory that describes the behavior of electrons is a cornerstone in modern chemistry. Quantum theory can be used to explain why atoms are stable, why things have the color they do, why the periodic table has the structure it does, why chemical bonds form, and why different elements combine in different ratios with each other.

Light and electrons both behave quantum mechanically. To understand the experimental basis for the quantum theory, we have to begin our discussion with light.

Waves

* Waves are an oscillation that moves outward from a disturbance (ripples moving away from a pebble dropped into a pond) Properties of waves | property | definition | symbol | SI units | velocity | distance traveled per second | c | m/s | amplitude | peak height above midline | A | varies with type of wave | wavelength | peak-to-peak distance | | m | frequency | number of peaks passing by per second | | s-1 (called Hertz) | | * relationship between frequency and wavelength * distance per cycle × cycles per second = distance per second = c * examples * The speed of sound in air is 330 m/s. Humans can hear sounds with wavelengths between 17 m and 17 mm. What is the highest sound frequency that is audible? * interference * constructive interference: amplitudes add * peaks, troughs of interfering waves occur in the same positions (waves are in phase ) * destructive interference: amplitudes cancel * peaks of one wave are in same position as troughs of the other (waves are out of phase ) * diffraction | | | Waves can bend around small obstacles… | …and fan out from pinholes. | particles effuse from pinholes. | * a wave can't bend around obstacles much larger than its wavelength * what does this imply about the wavelength of sound waves? radio waves? visible light? * waves are delocalized (spread out in space) Three ways to tell a wave from a particle | wave behavior | particle behavior | waves interfere | particles collide | waves diffract | particles effuse | waves are delocalized | particles are localized |

Is light a stream of particles or a wave? * Thomas Young, 1801 * pass light through two tiny adjacent slits * if light were particles: * target would be brightest where light passing through the slits overlapped * target would darken steadily moving away from the overlap region * this was not observed! * a pattern of light and dark stripes was observed instead * Young explained the stripes as a combination of diffraction and interference * these interference fringes are a sure sign of wave behavior * White areas are peak-peak or trough-trough overlaps (constructive interference) * black areas are peak-trough overlaps (destructive interference).

* width of interference bands suggested the wavelength of visible light was less than a millionth of a meter! * If light is a wave, what is oscillating? * light can move through a vacuum, so it isn't an oscillation of atoms as sound and water waves are.

Force Fields * force field: a region where forces act on an object; strength of forces vary with position * gravitational fields * larger mass at center of field = stronger forces * larger distance from center of field = weaker forces * electric fields * opposite charges attract each other, but like charges repel each other * larger charge at center of field = stronger forces * larger distance from center of field = weaker forces * magnetic fields * can be produced by moving charges (electromagnets) * a moving magnetic field can produce an electric field (electric generator)

Electromagnetic radiation | | * James Clerk Maxwell ca. 1855 * changes in electric and magnetic fields are always coupled: electromagnetism * making e/m waves with a vibrating charge * both electric and magnetic fields oscillate * oscillations are at right angles * electric oscillation produces magnetic oscillation, which produces another electric oscillation, …and on and on * vibrating charge creates a ripple in the electromagnetic field * The speed of electromagnetic radiation was computed to be around 3×108 m/s * The same speed had been determined experimentally for light! * hypothesis: light is a form of electromagnetic radiation (Maxwell, 1862)

The electromagnetic spectrum

Major regions of the electromagnetic spectrum. The rainbow band represents the visible region. Know the names of the regions and their order with increasing wavelength. | region of spectrum | wavelengths | typical source | Radio | more than 30 cm | radio, television | microwave | 3 mm to 30 cm | radar, microwave oven | infrared | 750 nm to 3 mm | hot objects | visible | 400 nm to 750 nm | very hot objects | ultraviolet | 20-400 nm | sun; black lights | x-rays | 3 pm to 20 nm | cathode ray tubes | gamma rays | less than 3 pm | |

Energy of electromagnetic radiation * radiation carries energy through space * work is done on charges in the e/m field * transmitter loses energy; reciever gains energy * for classical waves:

* higher amplitude means higher energy per peak * amplitude squared determines the intensity or brightness of light * therefore, brighter light should carry more energy per peak than dimmer light * an experiment to measure the energy carried by an electromagnetic wave * photoelectric effect: shining light on alkali metals knocks electrons out of metal * strategy: measure kinetic energy of ejected electrons; then measure light energy per ejected electron. * surprise: * brightness has NO EFFECT on the kinetic energy per ejected electron * brighter light ejects MORE electrons. * surprise #2: * red light can't eject any electrons, but blue light can! * below a threshold frequency (0), there are no ejected electrons! * 0 is a property of the metal being used * Albert Einstein's interpretion of the photoelectric effect (Nobel Prize, 1921) * maybe light is like a stream of massless particles (call them photons) * brighter light has more photons, but bluer light has higher energy photons * frequency-to-energy conversion factor is h (Planck's constant, 6.626×10-34 J/Hz) energy per photon | = | energy to pull one electron out of metal | + | kinetic energy per ejected electron | Ephoton | = | h0 | + | h(-0) | Ephoton | = | h | | | * examples * What is the energy of a photon of red light with wavelength 700 nm? * What is the wavelength of a photon which has an energy of 1×10-18J? * Shining light of 400 nm on a metal causes electrons with a kinetic energy of 5×1019 J to be ejected. What is the minimum energy required to eject an electron from the metal? summary: light moves like a wave, but transfers energy like a stream of particles; the particles (photons) have energy equal to h.

The collapsing atom paradox * what's the electron doing in an atom? * electrons within the atom can't be stationary: * positively charged nucleus will attract the negatively charged electron * electron will accelerate towards the nucleus * if electrons within the atom move, * moving charges emit electromagnetic radiation * emission will cause electrons to lose energy and spiral into the nucleus * the atom will collapse! * why don't atoms collapse? * classical physics has no answer! * key: electrons have wave/particle duality

Electrons as Waves * the de Broglie hypothesis (Nobel Prize, 1929) * connect wave and particle nature of matter using a relationship that applies to photons: = h/p where p is the momentum of the particle (p = mass times velocity). * examples

Compute the de Broglie wavelength of a tennis ball (m = 0.1 kg) and an electron (m = 9.1 x 10-31 kg), if both are moving at 1000 m/s. * experimental evidence of electron wave/particle duality * electron diffraction * C. J. Davisson and G. P. Thomson observed interference fringes when electron beams hit crystal surfaces and thin metal films (Nobel Prize, 1937) | Electron diffraction pattern collected from crystalline silicon

Semiconductor Surface Physics Group

Queens University | * electrons passing one at a time through a double slit. Each spot shows an electron impact on the detector. | 100 electrons | | 3000 electrons | | 70000 electrons

…interference fringes!!! | * applications * LEED surface analysis * electron microscopy

Bound electrons have quantized energies * model I: bead on a wire * kinetic energy of bead can have any value, because velocity can have any value * bead can be stationary * bead is equally likely to be found anywhere on the wire * exact position and velocity of the bead can be known simultaneously * model II: wave on a wire * there must be a whole number of peaks and troughs on the wire: n ( /2) = L, where: * n is an integer (1, 2, 3, ... ) * is the wavelength * L is the length of the wire * standing waves have quantized wavelengths * model III: electron on a wire * unite wave and bead models using the de Broglie relation: particle kinetic energy: | | de Broglie wave/particle relation: | | | standing wave allowed wavelengths: | | | * The fact that E depends on an integer n means that only certain energy states are allowed * the integer n labels each state; n is a quantum number * Notice that n can't be zero, so the lowest energy state is not zero: the electron is never at rest! * peaks and troughs correspond to buildup of negative charge- electron is not equally likely to be found anywhere on the wire.

Summary

* electrons behave like waves * bound waves have restricted wavelengths * therefore, electrons bound in atoms and molecules have restricted energies

The uncertainty principle * Quantum theory puts a limit on the precision of measurements * Werner Heisenberg's uncertainty principle (Nobel Prize, 1932) * You can never know both the exact position and the exact momentum of a particle. * Mathematical statement: x(mv) h/2 where x | is the error in a measurement of the particle's position, x | m | is the particle's mass, | v | is its velocity, | (mv) | is the error in a measurement of the particle's momentum, mv | h | is Planck's constant (6.626 × 10-34 Js). | * The uncertainty principle says that the act of measurement changes what you're trying to measure * You can bounce a photon off a brick wall, and the wall doesn't change much... * ...but bouncing a photon off an electron to measure its position will change its momentum * Why atoms don't collapse * The uncertainty in position is also the smallest space a particle can possibly be restricted to * The position of an electron can't possible be known to better than ± 4 × 10-13 m by the uncertainty principle * … so an electron can't be confined within the nucleus http://antoine.frostburg.edu/chem/senese/101/quantum/ -------------------------------------------------

Quantum Numbers

A total of four quantum numbers were developed to better understand the movement and pathway of electrons in its designated orbital within an atom. Each quantum number indicates an electron's trait within an atom, which satisfies to explain the movement of electrons as a wave function, described by the Schrodinger equation. Each electron in an atom has a unique set of quantum numbers; no two electrons can share the same combination of four quantum numbers. Quantum numbers are very significant because they can determine the electron configuration of an atom and a probable location of the atom's electrons. They can also aid in graphing orbitals. Quantum numbers can help determine other characteristics of atoms, such as ionization energy and the atomic radius. 1. 1. 1.1. The Four Quantum Numbers 2. 1.2. Principal Quantum Number 3. 1.3. Orbital Angular Momentum Quantum Number 4. 1.4. Magnetic Quantum Number 5. 1.5. Electron Spin Quantum Number 2. 2. A Closer Look at Shells, Subshells, and Orbitals 6. 2.1. Principal Shells 7. 2.2. Subshells 3. 3. Restrictions 8. 3.1. Pauli Exclusion Principle 9. 3.2. Hund's Rule 10. 3.3. Heisenberg Uncertainty Principle 11. 3.4. References 12. 3.5. Outside Links 13. 3.6. Problems 14. 3.7. Contributors

The Four Quantum Numbers

Quantum numbers designate specific levels, subshells, orbitals, and spins of electrons. This means that they are describing in detail the characteristics of the electrons in the atoms. They describe each unique solution to the Schrödinger Equation, or the wave function, of electrons in an atom. There are a total of four quantum numbers: the principal quantum number (n), the orbital angular momentum quantum number (l), the magnetic quantum number (ml), and the electron spin quantum number (ms). The principal quantum number, n, describes the energy of an electron and the most probable distance of the electron from the nucleus. In other words, it is referring to the size of the orbital and the energy level an electron is placed in. The number of subshells, or l, describes the shape of the orbital. You can also use l to find the number of angular nodes. The magnetic quantum number, ml, describes the amount of energy levels in a subshell. ms is referring to the spin on the electron, which can either be up or down.

Principal Quantum Number

The principal quantum number, n, designates the principal electron shell. Because n describes the most probable distance of the electrons from the nucleus, the larger the number n is, the farther the electrons are from the nucleus, the larger the size of the orbital, and the larger the atom is. n can be any positive integer starting at 1, as n=1 designates the first principal shell (the innermost shell). The first principal shell is also called the ground state, or lowest energy state. This explains why n can not be 0 or any negative integer, because there exists no atoms with zero or a negative amount of energy levels/principal shells. When an electron is in an excited state or it gains energy, it may jump to the second principle shell, where n=2. This is called absorption because the electron is "absorbing" photons, or energy. Known as emission, electrons can also "emit" energy as they jump to lower principle shells, where n decreases by whole numbers. As energy of the electron increases, so does the principal quantum number. n=3 designates the third principal shell, n=4 designates the fourth principal shell, and so on. n=1,2,3,4… Example: Given that n=7, what principal electron shell does this refer to?

Answer: We are referring to the 7th principal electron shell.

Example: If an electron shifted from energy level n=5 to energy level n=3, did absorption or emission of a photon occur?

Answer: Emission because energy was lost, or a photon was "emitted."

Orbital Angular Momentum Quantum Number

The orbital angular momentum quantum number l determines the shape of an orbital, and therefore the angular distribution. The number of angular nodes is equal to the value of the angular momentum quantum number l. (For more information about angular nodes, see Electronic Orbitals.) Each value of l indicates a specific s, p, d, f subshell (each unique in shape.) The value of l is dependent on the principal quantum number n. Unlike n, the value of l can be zero. It can also be a positive integer, but it cannot be larger than one less than the principal quantum number (n-1): l=0, 1, 2, 3, 4…, (n-1) Example | |

Example: If n=7, what are the possible values of l?

Answer: Since l can be zero or a positive integer less than (n-1), it can have a value of 0, 1, 2, 3, 4, 5 or 6. Example | |

Example: If l=4, how many angular nodes does the atom have?

Answer: The number of angular nodes is equal to the value of l, so the number of nodes is also 4.

Magnetic Quantum Number

The magnetic quantum number ml determines the number of orbitals and their orientation within a subshell. Consequently, its value depends on the orbital angular momentum quantum number l. Given a certain l, ml is an interval ranging from –l to +l, so it can be zero, a negative integer, or a positive integer.

ml= -l, (-l +1),( -l +2),…, -2, -1, 0, 1, 2, … (l – 1), (l – 2), +l

Example: If n=3, and l=2, then what are the possible values of ml ?

Answer: Since ml must range from –l to +l, then ml can be: -2, -1, 0, 1, or 2.

Electron Spin Quantum Number

Unlike n, l, and ml, the electron spin quantum number ms does not depend on another quantum number. It designates the direction of the electron spin and may have a spin of +1/2, represented by↑, or –1/2, represented by ↓. This means that when ms is positive the electron has an upward spin, which can be referred to as "spin up." When it is negative, the electron has a downward spin, so it is "spin down." The significance of the electron spin quantum number is its determination of an atom's ability to generate a magnetic field or not. (For more information, see Electron Spin.) ms= +1/2 or -1/2

Example: List the possible combinations of all four quantum numbers when n=2, l=1, and ml=0. ms.

Answer: The fourth quantum number is independent of the first three, allowing the first three quantum numbers of two electrons to be the same. Since the spin can be +1/2 or =1/2, there are two combinations: n=2, l=1, ml =0, ms=+1/2 and n=2, l=1, ml=0 ms=-1/2

Example: Can an electron with ms=1/2 have a downward spin?

Answer: No, if the value of ms is positive, the electron is "spin up."

A Closer Look at Shells, Subshells, and Orbitals

Principal Shells

The value of the principal quantum number n is the level of the principal electronic shell (principal level). All orbitals that have the same n value are in the same principal level. For example, all orbitals on the second principal level have a principal quantum number of n=2. When the value of n is higher, the number of principal electronic shells is greater. This causes a greater distance between the farthest electron and the nucleus. As a result, the size of the atom and its atomic radius increases.

Because the atomic radius increases, the electrons are farther from the nucleus. Thus it is easier for the atom to expel an electron because the nucleus does not have as strong a pull on it, and the ionization energy decreases.

Example: Which orbital has a higher ionization energy, one with n=3 or n=2?

Answer: The orbital with n=2, because the closer the electron is to the nucleus or the smaller the atomic radius, the more energy it takes to expel an electron.

Subshells

The number of values of the orbital angular number l can also be used to identify the number of subshells in a principal electron shell: * When n = 1, l= 0 (l takes on one value and thus there can only be one subshell) * When n = 2, l= 0, 1 (l takes on two values and thus there are two possible subshells) * When n = 3, l= 0, 1, 2 (l takes on three values and thus there are three possible subshells)

After looking at the examples above, we see that the value of n is equal to the number of subshells in a principal electronic shell: * Principal shell with n = 1 has one subshell * Principal shell with n = 2 has two subshells * Principal shell with n = 3 has three subshells

To identify what type of possible subshells n has, these subshells have been assigned letter names. The value of l determines the name of the subshell: Name of Subshell | Value of l | s subshell | 0 | p subshell | 1 | d subshell | 2 | f subshell | 3 |

Therefore:

* Principal shell with n = 1 has one s subshell (l = 0) * Principal shell with n = 2 has one s subshell and one p subshell (l = 0, 1) * Principal shell with n = 3 has one s subshell, one p subshell, and one d subshell (l = 0, 1, 2)

We can designate a principal quantum number, n, and a certain subshell by combining the value of n and the name of the subshell (which can be found using l). For example, 3p refers to the third principal quantum number (n=3) and the p subshell (l=1).

Example: What is the name of the orbital with quantum numbers n=4 and l=1?

Answer: Knowing that the principal quantum number n is 4 and using the table above, we can conclude that it is 4p.

Example: What is the name of the oribital(s) with quantum number n=3?

Answer: 3s, 3p, and 3d. Because n=3, the possible values of l = 0, 1, 2, which indicates the shapes of each subshell.

Orbitals

The number of orbitals in a subshell is equivalent to the number of values the magnetic quantum number ml takes on. A helpful equation to determine the number of orbitals in a subshell is 2l +1. This equation will not give you the value of ml, but the number of possible values that ml can take on in a particular orbital. For example, if l=1 and ml can have values -1, 0, or +1, the value of 2l+1 will be three and there will be three different orbitals. The names of the orbitals are named after the subshells they are found in: | s orbitals | p orbitals | d orbitals | f orbitals | L | 0 | 1 | 2 | 3 | Ml | 0 | -1, 0, +1 | -2, -1, 0, +1, +2 | -3, -2, -1, 0, +1, +2, +3 | Number of orbitals in designated subshell | 1 | 3 | 5 | 7 | In the figure below, we see examples of two orbitals: the p orbital (blue) and the s orbital (red). The red s orbital is a 1s orbital. To picture a 2s orbital, imagine a layer similar to a cross section of a jawbreaker around the circle. The layers are depicting the atoms angular nodes. To picture a 3s orbital, imagine another layer around the circle, and so on and so on. The p orbital is similar to the shape of a dumbbell, with its orientation within a subshell depending on ml . The shape and orientation of an orbital depends on l and ml.

To visualize and organize the first three quantum numbers, we can think of them as constituents of a house. In the following image, the roof represents the principal quantum number n, each level represents a subshell l, and each room represents the different orbitals ml in each subshell. The s orbital, because the value of ml can only be 0, can only exist in one plane. The p orbital, however, has three possible values of ml and so it has three possible orientations of the orbitals, shown by Px, Py, and Pz. The pattern continues, with the d orbital containing 5 possible orbital orientations, and f has 7:

Another helpful visual in looking at the possible orbitals and subshells with a set of quantum numbers would be the electron orbital diagram. (For more electron orbital diagrams, see Electron Configurations.) The characteristics of each quantum number are depicted in different areas of this diagram.

Restrictions

Pauli Exclusion Principle

In 1926, Wolfgang Pauli discovered that a set of quantum numbers is specific to a certain electron. That is, no two electrons can have the same values for n, l, ml, and ms. Although the first three quantum numbers identify a specific orbital and may have the same values, the fourth is significant and must have opposite spins.

Hund's Rule

Orbitals may have identical energy levels when they are of the same principal shell. These orbitals are called degenerate, or "equal energy." According to Hund's Rule, electrons fill orbitals one at a time. This means that when drawing electron configurations using the model with the arrows, you must fill each shell with one electron each before starting to pair them up. Remember that the charge of an electron is negative and electrons repel each other. Electrons will try to create distance between it and other electrons by staying unpaired. This further explains why the spins of electrons in an orbital are opposite (i.e. +1/2 and -1/2).

Heisenberg Uncertainty Principle

According to the Heisenberg Uncertainty Principle, we cannot precisely measure the momentum and position of an electron at the same time. As the momentum of the electron is more and more certain, the position of the electron becomes less certain, and vice versa. This helps explain integral quantum numbers and why n=2.5 cannot exist as a principal quantum number. There must be an integral number of wavelengths (n) in order for an electron to maintain a standing wave. If there were to be partial waves, the whole and partial waves would cancel each other out and the particle would not move. If the particle was at rest, then its position and momentum would be certain. Because this is not so, n must have an integral value. It is not that the principal quantum number can only be measured in integral numbers, it is because the crest of one wave will overlap with the trough of another, and the wave will cancel out.

References

1. Chang, Raymond. Physical Chemistry for the Biosciences. 2005, University Science Books. pp 427-428. 2. Petrucci, Ralph. General Chemistry. 2006, Prentice Hall. 3. Gillespie, Ronald. Demistifying Introductory Chemistry.The Forum: A Contribution from the Task Force on General Chemistry. 1996: 73;617-622. 4. Petrucci, Ralph. General Chemistry: Principles and Modern Applications, Tenth Edition.

Outside Links * A great overview of the topic at Purdue University chemistry page * Wikipedia entry for quantum numbers * Wikipedia entries for individual quantum numbers in more detail: * Principal quantum number n * Angular momentum quantum number l * Magnetic quantum number ml * Spin quantum number ms

Problems

1. Suppose that all you know about a certain electron is that its principal quantum number is 3. What are the possible values for the other four quantum numbers?

2. Is it possible to have an electron with these quantum numbers: n=2, l=1, ml=3, ms=1/2? Why or why not?

3. Is it possible to have two electrons with the same n, l, and ml?

4. How many subshells are in principal quantum level n=3?

5. What type of orbital is designated by quantum numbers n=4, l=3, and ml =0?

Solutions:

1. When n=3, l=0, ml = 0, and ms=+1/2 or -1/2 l=1 ml = -1, 0, or +1, and ms=+1/2 or -1/2 l=2 ml = -2, -1, 0, 1, or +2, and ms=+1/2 or -1/2

2. No it is not possible. ml=3 is not in the range of -l to +l. The value should be be either -1, 0, or +1.

3. Yes it is possible to have two electrons with the same n, l, and ml. The spin of one electron must be +1/2 while the spin of the other electron must be -1/2.

4. There are three subshells in principal quantum level n=3.

5. Since l=3 refers to the f subshell, the type of orbital represented is 4f (combination of the principal quantum number n and the name of the subshell).

http://chemwiki.ucdavis.edu/Physical_Chemistry/Quantum_Mechanics/Quantum_Theory/Trapped_Particles/Quantum_Numbers…...

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...Berners-Lee with his introduction of the World Wide Web in the early 1990’s, and constant developments in computer science and computational power made possible through advances in microprocessing capabilities which seem to be paralleling closely with forecasts given by Moore’s Law (Greenemeir, 2009). Moore’s Law, which is not exactly a law, is an estimation regarding the progression deemed possible in an integrated circuit. It was postulated by one of the founders of Intel, Dr. Gordon Moore, who stated that the number of transistors placed on a circuit will double roughly every 18 months (Colwell, 2013), and the capacity to follow this trend will end at a critical point when transistors become so close together that the manifestations of quantum mechanics will begin to transpire (Tally, 2012). Classical computers have undoubtedly reshaped our world and have provided incalculable progress unforeseen through the eyes of some of history’s greatest academics. Early computers were mammoth-sized mechanical devices utilized by the military for processing information that would take humans hours by manual calculation alone (Watson, 2012). Computers, in the days prior to digital computers, were actual people who would literally compute information for whatever their task or job may have been. Though these positions were often filled with clever mathematicians, their capacity to compute problems with increasing complexity became difficult as the number of variables in any given......

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...Quantum Computer August 13, 2014 Dr. Laura Wilson Quantum Computer As we progress in our technological world where everyone is interested in the next iPhone or Samsung Galaxy, quantum computers are still moving forward. It seems that only the smartest "nerds" seem to care and understand this wonder. What if all of the theories, concepts, and everything else that makes up what quantum computers are and will be, a way to have it so that everyone can understand. The way that quantum computers can be divided is into three main areas: quantum physics, quantum bits (qubits), and their future goals. To better understand how quantum computers work, you need to start with what clearly defines a quantum computer: "A quantum computer is a computer design which uses the principles of quantum physics to increase the computational power beyond what is attainable by a traditional computer" (Jones, 2014, p. 1). Quantum computers use two fundamental principles of quantum physics: superposition and entanglement. Quantum superposition is where the state of a physical system exists in all possible states at the same time. Then when the physical system is only one state is give to the collection device. A good example of this is the famous Schrödinger's Cat, a thought experiment purposed by Erwin Schrödinger in 1935, where Schrödinger talks about having a cat in a closed box with a vial of poisonous acid with a hammer that will break the vial if any radioactivity is detected (Schrödinger...

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...ELEC 9705 Quantum Devices Session 1, 2014 Assignment 1 Student ID:3402048 Name:Weitao Tang Part 1 – Controlling the spins of a hydrogen atom 1.1 4-dimentional Hilbert space: plus_plus = plus_minus = 0 1 0 0 minus_plus = 1 0 0 0 0 0 1 0 minus_minus = 0 0 0 1 Spin operator of electron: Ix = 0 0.5000 0 0 0.5000 0 0 0 0 0 0 0.5000 0 0 0.5000 0 Iy = 0.0000 + 0.0000i 0.0000 + 0.5000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 - 0.5000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.5000i Iz = 0.5000 0 0 -0.5000 0 0 0 0 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 - 0.5000i 0.0000 + 0.0000i 0 0 0 0 Spin operators of proton: Sx = 0 0 0.5000 0 0 -0.5000 0.5000 0 0 0 0 0 0 0.5000 0.5000 0 0 0.5000 0 0 Sy = 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.5000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.5000i 0.0000 - 0.5000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 - 0.5000i 0.0000 + 0.0000i 0.0000 + 0.0000i Sz = 0.5000 0 0 0 0 0.5000 0 0 0 0 -0.5000 0 0 0 0 -0.5000 1.2 The values of eigenvectors: phi_1 = phi_2 = 0 1 0.7071 0 -0.7071 0 0 0 phi_3 = phi_4 = 0 0.7071 0.7071 0 0 0 0 1 The values of eigenvalues: E_1 = E_2......

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...QUANTUM PHYSICS, SYNCHRONICITY, AND THE TAROT Just about anyone who has ever done a tarot reading and marveled at its accuracy has wondered, "How does it work?" The answer to this question depends greatly on our personal beliefs about spirituality, philosophy, and the way the world works, and there are many possible theories. On one end of the spectrum, many tarotists believe their readings are guided by a conscious force, whether that be God/dess, an angel or spiritual guide, or one's higher self. At the other end of the spectrum are tarotists of the psychological school, who believe that tarot cards can only reflect projections of what is in the mind of the client (and the reader), and who use a more interactive style of reading similar to counseling. Then there are the in between possibilities, which form the subject of this article – those who may not believe that a conscious force is guiding each reading, but who nevertheless believe there is more going on than simple psychological projection. I confess to falling in this middle ground. Since most of my readings are conducted via email, without the client even seeing the cards until after the reading is completed, there is little possibility that her projections are influencing my reading, nor that I am reading subtle bodylanguage clues that help me interpret the cards. Yet, I receive so much positive feedback for my readings, even for those that are very detailed and specific, that I cannot believe...

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...QUANTUM ELECTRODYNAMICS What is quantum electrodynamics? Sociology Anna C Jean – Guillaume Martin LCSW March 7, 2015 Professor Dr. Beverly Jackson Quantum electrodynamics (QED): Is the e area of physics devoted to the study of the interaction of matter (quantum particles) and light. It generally concerns itself with the wave (or field) aspects of nature. At the level of the quantum realm, QED demonstrates that everything is connected. The world is a vast web of relationships, with everything affecting everything else. The brain, blood, and bone of the body give way to invisible forces, fields, and particles whose interactions underlie not only the human body but all of matter. Molecules give way to atoms that dissolve into subatomic particles, so that our bodies are governed not only by the laws of everyday chemistry but also by the paradoxical principles of the physical body. History shows that most truly fundamental leaps forward in scientific understanding are shunned at first. Truly new insights are sometimes too radical an overturning of accepted theory or too threatening to business or academic interests to be evaluated impartially. You have only to think of germ theory, tectonic plate theory, quantum electrodynamics, and string theory to know that even ideas that are accepted widely in our day were dismissed as crackpot ideas by a previous generation of scientists. Over the years, as physicists’ explored wave-particle duality, the study of quantum......

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...Seminar Synopsis Topic Introduction to Quantum Computer Abstract Overview: A classical computer has a memory made up of bits, where each bit is represented by either a one or a zero. A quantum computer maintains a sequence of qubits. What is qubits A single qubit can represent a one, a zero, or any quantum superposition of those two qubit states; a pair of qubits can be in any quantum superposition of 4 states, and three qubits in any superposition of 8 states. In general, a quantum computer with n qubits can be in an arbitrary superposition of up to 2^n different states simultaneously (this compares to a normal computer that can only be in one of these 2^n states at any one time). A quantum computer operates by setting the qubits in a controlled initial state that represents the problem at hand and by manipulating those qubits with a fixed sequence of quantum logic gates. The sequence of gates to be applied is called a quantum algorithm. The calculation ends with a measurement, collapsing the system of qubits into one of the 2^n pure states, where each qubit is zero or one, decomposing into a classical state. The outcome can therefore be at most n classical bits of information. Quantum algorithms are often non-deterministic, in that they provide the correct solution only with a certain known probability. Quantum computing studies theoretical computation systems (quantum computers) that make direct use of quantum-mechanical phenomena, such as superposition and......

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...The word "quantum" comes from the Latin "quantus", meaning "how much". "Quanta", short for "quanta of electricity" (electrons) was used in a 1902 article on the photoelectric effect by Philipp Lenard, who credited Hermann von Helmholtz for using the word in the area of electricity. However, the word quantum in general was well known before 1900.[2] It was often used by physicians, such as in the term quantum satis. Both Helmholtz and Julius von Mayer were physicians as well as physicists. Helmholtz used "quantum" with reference to heat in his article[3] on Mayer's work, and indeed, the word "quantum" can be found in the formulation of the first law of thermodynamics by Mayer in his letter[4] dated July 24, 1841. Max Planck used "quanta" to mean "quanta of matter and electricity",[5] gas, and heat.[6] In 1905, in response to Planck's work and the experimental work of Lenard (who explained his results by using the term "quanta of electricity"), Albert Einstein suggested that radiation existed in spatially localized packets which he called "quanta of light" ("Lichtquanta").[7] The concept of quantization of radiation was discovered in 1900 by Max Planck, who had been trying to understand the emission of radiation from heated objects, known as black-body radiation. By assuming that energy can only be absorbed or released in tiny, differential, discrete packets he called "bundles" or "energy elements",[8] Planck accounted for the fact that certain objects change colour when......

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...Quantum mechanics Rohan Agarwal It is common knowledge that matter is comprised of particles. Or is it? We all think that light consists of waves. How true are these claims? Although it may seem counterintuitive on the surface, various experiments and supporting theories have proved that the line between what is particulate and what is wavelike is finer than we imagine. At a fundamental, sub-atomic level, matter and light are both wavelike and particulate; it is only on the macroscopic scale that the divisions become better defined. The study of the physical laws that govern the realm of the very small, such as atoms, protons, and the like; developed from Planck’s quantum principle and Heisenberg’s uncertainty principle are termed as quantum mechanics. The principles of quantum mechanics revolve around two major theories- the wave-particle duality, and, in extension to this, the theory that it is possible for absolutely small particles to be in two different states at the same time. Theoretical physicists have arrived at these conclusions after powerful observations of experiments, and these will be discussed in due course. Before doing so, it is imperative to understand that this quantum strangeness does not transfer to the macroscopic world, and so some claims in this article will seem absurd. In order to demonstrate this, Erwin Schrodinger published a paper in 1935, containing a thought experiment, now popularly known as Schrodinger’s cat. A cat is put in a steel......

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...Quantum mechanics (QM – also known as quantum physics, or quantum theory) is a branch of physics dealing with physical phenomena where the action is on the order of the Planck constant. Quantum mechanics departs from classical mechanics primarily at the quantum realm of atomic and subatomic length scales. QM provides a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. In advanced topics of quantum mechanics, some of these behaviors are macroscopic and only emerge at extreme (i.e., very low or very high) energies or temperatures. The name quantum mechanics derives from the observation that some physical quantities can change only in discrete amounts (Latin quanta), and not in a continuous (cf. analog) way. For example, the angular momentum of an electron bound to an atom or molecule is quantized.[1] In the context of quantum mechanics, the wave–particle duality of energy and matter and the uncertainty principle provide a unified view of the behavior of photons, electrons, and other atomic-scale objects. The mathematical formulations of quantum mechanics are abstract. A mathematical function called the wavefunction provides information about the probability amplitude of position, momentum, and other physical properties of a particle. Mathematical manipulations of the wavefunction usually involve the bra-ket notation, which requires an understanding of complex numbers and linear functionals. The......

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