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  • mbenkerumass 6:00 am on March 4, 2020 Permalink | Reply
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    de Broglie’s “Matter Waves” and Full Schroedinger’s Equation 

    Photons were originally assumed to be waves and even after they were found to be particles, photons still exhibited the same qualities that allowed them to be considered as waves before. de Broglie considered that, if a photon could be considered both a particle and a wave, then perhaps other forms of matter could be treated as waves.

    debro

    de Broglie’s finding later became a pillar of quantum mechanics. The Schroedinger equation becomes an equation for such ‘matter waves’ as proposed by de Broglie. The plane wave as mentioned above later becomes the ‘wavefunction’ ψ(x,t)  as is fundamental to Schroedinger’s equaiton. Is the wavefunction ψ(x,t) measurable? What is the meaning of the wavefunction ψ(x,t)? To come to this understanding, we should unpack some of de Broglie’s matter wave formulas.

    Consider that the wavelength is inversely proportional to the momentum in de Broglie’s formula. Further, the momentum is found to be equal to h_bar multiplied by the wavenumber.

    debrog

    de Broglie’s wave formula presented further complications however. It was found that certain aspects of the wave would in fact be correct, but other parts of the wave would not be accurate. For instance, the phase of de Broglie’s matter wave would be accurate. The phase of the wavelength is understood to adhere to Galilean, non-relativistic physics, meaning that the result is not altered according to perspective. An example of a system would be relitivistic is the case for instance where the speed of a vehicle is determined relative to another moving vehicle. The phase of de Broglie’s calculation is therefore is a type of ‘objective’ calculation.

    debro2

    Further calculations below show that the waves proposed by de Broglie are not directly measurable. The wavefunction  ψ(x,t) is also known to be multiplied by an imaginary number and this makes it difficult to measure. Further, the system is not Galilean invariant. This means that matter waves may differ according to reference. Finally it is concluded that the wavefunction ψ is not like sound waves, water waves, mechanical or electromagnetic waves. A difference according to reference however does not mean that all hope is lost in the calculation of the wavefunction, however. By taking account for the difference in reference, two points can be compared to allow for a wavefunction that may vary according to reference.

    debro4

    debro6

     

    What does a matter wave look like?

    Consider it should look like any other wave. It should have some kind of sinusoidal representation. Considering one aspect of matter, it should be that matter is not allowed to exist in any one place. This is the case for instance, when a particular piece of matter is not present in a location. However, to restrict a matter wave by time could be problematic. This is to say that, at a certain moment, no matter is allowed to exist anywhere or for all positions.

    There are four cases listed below that, being sinusoidal, may seen to be possible respresentations for matter waves. In the case however for a sin(kx-wt) or cos(kx-wt), it is implied then that for time wt eual to pi/2, 3*pi/2, etc matter is not allowed to exist anywhere. Therefore these functions are not acceptable representations.

    For the case however of e^(ikx-iwt) + e^(-kx-iwt) and it’s counterpart e(-ikx+iwt)+e^(ikx+iwt), this is not the case. Could both be used together in superposition? The answer is that if they were added together, matter would be restricted to one direction, which is undesireable as well. Therefore, either representations would be acceptable, but not both. The boxed answer below is the normative convention taken by physicists for the matter wave wavefunction for a particle. The other representation would work as well.

    mwave

     

    Principle of Stationary Phase

    Consider the case of a narrow peak, modulated by a sinusoidal function centered about zero. If the sinusoid frequency is too high with relation to the narrowness of the peak or if the phase is rapidly changing, the averaging of the system will cause the narrow peak to disappear. If however the frequency is low – or if the phase is stationary at the narrow peak, any averaging that is done will still allow the narrow peak to exist.

    yu

    As related to quantum mechanics, while determining characteristics of a wave, the principle of stationary phase becomes important. The wave Φ(k) as a function of wavenumber will be detected as a narrow peak. In order to properly detect this narrow peak, the modulating sinusoid must have a phase that varies much less with respect to (in this case, k) the x-axis. Otherwise, the narrow signal will be lost.

    yu7

     

    Towards developing a generalized Wavefunction

    Performing a specific operation to the wavefunction interestingly produces the momentum multiplied by the wavefunction. The operator used is termed the “momentum operator.”

    wf

    As we know from linear algebra, if a matrix multiplied by a vector is equal to a number multiplied by the vector, then the vector is termed an eigenvector of the matrix.

    la1

    Given the above momentum operator relationship, it may be concluded that ψ(x,t) is an eignevector, or more specifically an eigenstate of the momentum operator p_hat. p is then also the eigenvalue.

    la2

    And finally, using the eigenstate condition, Schroedinger’s Equation in general form for a free particle is derived:

    sch3

     

    Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

     
  • mbenkerumass 6:00 am on March 3, 2020 Permalink | Reply
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    Compton Wavelength of a Particle, Compton Scattering 

    Planck’s constant h is fundamental to Quantum Mechanics. But what is Planck’s constant?

    Separating the fundamental units from h, it is found to be equal to a distance multiplied by a momentum. If then we consider using the speed of light, c and a given mass, we arive at a new parameter equalling a distance. This parameter is the Compton wavelength of a particle.

    compton

    compton2

    The Compton wavelength of light:

    compton2

    Compton wavelength of an electron:

    compton-e

     

    Compton Scattering

    Physicists had considerable difficulty accepting the existence of the photon. For one, it introduced the non-deterministic nature which collided with classical mechanics and it also went against Maxwell’s theories, which at the time were considered highly successful. The concept of Compton Scattering was one of the final pieces of support that lead to a wider acceptance of the photon as a particle.

    Compton Scattering provided an image of a photon literally colliding with an electron, showing that like the electron, the photon was worthy of being considered a particle as well. The classical version of Compton Scattering was Thompson Scattering, which considered the photon as a wave. The Thompson Scattering experiment applied a photon wave to an electron, which caused the electron to be energized and radiate a photon wave.

    thom

    Compton Scattering, treating the photon as a particle provides a different picture at higher energy levels:

    compt5

    Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

     
  • mbenkerumass 6:00 am on March 2, 2020 Permalink | Reply
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    Entanglement, Mach-Zehnder Interferometer 

    It may be that one wishes to describe a quantum state as the existence of two separate, non-interacting particles each in a particular state. Imagine particle 1 can be in either state |u1> or |u2> and particle 2 can be in either |v1> or |v2>. We wish to describe the state where particle 1 is in state |u1> and particle 2 is in state |u2>. The notation for this state is:

    |u1> ⊗ |u2>

    sumstates

    Given a probability for each state, a general state formula can be written to describe both particles:

    gensumstate

    To describe a superposition of particular states however will result in a dependency of the particles on each other. This is called an engtangled state.

    entanglestate

    The following outlines an entangled state example. This shows how the fate of two particles becomes intertwined in such an entangled state, where there will exist no other combination other than the complete state combinations made available by the definition of the state.

    stateex

    Einstein has objected to the entangled pair hypothesis. John Bell had proposed an experiment to test entanglement using a three directions, such that a correlation would be more presentable. The results of his experiments however confirmed the possibility of this sort of entanglement on the quantum level, which appears to deny classical mechanics.

     

    Mach-Zehnder Interferometer Quantum Mechanical Calculation

    Let us model the Mach-Zehnder Interferometer using photon probability state matricies. First, we will consider the operation of the beam splitter. When a photon enters the beamsplitter from one direction, there is a given probability that the photon will be present at either the transmitted or refracted position. Given that the beamsplitter is balanced, meaning that the photon has an equal change (1/2) of exiting either side, the beamsplitter is modeled below:

    bspli

    Next, using the beamsplitter matrix, the Mach-Zehnder Interferometer can be modeled. Interestingly, the photon appears to exit (100%) from the side opposite which it entered.

    mz11

    mz2

    Let’s consider a case in which mirror 2 is blocked. Using the matrix for beamsplitter 1 and beamsplitter 2, the probabilities are calculated that the photon will either 1. be blocked by the concrete, 2. exit at detector 0 and 3. exit at detector 1.

    mz3

     

    Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

     
  • mbenkerumass 6:00 am on March 1, 2020 Permalink | Reply
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    Non-determinism & Superposition in Quantum Mechanics 

    A system is deterministic if the outcome is easily predictable. Ever since the beginning of Quantum Mechanics, precisely when Einstein had proposed the existence of photons, scientists were concerned that the quantum mechanical model of physics was not deterministic.

    Einstein had proposed that light was made of quanta called photons. Previously, light had always been considered a wave (as explained by Maxwell’s equations) and now it was being considered as particles.

    Einstein’s formula for the energy of a photon: E = h*ν, where ν is the frequency (ν = c/λ).

    One issue arose in the case of an experiment with a polarizer. A polarizer is a system that abosrbs energy of light that is not focused in the direction of the polarizer. The question arose then, how does one determine whether a photon in a beam that forms an angle with the polarizer will go through the polarizer? In vector mechanics, only the fraction of the magnitude that is in the direction would enter. In Quantum Mechanics, presenting an idea of a fraction of a photon went against the theory. In the end, it was decided that it could not predictably be determined whether or not a particular photon would be allowed through the system and it was a system represented by probability.

    pii

    The method of writing a photon moving in a particular direction is written in the following way in Quantum Mechanics:

    Photon in the x direction:

    • |photon; x>

    Photon in the y direction:

    • |photon; y>

    phot5

     

    Superposition in Quantum Mechanics

    The nature of superposition in Quantum Mechanics is different than in classical mechanics.

    The optical experiment using the Mach-Zehnder Interferometer bacome of interest to physicists dealing with Quantum Mechanics. Questions arose, such as how to understand the interference between photons in the interferometer. If two photons were able to interfere with each other in a cancelling manner, this would result in a violation of the conservation of energy. Likewise, a constructive interference would result in the creationof photons, which is also problematic. The current understanding that resulted from this debate was that photons interfered with themselves and that photons are unable to interfere with other photons. Further it was proposed that after moving through the beamsplitter, a single photon will exist in either path devised by the beamsplitter. As it comes to the second beamsplitter, the path of the photon towards either detector is a probabilistic expression. The existence of a single photon in both paths is the understanding of superposition in Quantum Mechanics.

    mz

    In the superposition of states in the quantum mechanical model, the result of the two states is the outcome of either of the two added states with a probability between either condition. The addition of both states with a scaling factor will effect the probability of the outcome state, however there is no intermediate, average or ‘total’ outcome state. States in a quantum mechanical model are notated as |A> and |B> for states A and B.

    986556799

     

    There is an assumption in quantum mechanics that the superposition of a state on itself does not change the outcome.

    986556798

    Using the above assumption, we are able to alter a system of photon states in two directions to simplify the expression to one complex parameter.

    986556797

    Consider you would like to design a new quantum state. It is measured that two quatum states are for one system, a positive particle spin in the z direction and a negative (downward) spin in the z direction. A superposition of these two states together is referred to as a new quatum state. An experiment could be done many times to determine the probability of outcome. The spin of the particle will however only be either in the positive z direction or negative z direction.

    particlespin1

     

    Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

     

     

     
  • mbenkerumass 6:00 am on February 29, 2020 Permalink | Reply
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    Linearity of Quantum Mechanics, Schödinger’s Equation 

    Linearity

    A linear function follows two properties:

    1. If a function is a solution and each variable is scaled or multiplied by the same number, then this is also a solution.
      lin
    2. If two solutions to a function are found, then a third solution of the function is the summation of each variable in the function.
      lin-1.png

     

    Given a linear operator L and an unknown variable u, the following properties apply:

    linw2

    Here is an example:

    linw23

     

    Linearity as related to Quantum Mechanics

    A linear system is far less complicated than a non-linear system.

    Maxwell’s equation is linear, for instance. Newton’s equations are not linear.

    Consider the example below that explains a particular scenario in which Newton’s equations are shwon to be non-linear:

    newton

    Quantum Mechanics is linear. Schroedinger’s Equation, devised in 1925 for a dynamic variable, ψ termed the wavefunction.

    Captureschr

    H_hat is the Hamiltonian, a linear operator as was L in the previous example for Linearity [link]. This means that in Quantum Mechanics, solutions can easily be scaled and added together. Thus, it is proven that Quantum Mechanics is actually simpler than classical mechanics. i is the complex number operator equal to the square root of negative one and h_bar is Planck’s constant.

    You might ask what the wavefunction is about, if there are any units, for instance. Interestingly, Schroedinger was not sure what the wavefunction referred to exactly. Max Born later proposed that it had to do with probability.

     

    Complex Numbers in Quantum Physics

    The complex operator i at the front of the formula notes a significant departure from classical mechanics in which almost all systems are primarily real. In the case of Quantum Mechanics, complex numbers are essential.

    Euler’s formula, e^(i*x) = cos(x) + i*sin(x) also proves useful in Quantum Mechanics.

    180px-Euler's_formula

    Below is some review relevant to Quantum Mechanics:

    tt

     

     

    Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

     
  • mbenkerumass 6:00 am on February 28, 2020 Permalink | Reply
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    Tape Measure Yagi Antenna 

    Radio Club Meeting – Tape Measure Yagi build

    image3image2image160082311118__8CCDCFDA-BFA7-4E2D-8351-9F4533FC0B00

     
  • mbenkerumass 6:00 am on February 25, 2020 Permalink | Reply
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    Optical Waveguides 

    Just as a metallic strip connects the various components of an electrical integrated circuit, optical waveguides connects components and devices of an optical integrated circuit. However, optical waveguides differ from the flow of current in that the optical waves travel through the a waveguide in a spatial distribution of optical energy, or mode. In contrast to bulk optics, which guide optical waves through air, optical waveguides guide light through dielectric conduits.

    Bulk Optical Circuit:

    waveguide2

    Optical Waveguides:

    waveguide1

    The use of waveguides allows for the creation of optical integrated circuits or photonic integrated circuits (PIC). Take for example, the following optical transmit and receive module:

    optical_transmitrecieve

    Planar Waveguides

    A planar waveguide is a structure that limits mobility in only one direction. If we consider the planar waveguide to be on the x axis, then the waveguide may limit the travel of light between two values on the x axis. In the y and z directions, light may travel infinitely. The planar waveguide does not serve many practical uses, however it’s concept is the basis for other tpyes of waveguides. Planar waveguides are also referred to as slab waveguides.Planar waveguides can be made out of mirrors or using a dielectric with a high refractive index slab. See also, Planar Boundaries, Total Internal Reflection, Beamsplitters.

    waveg1waveg2

    Rectangular Waveguides

    Rectangular waveguides can also be built either from mirrors or using a high refractive index rectangular waveguide.

    waveg3waveg4.png

    The following are useful waveguide geometries:

    waveg5

    Various combinations of waveguides may produce different and useful configurations of waveguides:

    waveg6

     

     

     

     
  • mbenkerumass 6:00 am on February 24, 2020 Permalink | Reply
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    Optoelectronic Integrated Circuit Substrate Materials 

    The substrate material used on an optical integrated circuit (OIC) is dependent primarily on the function performed by the circuit. An optical integrated circuit may consist of sources, modulators, detectors, etc and no one substrate will be optimal for all components, which means that a compromise is needed when building an integrated circuit. There are two main approaches that taken to deciding on a solution to this compromise: hybrid and monolithic approaches.

     

    Hybrid Approach

    The hybrid approach attempts to bond more than one substrate together to obtain an optimization for each device in the integrated circuit. This approach allows for a more optimized design for each component in theory, however the process of bolding the various elements together is prone to misalignment and damage from vibration and thermal expansion. For this reason, although the hybrid approach is a theoretically more otpimized approach, it is more common to use the monolithic approach for OIC.

     

    Monolithic Approach

    The monolithic OIC uses a single substrate for all devices. There is one complication in this approach which is that most OIC will require a light source, which can only be fabricated in optically active materials, such as a semiconductor. Passive materials, such as Quartz and Lithium Niobate are effective as substrates, however an external light source would need to be coupled to the substrate to use it.

     

    Optically Passive and Active Materials

    Optically active materials are capable of light generation. The following are examples of optically passive materials:

    • Quartz
    • Lithium Niobate
    • Lithium Tantalate
    • Tantalum Pentoxide
    • Niobium Pentoxide
    • Silicon
    • Polymers

    The following are optically active materials:

    • Gallium Arsenide
    • Gallium Aluminum Arsenide
    • Gallium Arsenide Phosphide
    • Gallium Indium Arsenide
    • Other III-V and II-VI semiconductors

     

    Losses in Substrate due to Absorption

    Monolithic OICs are generally limited to the active substrates above. Semiconductors emit light at a wavelength corresponding to their bandgap energy. They also absorb light at a wavelength equal to or less than their bandgap wavelength. It follows then, for example, if a light emitter, a waveguide and a detector are all fabricated in a single semiconductor, there is a considerable issue of light being absorbed into the substrate, meaning that not enough light will be present for the detector. Thus, reducing losses due to absorbtion is one of the main concerns in substrate materials.

    substrate

     
  • mbenkerumass 6:00 am on February 22, 2020 Permalink | Reply
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    Gas Laser and Semiconductor Lasers 

    heliumconstruction

    The Gas Laser

    In laboratory settings, gas lasers (shown right) are often used to eveluate waveguides and other interated optical devices. Essentially, an electric charge is pumped through a gas in a tube as shown to produce a laser output. Gasses used will determine the wavelength and efficiency of the laser. Common choices include Helium, Neon, Argon ion, carbon dioxide, carbon monoxide, Excimer, Nitrogen and Hydrogen. The gas laser was first invented in 1960. Although gas lasers are still frequently used in lab setting sfor testing, they are not practical choices to encorperate into optical integrated circuits. The only practical light sources for optical integrated circuits are semiconductor lasers and light-emitting diodes.

     

    The Laser Diode

    ladio

    The p-n junction laser diode is a strong choice for optical integrated circuits and in fiber-optic communications due to it’s small size, high reliability nd ease of construction. The laser diode is made of a p-type epitaxial growth layer on an n-type substrate. Parallel end faces may functions as mirrors to provide the system with optical feedback.

     

    The Tunnel-Injection Laser

    The tunnel-injection laser enjoys many of the best features of the p-n junction laser in it’s size, simplicity and low voltage supply. The tunnel-injection laser however does not make use of a junction and is instead made in a single crystal of uniformly-doped semiconductor material. The hole-electron pairs instead are injected into the semiconductor by tunneling and diffusion. If a p-type semiconductor is used, electrons are injected through the insulator by tunneling and if the semiconductor is n-type, then holes are tunneled through the insulator.

     
  • mbenkerumass 6:00 am on February 20, 2020 Permalink | Reply
    Tags: , Quantum Wells,   

    The Quantum Well 

    What is a Quantum Well

    Optical Integraded devices are normally built with the consideration that the device size will be large compared to the wavelength of the beams in the system. When however, the device size is reduced to a size of the same order of magnitude as the wavelength of light in the system, unique properties can be observed. The class of device that operates under the unique properties of this arrangement is the “quantum well.”

     

    Uses of Quantum Wells

    Quantum wells may be integrated to other optical and opt-electronic integrated circuits. Uses of quantum wells include improved lasers, photodiodes, modulators and switches.

     

    Building a Quantum Well

    A quantum well structure features one or more very thin layers of narrow bandgap semiconductor material, interleaved with layers of wider bandgap semiconductors. The thickness of the layers in a quantum well are typically 100 Angstroms or smaller. Quantum wells with many layers are termed a “Multiple Quantum Well” (MQW) structure and quantum wells with only one layer are termed a “Single Quantum Well (SQW) structure. A typical MQW structure may have around 100 layers. The GaAs-AlAs material system or GaInAsP are common choices for materials in quantum well structures.

    quantumwell1

    Superlattice Structure

    A superlattice structure is a term for a case in whic a multiple quantum well structure is built with barrier wals that are thin enough that electrons are able to tunnel through the structure.

     

    The Quantum Well and Quantum Dot

    qwell1

    The quantum well reduces the separation between an electron and hole in a semiconductor, altering the wavefunction and allowing a strong exciton bonding effect at room temperature. The semiconductor laser results from this process. Wave functions in the well are shown to the right.

    When a field is applied across the well, this can result in the tilting of the wells. This can reduce the effective band gap of the material. The process of tilting the wells the alter the band gap is called the Quantum Confined Stark Effect.

    qwell3

     

    Quantum wells are generally understood in two dimensions. The conduction band is forced to be closer the valence band. When this is done in three dimensions to create a small box, where this squeezing effect can be emulated in all dimensions, this is termed a Quantum Dot. A Quantum Dot it turns out is highly effective at producing a high level of energy and as a result there is a high probability that it works as a coherent light source (laser). Quantum dots are readily used today, however since the process of fabrication employs the use of defects in a material to create a quantum dot, the coherency of the light produced is not perfect. Quantum dots are used in data centers for light transmission at a distance of meters. Quantum dots remain a low cost and reasonably efficient light transmission source for small distances. One reason for the low cost of quantum dots is that they can be grown on silicon wafers. A quantum well is not easily (highly unreliably, but perhaps not impossible) grown on Silicon wafers. The issue that arises with quantum wells when being grown on silicon wafers is that the size of atoms in the wafers and thereby the lattice constant is not readily compatible.

     
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