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  • mbenkerumass 6:00 am on February 29, 2020 Permalink | Reply
    Tags: Physics,   

    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 January 23, 2020 Permalink | Reply
    Tags: , , Physics   

    The Oscillator 

    The oscillator is an important concept used in a variety of applications. One basic use of an oscillator is that of signal generation.

    An oscillator is a system with a gain and positive feedback. The gain must be greater than the loss in the feedback system, so that each time the signal goes through the aplifier in the system, a net gain is produced. The phase shift of a single round trip in the gain-feedback loop must also be a multiple of 2*pi so that a pure signal is repeatedly amplified.

    When these conditions are satisfied, the system is unstable and oscillation begins. Eventually, the amplifier gain becomes saturated and rather than a further increase of amplification, the added gain only compensates for system losses.oscillator

    Since the system is dependen upon a 2*pi phase shift (the period), an oscillator may be designed for a specific frequency. An oscillator generate a signal from noise by repeatedly amplifying the noise periodically.

    Although there are many applications for oscillators, a laser is fundamentally an optical oscillator, an optical signal generator. The maser, which stands for microwave amplification by stiumulated emission of radiation was used before the laser. The saser is an acoustic version of the laser, in which instead of emitting a beam of photons or electromagnetic radiation, an acoustic beam or signal is generated.

    The following outlines the operation of a laser; an optical amplifier placed inside of a resonator with a partially transmitting mirror as the output of the system.

    laser2

    B. E. A. Saleh and M. C. Teich, Fundamentals of photonics. Hoboken: Wiley, 2019.

     

     
  • mbenkerumass 10:02 am on November 27, 2019 Permalink | Reply
    Tags: , , Physics, RADAR   

    Doppler Effect 

    RF/Photonics Lab
    November 2019
    Michael Benker

     

    Doppler Effect

                    The Doppler Effect is an important principle in communications, optics, RADAR systems and other systems that deal with the propagation of signals through space. The Doppler Effect can be summarized as the resultant change to a signal’s propagation due to movement either by the source or receiving end of the signal. As the distance between two objects changes, so does the frequency. If, for instance, a signal is being propagated towards an object that is moving towards the source, the returning signal will be of a higher frequency.

    260px-Dopplerfrequenz

    The Doppler Effect is also applied to rotation of an object in optics and RADAR backscatter scenarios. A rotating target of a radar or optical system will return a set of frequencies which reflect the distances of each point on the target. If one side of the target is moving closer while the other side is moving away, there will be both a higher and lower frequency component to the return signal.

    Dopplereffectsourcemovingrightatmach0.7

     
  • mbenkerumass 10:08 am on November 22, 2019 Permalink | Reply
    Tags: , , Physics   

    Interferometry – Introduction 

    RF/Photonics Lab
    Jared Alves
    November 2019

    Interferometry – Introduction

                    Interferometry is a family of techniques in which waves are superimposed for measurement purposes. These waves tend to be radio, sound or optical waves. Various measurements can be obtained using interferometry that portray characteristics of the medium through which the waves propagate or properties of the waves themselves. In terms of optics, two light beams can be split to create an interference pattern when the waves combine (superimpose). This superposition can lead to a diminished wave, an increased wave or a wave completely reduced in amplitude. In an easily realizable physical sense, tossing a stone into a pond creates concentric waves that radiate away from where the stone was tossed. If two stones are thrown near each other, their waves would interfere with each other creating the same effect described previously. Constructive interference is the superposition of waves that results in a larger amplitude whereas destructive interference diminishes the resultant amplitude. Normally, the interference is either partially constructive or partially destructive, unless the waves are perfectly out of phase. The following image displays total constructive and destructive interference.

    interferrometry1

    A simple way to explain the operation of an interferometer is that it converts a phase difference to an intensity. When two waves of the same frequency are added together, the result depends only on the phase difference between them, as explained previously.

    interferrometry2The image above shows a Michelson interferometer which uses two beams of light to measure small displacements, refractive index changes and surface irregularities.  The beams are split using a mirror that is not completely reflective and angled so that one beam is reflected, and one is not. The two beams travel in separate paths which combine to produce interference. Whether the waves combine destructively or constructively depends on distancing between the mirrors. Because the device shows the difference in path lengths, it is a differential device. Generally, one leg length is kept constant for control purposes.

     
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