Introduction to the Dirac Delta Function

The Dirac delta function is the name given to a mathematical structure that is intended to represent an idealized point object, such as a point mass or point charge. It has broad applications within quantum mechanics and the rest of quantum physics, as it is usually used within the quantum wavefunction. The delta function is represented with the Greek lowercase symbol delta, written as a function: ÃŽ ´(x). How the Delta Function Works This representation is achieved by defining the Dirac delta function so that it has a value of 0 everywhere except at the input value of 0. At that point, it represents a spike that is infinitely high.  The integral taken over the entire line is equal to 1. If youve studied calculus, youve likely run into this phenomenon before. Keep in mind that this is a concept that is normally introduced to students after years of college-level study in theoretical physics. In other words, the results are the following for the most basic delta function ÃŽ ´(x), with a one-dimensional variable x, for some random input values: ÃŽ ´(5) 0ÃŽ ´(-20) 0ÃŽ ´(38.4) 0ÃŽ ´(-12.2) 0ÃŽ ´(0.11) 0ÃŽ ´(0) ∞ You can scale the function up by multiplying it by a constant. Under the rules of calculus, multiplying by a constant value will also increase the value of the integral by that constant factor. Since the integral of ÃŽ ´(x) across all real numbers is 1, then multiplying it by a constant of would have a new integral equal to that constant. So, for example, 27ÃŽ ´(x) has an integral across all real numbers of 27. Another useful thing to consider is that since the function has a non-zero value only for an input of 0, then if youre looking at a coordinate grid where your point isnt lined up right at 0, this can be represented with an expression inside the function input. So if you want to represent the idea that the particle is at a position x 5, then you would write the Dirac delta function as ÃŽ ´(x - 5) ∞ [since ÃŽ ´(5 - 5) ∞].   If you then want to use this function to represent a series of point particles within a quantum system, you can do it by adding together various dirac delta functions. For a concrete example, a function with points at x 5 and x 8 could be represented as ÃŽ ´(x - 5) ÃŽ ´(x - 8). If you then took an integral of this function over all numbers, you would get an integral that represents real numbers, even though the functions are 0 at all locations other than the two where there are points. This concept can then be expanded to represent a space with two or three dimensions (instead of the one-dimensional case I used in my examples). This is an admittedly-brief introduction to a very complex topic. The key thing to realize about it is that the Dirac delta function basically exists for the sole purpose of making the integration of the function make sense. When there is no integral taking place, the presence of the Dirac delta function isnt particularly helpful. But in physics, when you are dealing with going from a region with no particles that suddenly exist at only one point, its quite helpful. Source of the Delta Function In his 1930 book, Principles of Quantum Mechanics, English theoretical physicist Paul Dirac laid out the key elements of quantum mechanics, including the bra-ket notation and also his Dirac delta function. These became standard concepts in the field of quantum mechanics within the Schrodinger equation.

A Comparison of Haydns Symphony No 95 With Beethovens...

A Comparison of Haydns Symphony No. 95 with Beethovens Symphony No. 5 Introduction As noted by Robert Hughes, Beethoven was not only the embodiment of all that was before him, but also of that which was yet to come (Hughes 486). The truth of this may be seen by comparing Beethovens 5th Symphony in C Minor to Haydn, the father of Symphony, and his 95th in C Minor. While Haydns symphony is both playful and dramatic, Beethovens symphony is grander both in terms of scale and vision. He expands the size of the orchestra to incorporate the sounds swirling around, underlying, and depicting the arrival of Fate in a rhythm-driven, thematic symphony that takes Haydns form and runs with it as though to the top of a mountain peak. This paper will analyze the symphonies by movement, according to form, size, structure, tonalities, melodies, orchestral sound and overall mood and effect. 1st Movement Both Haydns and Beethovens first movements are composed in sonata form, with an exposition stated in the opening that is then developed and recapitulated. The size of Haydns opening movement is somewhat smaller than Beethovens, whose opening exposition itself seems larger than life. Haydns first movement is roughly six minutes long, and the length of Beethovens first movement only exceeds Haydns by about a minute and a half to two minutes. Beethoven does not introduce too many structural changes to the traditional sonata form, which he learned from Haydn, in the first

Chivalric Code in Beowulf Free Essays

The Importance of the Retainer and King in the Heroic Code Members of the Anglo-Saxon warrior society subscribed to an ethos that celebrated the heroic code. In the passage from Beowulf, the poet’s interest in the duties of a loyal retainer and the duties of a great king are evident in the specific language he uses to describe Beowulf’s encounter with the dragon. In one specific passage of this poem, Beowulf is portrayed as an ideal retainer by the loyalty, courage and fealty to the king he possesses. We will write a custom essay sample on Chivalric Code in Beowulf or any similar topic only for you Order Now At the beginning of this passage, Beowulf reflects on King Hygelac and the many sacrifices and deeds he provided to Beowulf during his youth. A sorrowful mood is brought upon this specific text, as Beowulf reminisces on the death of Hygelac, â€Å"I marched ahead of him, always there/ at the front of the line; and I shall fight like that/ for as long as I live†¦Ã¢â‚¬  (Beowulf 2497-2499). Beowulf acknowledges how privileged he is to have a life of luxury and obtain such wisdom inherited by Hygelac. If it was not for Hygelac, Beowulf would not have survived such a long, prosperous life, mentoring Hygelac’s son and soon holding the throne himself. As Beowulf prepares to fight the dragon, he easily allows the warriors to stand their place and not go any further. Demonstrating his courage, Beowulf states: This fight is not yours, nor is it up to any man except me to measure his strength against the monster or to prove his worth. I shall win the gold by my courage, or else mortal combat, doom of battle, will bear your lord away. (2532-2535) He believes that terminating the dragon is the duty only he can accomplish as well as longing for a feeling of satisfaction as he seeks the glory of winning the battle and knowing he has demolished the risk of danger his people will have to face. Based on the duration of this poem, Beowulf is accustomed to the warrior duties and later on the lifestyle of a king. These two titles have very distinct roles in Anglo-Saxon society. Beowulf, as a young warrior need not much to worry about status and having the knowledge that his people are safe and satisfied. He then gains wisdom as he witnesses Hygelac’s form of generosity towards him and his people, â€Å"The treasures that Hygelac lavished on me†¦He gave me land/ and the security land brings, so he had no call/ to go looking for some lesser champion,† (2490-2494). Beowulf demonstrates Hygelac as a loyal lord, true to his people; in return Beowulf brings himself forward as a tribute to fight the dragon. As Beowulf earns the chance to hold the throne as king later on in the poem, it is evident that the responsibilities he held were much to his advantage when the time of battle arose against the dragon. His full awareness, â€Å"‘†¦as king of the people I shall pursue this fight/ for the glory of winning, if the evil one will only/ abandon his earth-fort and face me in the open. ’† (2513-2515) Beowulf, as king and warrior only interprets as a right-doing to go into battle with the dragon, knowing his life is at risk. Towards the conclusion of this passage, Beowulf falls and has no choice but, â€Å"to give ground like that and go/ unwillingly to inhabit another home/ in a place beyond;† (2588-2590). Beowulf is true to his fellow citizens and attempts to provide to their needs even if it means losing the battle and sacrificing his life, which was the case in this poem. From this passage, it is evidently shown that Beowulf can not only hold a title of a loyal retainer with his endless amount of courage but also a grand lord, with unlimited sacrifice to his citizens. Word Count: 578 How to cite Chivalric Code in Beowulf, Papers