The solutions and tunneling properties for a linear potential and an inverted harmonic oscillator in an infinite well

In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.

The solutions and tunneling properties for a linear potential and an inverted harmonic oscillator in an infinite well

In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.

Pienitaajuinen radiosekoitin

Normaalisti radiovastaanottimet on luokiteltavissa suoriin vastaanottimiin ja superheterodynevastaanottimiin. Jälkimmäistä nimitetään tavallisesti supervastaanottimeksi. Molemman vastaanottimen oleellisiin osiin kuuluu antennin virityspiiri, supervastaanottimelle lisäksi paikallisoskillaattorin virityspiiri, mikä pitää virittää antennipiirin kanssa samanaikaisesti. Pienillä taajuuksilla, taajuudet luokassa kilo-Hertzejä tai pienemmillä, on antennipiirin viritys resonanssipiirin ominaisuuksista johtuen sitä kapeammalla kaistalla ja sitä hitaampaa mitä pienemmällä taajuudella vastaanotto tapahtuu. Lisäksi virityspiiri hyvyysluku Q on vaikea saada sopivaksi, mikäli viritys on muuten käytännöllinen, säädettävä resonanssipiiri. Vaadittaessa kiinteätaajuista viritystä on käytännöllistä hyödyntää sähkömekaanisia osia, siis keraamisia tai kvartsikiteitä. Koska kiteitten ja korkean hyvyysluvun piirin värähtely jatkuu useita värähtelyjaksoja ennen saapuneitten värähtelyjen sammumista, kestää myös kauan aikaa, ennen kuin värähtely piirissä on loppu. Pienitaajuinen resonanssipiiri saavuttaa maksimivirtansa hitaasti, jos hyvyysluku on iso, kun piiri alkaa johtaa resonanssitaajuista virtaa. Tässä työssä pyritään vastaanotinjärjestelyyn ongelmallisen, pientaajuisen virityspiirin käytön välttämiseksi. Toisena tavoitteena on saada aikaan vastaanotto siten, että tietty pienitaajuinen radiotaajuusalue voidaan kokonaisuudessaan vastaanottaa jatkuva-aikaisesti, ilman antennipiirin jatkuvaa virittämistä erillisille taajuuksille. Laaditaan kytkentä, joka mitoitetaan, simuloidaan ja mitataan.

Satelliittioskillaattorin jyrsintätyövaiheiden valmistusystävällisyyden analysointi

Tämän tutkimuksen aiheena on satelliittioskillaattorin jyrsintätyövaiheiden valmistusystävällisyyden analysointi. Tutkimuksen viitekehyksenä on DFM (design for manufacturing), johon kaikki päätelmät ja tulokset sidotaan. Tutkimuksen päätavoite on etsiä valmistusteknisiä ratkaisuja, joilla samanaikaisesti parannetaan, sekä oskillaattorin suorituskykyä, että sen valmistettavuutta noudattamalla DFM-sääntöjä. Suorituskyvyn näkökulmasta tärkeintä on oskillaattorin hyvyysluvun maksimointi. Tutkimuksessa käsitellään lyhyesti satelliittioskillaattorin porauksia, pinnoitusta ja asennusta, mutta pääpaino tutkimuksessa on oskillaattorin rungon jyrsintätyövaiheissa. Tässä työssä toteutettiin DFM-analyysi, jonka avulla pystyttiin helpottamaan lukuisia tuotannollisia ongelmia ja onnistuttiin löytämmään keinoja oskillaattorin suorituskyvyn parantamiseksi.