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The scattering matrix is used to relate particle states before and after undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory. When quantum theory got developed, the classical descriptions for scattering were no longer sufficient and other methods began to develop. One of them being the S-matrix. In this thesis we present assumptions made in order to define the S-matrix, as well as its different properties and some one-dimensional examples. The definition of the S-matrix relates to the probability amplitude for different outcomes in a scattering experiment and the elements of the matrix are called scattering amplitudes. The S-matrix is unitary and if the scattering potential V (x) is real, it is also time-reversal invariant. In one dimension it is showcased how particles can transmit through or reflect off of a potential as well as how the symmetry of the potential effects the S-matrix. Lastly we give some examples of the S-matrix in one dimension and we end with a brief outlook of how it is defined and applied in quantum field theory where special relativity is taken into account.

This thesis presents the results of an experimental study onsmall-capacitance Josephson junctions in a high impedanceenvironment. In small-capacitance Josephson junctions the charging energyEC=e2/2C becomes relevant at low temperature and thecharging effects influence the transport properties. In theobservation of single-charge effects, a fundamental role isplayed by the impedance Z(ω) of the electromagneticenvironment surrounding the junction. To detect the tunnelingof single Cooper pairs in one junction, it is essential toembed the junction in a high impedance environment at all thefrequencies, such that Z(ω)>>RQ, where RQ=h/(2e)2= 6.45 kΩ is the quantum resistance forCooper pairs. This requirement is necessary in order to avoidthe fast discharge of the junction capacitance through theexternal circuit. In this work the effect of the high impedance environment isstudied on Cooper pair transistors (CPT) and single junctionswith tunable coupling (SQUID). The high impedance environmentis obtained by biasing the sample under test with four SQUIDarrays. The SQUID geometry, besides the modulation of thejunctions Josephson energy (EJ), allows the tuning in situ of the effective impedanceof the environment. By applying a magnetic field perpendicularto the SQUIDs loop, the zero bias resistance of the leads (R0) can be varied over several orders of magnitude(104<R0 (Ω)<108), providing the possibility to study the samesample in different electromagnetic environments. Measurements of the current-voltage (I-V) characteristic ofa single junction with SQUID geometry allow the investigationof the influence of both the electrodynamic environment and theEJ/ECratio in the same sample. As the impedance of thebiasing leads is increased (R0>>RQ), the I-V curve of the junction exhibits a welldefined Coulomb blockade feature with a region of negativedifferential resistance, evidence of the coherent tunneling ofsingle Cooper pairs. Measurements on CPT samples also show how the Coulombblockade feature in the I-V curve becomes sharper as theimpedance of the environment is increased. Furthermore theyindicate how the Coulomb blockade threshold voltage is modifiedwith the DC signal applied to the gate. When the SQUID arraysare highly resistive, the gate-induced modulation of the I-Vcurve changes frome-periodic to 2e-periodic, suggesting that high impedance leadseffectively reduce non-equilibrium quasiparticle poisoning. / QC 20120320

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Understanding the dynamics of proteins is crucial in order to understand life on a molecular level. In this Bachelor’s thesis large scale motions of the membrane protein SERCA has been investigated with coarse-grained methods. The goal was to find certain places in the structure where mutations could alter and affect the functionality of the protein, possibly leading to disease. The protein was modeled with an elastic network model and simulated with Langevin dynamics. Data from the simulations was used to calculate all the places were formation of stabilizing salt bridges were possible and these were then compared with mutations found in real cancerous cells. Promising results were obtained but need to be investigated further with more sophisticated methods before any conclusions can be drawn regarding their significance for the protein functionality. Likewise, the use of elastic network models together with Langevin dynamics showed to be a good alternative to sample large conformational changes in proteins. / Kunskap om proteiners dynamik är av stor vikt för att kunna förstå mekanismerna bakomlivet på en molekylär nivå. I denna kandidatexamensuppsats har storskaliga rörelser i membranproteinet SERCA undersökts med hjälp grovkorniga metoder. Målet var att hitta särskilda platser i strukturen där mutationer kunde införas och påverka proteinetsfunktionalitet, något som möjligen skulle kunna leda till sjukdom. Proteinet modellerades som ett elastiskt nätverk och simulerades med Langevin-dynamik. Data från simuleringarna användes sedan för att beräkna samtliga platser där bildandet av stabiliserandesalt bryggor var möjlig, varpå dessa jämfördes med mutationer funna i riktiga cancerceller. Lovande resultat er hölls, men dessa måste dock undersökas vidare med mer sofistikerade metoder innan slutsatser kan dras gällande deras betydelse för proteinets funktionalitet. Där till visade sig användan det av elastiska nätverksmodeller tillsammans med Langevindynamik vara ett bra alternativ för att fånga stora strukturella förändringar i proteiner.