We determine the intermediate integral separately:
where F(θ, m) is the incomplete elliptic integral of the first kind [57], and C is a constant which we will let be absorbed by f(v, w). We note that for v >0, w >0,F(0, m)≡0 so that
where K(m) is the complete elliptic integral of the first kind [57]. In all, we have managed to find the integral
I(u, v, w) = 2π3/2i We can now find the original integral by differentiating. Especially we need
Z e−ax2−by2−cz2x2 As a bonus, we also note thatI(a, b, c) corresponds to the Coulomb matrix element of two electrons in a harmonically confined quantum dot.
The Probability Distribution of the Overhauser Field in a Quantum Dot
In this section we show that probability distribution of a weighted sum of N identical stochastic variables approaches a Gaussian distribution when N → ∞. The aim is to find the probability density function for the sum
Y = XN n=1
anXn, (C.1)
where Xn are identically distributed strochastic variables with expectation value E[X] and variance σ2, and an are finite coefficients which, in general, also depend on N. For a QD we may choose the coefficients to match the electron position probability density function:
an= |Ψ (nl/N)|2
N , (C.2)
which implies that
N→∞lim XN n=1
an = lim
N→∞
XN n=1
|Ψ (nl/N)|2 N
= Z l
0 |Ψ (x)|2dx= 1,
(C.3)
wherel is the size of the quantum dot. We introduce the new variables with vanishing expectation value
X˜n =Xn−E[Xn], (C.4)
109
and form the new sum
Y˜ = XN n=1
anX˜n, (C.5)
which is related to Y by
Y = ˜Y + Σ(N)E[X], (C.6)
The characteristic function of ˜Y is given by ϕY˜(t) =
where ϕX˜ is the characteristic function of any of the ˜Xn which we expand Taylor series to the second order in the second step. Now we consider
ϕY˜(t)≈exp
We make the restriction that |Ψ(x)|2 is bounded on [0, l] which means that there is some contant C such that |Ψ(x)|2 ≤ C. This also ensures the ex-istence of all Sk and for N approaching infinity we may keep only the first term in Eq. (C.11) and we get
and we obtain the probability density function of ˜Y as fY˜(y) =
√N σ√
2πS1
exp
− Ny2 2S1σ2
, (C.14)
which is a Gaussian distribution with variance S1σ2/N. S1 depends on the the coefficients an but the general form is always a Gaussian distribution regardless of what wave function is considered.
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July 2014 Dissertation in Theoretical Physics aboutSpin and Photon Coherence and Entanglement in Semiconductor Quantum Dots under the supervision of Prof. Dr. Guido Burkard.
2009.11 - 2014.05 Graduate student at the University of Konstanz under the supervision of Prof. Dr. Guido Burkard.
2009.08 - 2009.11 Stipendium work and studies in the Theory and Modelling group, IFM, Link¨oping University
June 2009 Master of Science in Applied Physics with Diploma Thesis Electron Localization and Spin Polarization in a Quantum Circle
2005.08 - 2009.06 Studies of Applied Physics and Electrical Engineering at Technical University of Vienna and atLink¨oping Univeristy 2003.06 - 2005.08 Employment at Esca Food Solutions
2002.05 - 2003.06 Studies of Applied Physics and Electrical Engineering,Link¨oping University
2001.06 - 2002.04 Military service atEngineer Corps Bodens
2000.08 - 2001.06 Studies of Applied Physics and Electrical Engineering,Link¨oping University
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1997.08 - 2000.06 High school, Scientific profile, atBerzeliusskolan, Link¨oping
1997.08 - 2000.06 High school, Scientific profile, atBerzeliusskolan, Link¨oping