PowerPoint Presentation - Inflation, String Theory,
... “unnatural” state will have more volume (and contain more observers like us) than all other string theory vacua combined. ...
... “unnatural” state will have more volume (and contain more observers like us) than all other string theory vacua combined. ...
Tight bounds on quantum searching
... We shall estimate the expected number of times that a Grover iteration is performed: the total time needed is clearly in the order of that number. On the s–th time round the main loop, the value of m is λs−1 and the expected number of Grover iterations is less than half that value since j is chosen ...
... We shall estimate the expected number of times that a Grover iteration is performed: the total time needed is clearly in the order of that number. On the s–th time round the main loop, the value of m is λs−1 and the expected number of Grover iterations is less than half that value since j is chosen ...
Brute – Force Treatment of Quantum HO
... * There is a PROBLEM with our discussion however since NOT all the solutions obtained in this way can be normalized! ⇒ At LARGE i the recursion formula becomes ...
... * There is a PROBLEM with our discussion however since NOT all the solutions obtained in this way can be normalized! ⇒ At LARGE i the recursion formula becomes ...
Uncertainty relations for information entropy in wave mechanics
... every Gaussian wave function. Moreover, it follows from our variational calculation that the first and the third inequalities become equalities only for Gaussian functions. We conjecture that this is also the case for the second inequality. If we take only the first and the third term in (8) we obta ...
... every Gaussian wave function. Moreover, it follows from our variational calculation that the first and the third inequalities become equalities only for Gaussian functions. We conjecture that this is also the case for the second inequality. If we take only the first and the third term in (8) we obta ...
qm-cross-sections
... In a practical scattering situation we have a finite acceptance for a detector with a solid angle W. There is a range of momenta which are allowed by kinematics which can contribute to the cross section. The cross section for scattering into W is then obtained as an integral over all the allowed m ...
... In a practical scattering situation we have a finite acceptance for a detector with a solid angle W. There is a range of momenta which are allowed by kinematics which can contribute to the cross section. The cross section for scattering into W is then obtained as an integral over all the allowed m ...
Learning Theory 1 Introduction 2 Hoeffding`s Inequality
... Most of the methods we have talked about in the course have been introduced somewhat heuristically, in the sense that we have not rigorously proven that they actually work! Roughly speaking, in supervised learning we have taken the following strategy: • Pick some class of functions f (x) (decision t ...
... Most of the methods we have talked about in the course have been introduced somewhat heuristically, in the sense that we have not rigorously proven that they actually work! Roughly speaking, in supervised learning we have taken the following strategy: • Pick some class of functions f (x) (decision t ...
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... electrons, when an electron moves from the n = 1 level to the n = 3 level, the circumference of its orbit becomes 9 times greater. This occurs because (a) there are 3 times as many wavelengths in the new orbit, (b) there are 3 times as many wavelengths and each wavelength is 3 times as long, (c) the ...
... electrons, when an electron moves from the n = 1 level to the n = 3 level, the circumference of its orbit becomes 9 times greater. This occurs because (a) there are 3 times as many wavelengths in the new orbit, (b) there are 3 times as many wavelengths and each wavelength is 3 times as long, (c) the ...
9 Radon-Nikodym theorem and conditioning
... Proof. “⇐=” is easy: µ(A) = 0 implies ∀ε ν(A) < ε. “=⇒”: Otherwise P we have ε and An ∈ S such that µ(An ) → 0 but ν(An ) ≥ ε. WLOG, n µ(An ) < ∞. Taking Bn = An ∪ An+1 ∪ . . . we have µ(Bn ) → 0, ν(Bn ) ≥ ε, and Bn ↓ B for some B. Thus, µ(B) = 0, but ν(B) ≥ ε (due to finiteness of ν), in contradict ...
... Proof. “⇐=” is easy: µ(A) = 0 implies ∀ε ν(A) < ε. “=⇒”: Otherwise P we have ε and An ∈ S such that µ(An ) → 0 but ν(An ) ≥ ε. WLOG, n µ(An ) < ∞. Taking Bn = An ∪ An+1 ∪ . . . we have µ(Bn ) → 0, ν(Bn ) ≥ ε, and Bn ↓ B for some B. Thus, µ(B) = 0, but ν(B) ≥ ε (due to finiteness of ν), in contradict ...
Quantum and classical statistics of the electromagnetic zero
... the basis for theoretical investigations in the discipline known as random or stochastic electrodynamics ~SED!. In SED the statistical character of quantum measurements is imitated by the introduction of a stochastic classical background electromagnetic field. Random electromagnetic fluctuations are ...
... the basis for theoretical investigations in the discipline known as random or stochastic electrodynamics ~SED!. In SED the statistical character of quantum measurements is imitated by the introduction of a stochastic classical background electromagnetic field. Random electromagnetic fluctuations are ...
Scattering model for quantum random walks on a hypercube
... Turing machine can be simulated using a cellular automaton. The CQRW is usually defined on regular graphs (each vertex having the same number of outgoing edges). The definition on nonregular graphs is also possible, and some interesting algorithms are based on this version [13]. However, the latter ...
... Turing machine can be simulated using a cellular automaton. The CQRW is usually defined on regular graphs (each vertex having the same number of outgoing edges). The definition on nonregular graphs is also possible, and some interesting algorithms are based on this version [13]. However, the latter ...
continuous vs discrete processes: the
... In this section we show that superpositions of atomic energy states can be incorporated into Markov processes. Consider an arbitrary linear combination v = a1u1 + a2u2 + ... where | v | = 1 and the basis vectors u1, u2, ... are chosen so that each represents one of the atom’s discrete energy levels. ...
... In this section we show that superpositions of atomic energy states can be incorporated into Markov processes. Consider an arbitrary linear combination v = a1u1 + a2u2 + ... where | v | = 1 and the basis vectors u1, u2, ... are chosen so that each represents one of the atom’s discrete energy levels. ...
Identical particles
... called their statistics. The statistics of a particle can be boson or fermion. By the way, the names come from two big-time physicists: Bose and Fermi. Also sometimes more names get into the act and Bose statistics is also called Bose-Einstein statistics, while Fermi becomes Fermi-Dirac. Einstein yo ...
... called their statistics. The statistics of a particle can be boson or fermion. By the way, the names come from two big-time physicists: Bose and Fermi. Also sometimes more names get into the act and Bose statistics is also called Bose-Einstein statistics, while Fermi becomes Fermi-Dirac. Einstein yo ...
Probability amplitude
In quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability or probability density.Probability amplitudes provide a relationship between the wave function (or, more generally, of a quantum state vector) of a system and the results of observations of that system, a link first proposed by Max Born. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics. In fact, the properties of the space of wave functions were being used to make physical predictions (such as emissions from atoms being at certain discrete energies) before any physical interpretation of a particular function was offered. Born was awarded half of the 1954 Nobel Prize in Physics for this understanding (see #References), and the probability thus calculated is sometimes called the ""Born probability"". These probabilistic concepts, namely the probability density and quantum measurements, were vigorously contested at the time by the original physicists working on the theory, such as Schrödinger and Einstein. It is the source of the mysterious consequences and philosophical difficulties in the interpretations of quantum mechanics—topics that continue to be debated even today.