From Brownian Motion to Schrödinger’s Equation


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Brownian motion

Flash and JavaScript are required for this feature. About this Video Playlist Download this Video Description: This lecture covers topics, including the basic wave characteristics, wave-particle duality in both electromagnetic waves and material waves, and also the fundamentals in the mathematical description of wave mechanics. Instructor: Prof. Gang Chen.

Lecture 1: Intro to Nanotec Lecture 2: Characteristic T Lecture 4: Solutions to Sch Lecture 5: Electronic Level Lecture 6: Crystal Bonding Lecture 7: Phonon Energy Le Lecture 8: Density of State Lecture 9: Specific Heat an Lecture Fundamental of Lecture Energy Transfer Lecture EM Waves: Refle Lecture EM Wave Propaga Lecture Wave Phenomena Lecture Particle Descri Lecture Fermi Golden Ru The Malliavin derivatives can be used to calculate condi- tional expectations or chaos decompositions of stochastic processes see [3], [7].

This paper has the following structure: In Section 2 we introduce the list of assumptions and give the definition of the solution. In Section 3 we briefly present the two stochastic integrals that appear in the equation which is investigated.

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The existence of the solution is derived in Section 4. Section 5 contains results about infinite dimensional Malliavin derivatives and the existence of the Malliavin derivative of the solution is proved. Let K be a separable real Hilbert space. The stochastic integrals In this section we briefly present the definitions of the stochastic integrals we considered in 2.

Let en n be an orthonormal basis in K. Let kn n be an orthonormal basis in K.

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For a. Proposition 3. The stochastic integral Z t can also be represented by a sto- chastic integral with respect to the cylindrical Wiener process W see [3], [6]. If f is concentrated on [0, T ], then we consider [0, T ] instead of R.

Schrödinger's Equation and Classical Brownian Motion

Existence of the solution Theorem 4. Assume that [I], [A], [f], [g], [b] are satisfied. Equation 2. In order to prove the existence of the solution of 2. Observe that for a. We rewrite 2.

The proof of the existence of a unique solution U for 4. The existence of Malliavin derivative of the solution We briefly present some results about infinite dimensional Malliavin deriva- tives: We consider the random variable Y with values in a complex Hilbert space H. The following result is known see Lemma 5. By using Proposition 5. Let M be a further separable Hilbert space. We will use the following well-known properties of Dt see, for example [7], [3] : Proposition 5.

The assumption in Remark 3. The functions f and g are deterministic. The initial condition X0 is deterministic.


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Theorem 5. We process the proof in two steps: Step 1: It follows from the above assumption 3 that the functions f and g are globally Lipschitz continuous.

Equation 4. In this case it follows from Proposition 5.

From Brownian Motion to Schrödinger’s Equation

Then, by Proposition 5. Since the spaces are finite dimensional, the operators are also Hilbert-Schmidt operators. The constant C does not depend on n. Then we get by Lemma 5. Consider that the assumptions of this section hold.

From Brownian Motion to Schrödinger’s Equation From Brownian Motion to Schrödinger’s Equation
From Brownian Motion to Schrödinger’s Equation From Brownian Motion to Schrödinger’s Equation
From Brownian Motion to Schrödinger’s Equation From Brownian Motion to Schrödinger’s Equation
From Brownian Motion to Schrödinger’s Equation From Brownian Motion to Schrödinger’s Equation
From Brownian Motion to Schrödinger’s Equation From Brownian Motion to Schrödinger’s Equation
From Brownian Motion to Schrödinger’s Equation From Brownian Motion to Schrödinger’s Equation
From Brownian Motion to Schrödinger’s Equation From Brownian Motion to Schrödinger’s Equation

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