文章基本信息

标题：Heisenberg Uncertainty Relation in Quantum Liouville Equation
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作者：Davide Valenti
期刊名称：International Journal of Mathematics and Mathematical Sciences
印刷版ISSN：0161-1712
电子版ISSN：1687-0425
出版年度：2009
卷号：2009
DOI：10.1155/2009/369482
出版社：Hindawi Publishing Corporation
摘要：We consider the quantum Liouville equation and give a characterization of the solutions which satisfy the Heisenberg uncertainty relation. We analyze three cases. Initially we consider a particular solution of the quantum Liouville equation: the Wigner transform 𝑓(x,v,𝑡) of a generic solution 𝜓(x;𝑡) of the Schrödinger equation. We give a representation of 𝜓(x, 𝑡) by the Hermite functions. We show that the values of the variances of x and v calculated by using the Wigner function 𝑓(x,v,𝑡) coincide, respectively, with the variances of position operator 𝑋 and conjugate momentum operator 𝑃 obtained using the wave function 𝜓(x,𝑡). Then we consider the Fourier transform of the density matrix 𝜌(z,y,𝑡) = 𝜓∗(z,𝑡)𝜓(y,t). We find again that the variances of x and v obtained by using 𝜌(z, y,𝑡) are respectively equal to the variances of 𝑋 and 𝑃 calculated in 𝜓(x,𝑡). Finally we introduce the matrix ‖𝐴𝑛𝑛′(𝑡)‖ and we show that a generic square-integrable function 𝑔(x,v,𝑡) can be written as Fourier transform of a density matrix, provided that the matrix ‖𝐴𝑛𝑛′(𝑡)‖ is diagonalizable.