(6) It is a deeper quantum mechanical requirement than the condition about a minimal scale. Example 21 shows that it is not equivalent to it. Indeed, inequality ( 11) implies immediately the Robertson–Schrödinder uncertainty principle. Indeed, inequality ( 15) is an equality iff the state is pure. 1 (4) It includes a pure state condition. Here we use the squares | W ρ ~ | 2 and | F σ ( W ρ ~ ) | 2, and so we are treating W ρ as a wave function in ordinary quantum mechanics on a 2 n-dimensional configuration space. (3) It makes a direct connection with harmonic analysis, as it amounts to an inequality relating W ρ and its Fourier transform F σ ( W ρ ). (2) It is invariant under linear symplectic and anti-symplectic transformations (see Theorem 19). In fact, we only have to compute the covariance matrices of W ρ, | W ρ ~ | 2 and | F σ ( W ρ ~ ) | 2 and check inequalities ( 11). (1) It is parsimonious, in the sense that it is a computable test as the RSUP, but not a complicated one as sets of necessary and sufficient conditions such as the Kastler, Loupias, Miracle-Sole (KLM) conditions. Roughly speaking, it can be obtained from the ordinary Fourier transform F ( F ) by a symplectic rotation and a dilation ( F σ F ) ( z ) = 1 ( 2 π ħ ) n ( F F ) Jz 2 π ħ.įor a given measurable phase-space function F, satisfying In the sequel F σ ( F ) denotes the symplectic Fourier transform of the function F. In order to state our results precisely, let us fix some notation. This prompted us to look for an alternative uncertainty principle which goes beyond the RSUP. This means that being a quantum state is not only a question of scale but also of shape. More emphatically, we will show that any measurable function in phase space F with a positive definite covariance matrix Cov ( F ) > 0 satisfies ( 4) after a suitable dilation F ( z ) ↦ λ 2 n F ( λ z ), while most of them remain non quantum. We shall give an example of a function in phase space which saturates the RSUP, but which is manifestly not a Wigner function. This condition is not sufficient to ensure that the state is quantum mechanical (not even if saturated). In fact, ( 4) is only a requirement about a minimal scale related to ħ. Having said that, there is nothing about inequality ( 4) which is particularly quantum mechanical, with the exception of the presence of Planck’s constant. More specifically, the RSUP ( 4) is saturated, whenever all the Williamson invariants of Cov ( W ρ ) are minimal : In particular, we say that the RSUP is saturated if we can find n two-dimensional symplectic planes, where the uncertainty is minimal. By a suitable linear symplectic transformation, the RSUP makes it a simple task to determine directions in phase space of minimal uncertainty. It has a nice geometric interpretation in terms of Poincaré invariants, and it is intimately related with symplectic topology and Gromov’s non-squeezing theorem. It is invariant under linear symplectic transformations (unlike the more frequently used Heisenberg uncertainty relation). For a Gaussian measure G it is both a necessary and sufficient condition for G to be a Wigner distribution. Nevertheless it has many interesting features. It can be shown that condition ( 4) is a necessary but not sufficient condition for a phase space function to be a Wigner distribution.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |