If not all " N " particles are identical, but some of them are, then the function must be ( anti ) symmetrized separately over the variables corresponding to each group of identical variables, according to its statistics ( bosonic or fermionic ).
22.
Due to inherent limitations of the semiclassical ( stationary phase ) approximation the physically correct procedure is to use the non-symmetrized quantum versions of \ tilde { S } ( q, \ omega ) and \ tilde { P } ( q, \ omega ).
23.
M . Sc . ( Engg . ) Research in computer architecture for F . Remo Kilan Kumar in Supercomputer Education and Research Centre ( SERC ), IISc ( India ), Designs for VLSI Parallel Processor Arrays for Symmetrizing Hessenberg Matrices and Systolic Algorithms for Linear Systems around 1995.
24.
I am somewhat familiar with representations and characters of groups, and I know that by taking the Kronecker product of two representations of the same group, one ends up with a character which is the product of the two original characters, but I bumped into " symmetrized Kronecker product ".
25.
In fact, for the commutativity of the multiplication it is always possible to symmetrize it by forming a new kernel taken as the average of the kernels for all permutations of the variables \ tau _ { 1 }, . \, . \,, \ tau _ { p }.
26.
When symmetrizing " p " indices using ? to range over permutations of the numbers 1 to " p ", one takes a sum over the permutations of those indices \ alpha _ { \ sigma ( i ) } for, and then divides by the number of permutations:
27.
Then, even though we have to explicitely symmetrize or anti-symmetrize the total wavefunction, you can still say that there is a Gaussian peak somewhere that wasn't there before the molecule was put in and that extra Gauusian peak will move due to the time evolution described by the Schr�dinger equation.
28.
Then, even though we have to explicitely symmetrize or anti-symmetrize the total wavefunction, you can still say that there is a Gaussian peak somewhere that wasn't there before the molecule was put in and that extra Gauusian peak will move due to the time evolution described by the Schr�dinger equation.
29.
Similarly, the symmetric algebra can also be given the structure of a Hopf algebra, in exactly the same fashion, by replacing everywhere the tensor product \ otimes by the symmetrized tensor product \ otimes _ \ mathrm { Sym }, i . e . that product where v \ otimes _ \ mathrm { Sym } w = w \ otimes _ \ mathrm { Sym } v.
30.
This work culminated in the " Sharp and Shanks proof " that counterfactual interpretations of quantum spin measurements could not be reconciled with the validated predictions of quantum mechanics ( see W . D . Sharp and N . Shanks, 1993, " The Rise and Fall of Time-Symmetrized Quantum Mechanics ", Philosophy of Science, vol . 60 : 488-499 ).
