Commutation rule Heisenberg

We shall again postulate commutation rules which have the property that the equations of motion of the matter field and of the electromagnetic field are consequences of the Heisenberg equation of motion ... 

Now in quantum theory the description of a physical system in the Heisenberg picture for a given observer O is by means of operators Q, which satisfy certain equations of motion and commutation rules with respect to O s frame of reference (coordinate system x). The above notion of an invariance principle can be stated alternatively as follows If, when we change this coordinate frame of reference (i.e., for observer O ) we are able to find a new set of operators that obeys the same equations of motion and the same commutation rules with respect to the new frame of reference (coordinate system x ) we then say that these observers are equivalent and the theory invariant under the transformation x - x. The observable consequences of theory in the new frame (for observer O ) will then clearly be the same as those in the old frame. 

In the Heisenberg-type description the existence of such a unitary or anti-unitary operator U is inferred from the fact that the set of observable Q and Q satisfy the same commutation rules. 

Starting from the standard Heisenberg commutator [p, q] — —hi, it is readily shown [119] that the coefficients satisfy the following commutation rules... 

In this manner the mathematical formalism incorporates the important experimental result that not all physical observables can be known simultaneously for a quantum system, the incompatibility of a knowledge of the position and its conjugate momentum for a particle being an example of this behaviour. The non-compatibility of these two observables is enshrined in the Heisenberg commutation rules... 

Recognizing the fact that representation of observable quantum operators, in various bases, is made by (hermitic) matrices, Heisenberg had generalized the commutation rules to operators and thus to matrix level, while this way constructing the so-called quantum matrix mechanics. It is basically founded by the commutation rules among the coordinate [2] momentum [- ] matrices. 

However, the seeond (explicit) level of quantum theory in the Heisenberg matrix approach regards the evaluation of the diagonal components of the total eneigy matrix through employment of the commutation rule in the matrix forms specialized to actual harmonic oscillator case it looks equivalently like ... 

Projecting the Hamiltonian and other quantum-mechanical operators onto a finite basis sets has serious consequences, even if the basis is large. It can be shown that certain quantum-mechanical rules eo ipso cannot be represented in finite basis. This leads to serious inconsistencies inherent in practical quantum chemistry where the basis set is finite in nearly all calculations. One such example is given by the Heisenberg commutation rule between the coordinate operator q and the canonically conjugated momentum operator p ... 

Substituting these expressions into the Heisenberg commutation rule of Eq. (8.19) we obtain ... 

I use temporarily roman n, m to include zero.) Show that the Heisenberg operators an(t), a (t) obey the same rules provided they are taken at the same f. The commutation relations of two of these operators taken at different times are not simple they involve the solution of the equations of motion. 

It is important to note that, of the seven rules given above, fiiis seventh rule should be stressed as the most basic matrix property, since as a natural consequence of this method of multiplication, we find that matrix multiplication is not necessarily commutative. That is, AB does not necessarily equal BA. This basic property of matrices ultimately led Werner Heisenberg and Max Bom to what we now refer to as the Heisenberg uncertainty principle. 

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