List All Possible Combinations

jewels of mathematics (I)

Today we are speaking part Unbeschränktheit Über die Quantenmechanik der der Operatoren Helmut Wielandt, appeared in the Mathematische Annalen in 1949.

The title, translated into Italian, means "Sull'illimitatezza operators of quantum mechanics." What are we talking about? Simplifying a bit, in quantum mechanics all the variables become operators. And if two sizes A, B are the Fourier transform of the other, then they must be such that

 [A, B] = i \\ hbar 

where we have denoted [A, B] = AB-BA the commutator of two operators.

This relationship is called the canonical commutation relation, more here. Wintner proved in 1947, using a technique developed by Rellich in 1946, that two operators satisfying the commutation relations must be unlimited. This demonstration was, however complicated, and in 1949 gave an elementary proof of Wielandt this fact. How does it work?

Wielandt on the observation that if

 [A, B] = 1 then 

 [A, B ^ {n +1}] = (N +1) B ^ n 

The proof of this fact is a simple algebraic manipulation and with an easy induction.

From this fact, and the reverse triangle inequality operatoriale applied the standard of A and B, yields an estimate of the standard

 B (n +1)