Why do q's form a complete commuting set of observables?

In summary, Dirac's book discusses the equation Pr = -iħ∂/∂qr, concluding that this linear operator satisfies the same commutation relations with q's and with each other as p's do. This leads to the understanding that q's must form a complete commuting set of observables. The statement that functions of q's and p's can be taken as functions of q's and iħ∂/∂q's and still commute with q's, unless they are solely dependent on q's, further supports this understanding. The equation ∂fr/∂qs = ∂fs/∂qr is also mentioned, showing that functions fr have the form ∂F/∂qr
  • #1
bikashkanungo
9
0
I have a doubt from Dirac’s book which states that , Pr =-iħ∂/∂qr
This equation was concluded after stating that the linear operator iħ∂/∂qr satisfy the same commutation relations with the q’s and with each other as p’s do.
Next it is stated that(Dirac, p-92) “ This possibility enables us to see that the q’s must form a complete commuting set of observables since it means that any function of the q’s and p’s could be taken to be a function of q’s and iħ∂/∂q ‘ s and then could not commute with all the q’s unless it is a function of the q’s only.
I could not understand the above statement. Can anyone help me in the explanation of it ??

Also there is stated that ∂fr/∂qs =∂fs /∂qr showing that the functions fr are all of the form :
fr = ∂F/∂qr, where F is independent of r . How is this equation concluded ??
 
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  • #2
That q's commute is assumed by hypothesis. That they form a CSCO, well, Dirac's heuristic argument is quite self explanatory. Q's commute with an arbitrary function of P and Q, iff the function is dependent of Q's only.

Please, see another good heuristic argument in Ballentine's book, i.e. his chapter on dynamics derived from the Galileo group and especially the appendix on the irreducibility of Q's and P's based on Schur's lemma.
 

What is Schrodinger's Representation?

Schrodinger's Representation, also known as the Schrodinger picture, is a mathematical framework used in quantum mechanics to describe the time evolution of a quantum system.

How does Schrodinger's Representation differ from other representations?

Unlike the Heisenberg picture, where the operators are time-dependent and the states are time-independent, in Schrodinger's Representation, the operators are time-independent and the states are time-dependent.

What is the Schrodinger equation?

The Schrodinger equation is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system. It is a partial differential equation that relates the time derivative of the state vector to the Hamiltonian of the system.

How is Schrodinger's Representation used in practice?

Schrodinger's Representation is used to calculate the time evolution of a quantum system and make predictions about the behavior of particles at different points in time. It is a useful tool for understanding quantum systems and designing experiments to test quantum phenomena.

What are the advantages of using Schrodinger's Representation?

Schrodinger's Representation allows for a more intuitive understanding of the quantum system, as the states are time-dependent and can be visualized more easily. It also simplifies calculations and makes it easier to solve certain problems in quantum mechanics.

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