- #1
fog37
- 1,568
- 108
Hello Forum,
When a system is in a particular state, indicated by a |A>, we can use any basis of eigenvectors to represent it. Every operator that represents an observable has a set of eigenstates. I bet there are operators with only one eigenstate or no eigenstates. There are operators, like the evolution operator, that does not represent an observable.
The position operator has an infinite spectrum of eigenvalues and infinite eigenstates. When we see the wavefunction written as Psi(x), the position representation is implicit.
The energy operator (called Hamiltonian) also has an infinite spectrum and infinite eigenvectors.
The momentum operator has an infinite spectrum of eigenvalues and infinite eigenstates.
What about the angular momentum operator?
The spin operator has only two eigenvalues and two eigenstates, correct? How is it possible to represent any general state using only two eigenvalues and two eigenvectors?
So, in general, the generic state |A> can be written as a sum of any of those eigenvectors, correct?
What is the difference between a pure state and a mixed state?
Why do we call "1st quantization" the process of converting observables to operators? Does "2nd quantization" represent the conversion of the wavefunctions, i.e. states, to operators?
Thanks,
Fog37
When a system is in a particular state, indicated by a |A>, we can use any basis of eigenvectors to represent it. Every operator that represents an observable has a set of eigenstates. I bet there are operators with only one eigenstate or no eigenstates. There are operators, like the evolution operator, that does not represent an observable.
The position operator has an infinite spectrum of eigenvalues and infinite eigenstates. When we see the wavefunction written as Psi(x), the position representation is implicit.
The energy operator (called Hamiltonian) also has an infinite spectrum and infinite eigenvectors.
The momentum operator has an infinite spectrum of eigenvalues and infinite eigenstates.
What about the angular momentum operator?
The spin operator has only two eigenvalues and two eigenstates, correct? How is it possible to represent any general state using only two eigenvalues and two eigenvectors?
So, in general, the generic state |A> can be written as a sum of any of those eigenvectors, correct?
What is the difference between a pure state and a mixed state?
Why do we call "1st quantization" the process of converting observables to operators? Does "2nd quantization" represent the conversion of the wavefunctions, i.e. states, to operators?
Thanks,
Fog37