- #1
John Greger
- 34
- 1
Let's consider two observables, H (hamiltonian) and P (momentum).
These operators are compatible since [H,P] = 0.
Let's look at the easy to prove rule:
1: "If the observables F and G are compatible, that is, if there exists a simultaneous set of eigenfunctions of the operators F and G, then these operators must commute; [F , G] = 0."
This can be turned around to yield,
2: " If the operators F and G commute, then it is possible to find a simultaneous set of eigenfunctions."
How would the simultaneous eigenfunctions look like for H and P? Would it be something on matrix form?
Or is the following functions simply the "simultaneous eigenfunctions":
$$\hat{H} u_{n,m}(x) = E_n u_{n,m}(x)$$
$$\hat{P} u_{n,m}(x) = P_m u_{n,m}(x)$$
where ## \psi(x) = \Sigma_{n,m} C_{n,m} u_{n,m}(x)## ?
And I think the spectrum cannot be continuous right...
EDIT: How would I express a generic wave function as a superposition of the above eigenstates if they are "equivalent"?
These operators are compatible since [H,P] = 0.
Let's look at the easy to prove rule:
1: "If the observables F and G are compatible, that is, if there exists a simultaneous set of eigenfunctions of the operators F and G, then these operators must commute; [F , G] = 0."
This can be turned around to yield,
2: " If the operators F and G commute, then it is possible to find a simultaneous set of eigenfunctions."
How would the simultaneous eigenfunctions look like for H and P? Would it be something on matrix form?
Or is the following functions simply the "simultaneous eigenfunctions":
$$\hat{H} u_{n,m}(x) = E_n u_{n,m}(x)$$
$$\hat{P} u_{n,m}(x) = P_m u_{n,m}(x)$$
where ## \psi(x) = \Sigma_{n,m} C_{n,m} u_{n,m}(x)## ?
And I think the spectrum cannot be continuous right...
EDIT: How would I express a generic wave function as a superposition of the above eigenstates if they are "equivalent"?
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