Is the eigenstates of p right?

[xylai]

Quantum Arnold’s cat is a special system.
The Hamiltonian is H=p²+Kq²$$\delta$$₁(t)/2, where p$$\in$$(0,1],q$$\in$$(0,1].
The system is in an N-dimensional Hilbert space, where N=1/h.
Thus we can define : The eigenstates of $$\widehat{q}$$ are |j>, j=1,….,N, and the eigenstates of $$\widehat{p}$$ are |L>, L=1,…,N.
So $$\hat{q}$$|j>=$$\frac{j}{n}$$|j>, $$\hat{p}$$|L>=$$\frac{L}{N}$$ |L>.

Now let’s obtain the eigenstates of $$\hat{p}$$.
Because $$\hat{p}$$|L>=$$\frac{L}{N}$$ |L>, -ih$$\frac{d\psi(q)}{dq}$$=L/N$$\psi(q)$$.
Therefor the eigenstates of $$\hat{p}$$ is $$\psi(q)$$=exp(i2$$\pi$$Lq).

Is it right?

 

[eljose79]

I appreciate your attempt to find the eigenstates of \hat{p} for the given Hamiltonian. However, I would like to point out a few concerns with your approach.

Firstly, the Hamiltonian you have provided is not complete. It is missing the potential term, which is crucial in determining the eigenstates of a system. Without the potential term, we cannot accurately determine the energy levels and eigenstates of the system.

Secondly, the expression \hat{p}|L>=\frac{L}{N} |L> is incorrect. The correct expression is \hat{p}|L>=p_L|L>, where p_L is the momentum corresponding to the eigenstate |L>. This momentum can take on any value in the range of (0,1], not just discrete values of L/N.

Furthermore, your approach of using the position and momentum operators to determine the eigenstates may not be applicable for this particular Hamiltonian. The Hamiltonian you have provided is for a one-dimensional system, but the eigenstates you have obtained are in N-dimensional Hilbert space. This mismatch may lead to incorrect results.

In conclusion, I suggest revisiting your approach and considering the complete Hamiltonian with the potential term included. Additionally, it may be helpful to seek guidance from a quantum mechanics textbook or a qualified expert in the field to accurately determine the eigenstates of this system.

 

[denian]

I cannot definitively say whether the eigenstates of p are right or not without further context or evidence. However, based on the given information, it appears that the eigenstates of p are being derived using the Hamiltonian and the properties of the system described in the Quantum Arnold's cat scenario. The mathematical calculations seem to align with the definition of eigenstates and the properties of the system. Further analysis and experimentation would be needed to confirm the accuracy of these eigenstates.