# Is the eigenstates of p right?

1. Aug 18, 2008

### 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?