Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision | |||
quantum_integer [2014/12/10 12:02] nikolaj |
quantum_integer [2016/07/22 18:36] nikolaj |
||
---|---|---|---|
Line 15: | Line 15: | ||
The case $f=0$ is often considered. | The case $f=0$ is often considered. | ||
- | Quantum aspect: $f=n-1$ gives $[n]_{q^2}$=n+\mathcal{O}\left((q-1)^2\right)$. (The $q^2$ isn't necessary.) In the imaginary direction, $q\propto\mathrm{e}^{i\varphi}$, this corresponds to $\lim_{\varphi\to 0}\frac{\sin(n\varphi)}{\sin(\phi)}=n$. | + | Quantum aspect: $f=n-1$ gives |
+ | |||
+ | $[n]_{q^2} = n + \mathcal{O}\left((q-1)^2\right)$. | ||
+ | |||
+ | (The $q^2$ isn't necessary.) In the imaginary direction, $q\propto\mathrm{e}^{i\varphi}$, this corresponds to $\lim_{\varphi\to 0}\frac{\sin(n\varphi)}{\sin(\phi)}=n$. | ||
With $q=r\mathrm{e}^{i\varphi}$, along the positive real axis number $[n]_q$ is a valley with bottom at $q=1$, where $[n]_{1}=n$, and along $\varphi$ you have harmonic oscillations with period depending on $n$. | With $q=r\mathrm{e}^{i\varphi}$, along the positive real axis number $[n]_q$ is a valley with bottom at $q=1$, where $[n]_{1}=n$, and along $\varphi$ you have harmonic oscillations with period depending on $n$. | ||