 In fact, the particle momentum around the circle - and the contribution to its energy - is of the form n/R (in standard units, for an integer n), so that at large R there will be many more states compared to small R (for a given maximum energy).A string, in addition to traveling around the circle, may also wrap around it.The number of times the string winds around the circle is called the winding number, and that is also quantized (as it must be an integer).