We present a procedure for a precise power spectral analysis of optical tweezers data. This procedure uses the entire frequency range of the experimental power spectrum. We apply this procedure to experimental data from a biological system, data for the motion of a microsphere attached with a biotin-streptavidin linker to a single protein, the λ-receptor of E. coli. As in [Oddershede et al, Biophys. J. 83, 3152 (2002)] and [Oddershede et al, J. Phys.: Condens. Mat. 15, S1737 (2003)], we find that the λ-receptor moves diffusively in the outer membrane, but is confined to a particular region by a harmonic potential. Here, we show that since the power spectrum is known with precision up to its Nyquist frequency of 8 kHz, one can resolve the relative motion of the λ-receptor and the microsphere attached to it. We find that the biotin-streptavidin link between them can be described as a Hookean spring. Since we can see only the microsphere, these results are based on an interpretation of the power spectrum of its motion. This interpretation is based on a model. We use the simplest model that is necessary and sufficient to interpret the power spectrum in its full range. This model fits the spectrum well and the fit yields the model's parameters with some precision. These results indicate that optical tweezers have the potential to become a tool of precision. We discuss improvements of experiments and analysis that are necessary in order to realize this potential.