We apply the scale-relativity theory of turbulence to the turbulent boundary layer problem. On the basis of Kolmogorov's scaling, the time derivative of the Navier-Stokes equations can be integrated under the form of a macroscopic Schr\"odinger equation acting in velocity-space. In this equation, the potential coming from pressure gradients takes the form of a quantum harmonic oscillator (QHO) in an universal way. From the properties of QHOs we can then derive the possible values of the ratio of turbulent intensities in the shear flow $R=\sigma_u/\sigma_v=1.35\pm0.05$. We show that the Karman constant is theoretically predicted to be $\kappa=1/R^3$, in good agreement with its typical value $\kappa\approx 0.4$ and its observed possible variations. Then we find a generic solution of our equations for the normal Reynolds stress pure profile, which closely fits the data from laboratory and numerical experiments. Its amplitude $\mu_B$ is solution of an implicit equation that we solve numerically and analytically through power series, yielding to lowest order $\mu_B-1.35\approx -2(R-1.35)$, plus smaller contributions from other parameters. Consequently the correlation coefficient of velocities is given by $\rho \approx 1/R \,\mu_B^2 \approx 1/R^3 \approx 0.4$ and is therefore equal to the Karman constant to lowest order, in agreement with its universally measured value $\approx0.4$ for all shear flows. We also find a general similarity between turbulent round jets and boudary layers in their outer region. These results therefore apply to a wide set of turbulent flows, including jets, plane boundary layers, and to some extent channels and pipes.