Recently it has been found that for several two-dimensional classical models the amplitude of the correlation length of a finite system at a critical point is universal and related to the correlation-function exponent $\ensuremath{\eta}$. Here we propose that for quantum systems the ratios of two such amplitudes are universal and are related to the ratios of the corresponding exponents. This proposition is confirmed numerically for the transverse Ising model for spin
S=21,1,23,and2. It is used to calculate $\ensuremath{\eta}$ in a simplified way.
The phase transitions in the ground state of (1+1)-dimensional Ising models in a transverse field are investigated for spin
S=21,1,23,and2. The critical fields and "thermal" and correlation exponents are calculated as a function of
S by the finite-size scaling method. The universality of the transition with respect to
S is fully confirmed. Furthermore, we elaborate on a recent conjecture by Luck and others relating the correlation exponent $\ensuremath{\eta}$ to the amplitude of a correlation length for a two-dimensional (2D) classical isotropic system. The extension of this relation to quantum chains as extremely anisotropic 2D systems has been considered. We find that while no simple prescription can be given to calculate various correlation exponents from the amplitudes, the ratios of the amplitudes of different quantities still have universal character.