In recent inelastic neutron scattering experiments, Rüegg et al. investigated a pressure-induced magnetic quantum phase transition in the dimer spin system TlCuCl₃. The spin-triplet excitation spectra were measured on both sides of the quantum critical point, and the emergence of a well-defined, massive amplitude mode, or longitudinal excitation of the magnetic moment, demonstrated in the ordered phase.  We discuss the pressure-induced magnetic quantum phase transition and compare this with its field-induced counterpart. The splitting of the excitation mode gaps suggests a finite exchange anisotropy, which is shown to be of the easy-plane type; its quasi-uniaxial nature is responsible for the presence of one effectively massless spin-wave mode in the ordered phase. Also fitted is the pressure-dependence of the interdimer coupling parameters required to explain the vanishing of the gap. The longitudinal mode arises naturally at the quantum phase transition, and is reproduced with no additional parameters.