The appearance of a unique function in the energy transfer from one system to the other in different physical situations such as electrical, mechanical, optical, and quantum mechanical processes is established in this work. Though the laws governing the energy transformation and its transfer from system to system are well known, here we notice a unity in diversity; a unique function appears in various cases of energy transfer whether it is a classical or a quantum mechanical process. We consider four examples, well known in elementary physics, from the fields of electricity, mechanics, optics, and quantum mechanics. We find that this unique function is in fact the transfer function corresponding to all these physical situations, and the interesting and intriguing finding is that the inverse Laplace transform of this transfer function, which is the impulse-response function of the systems when multiplied by a factor of -0.5, is the solution of a linear differential equation for an "instantly forced critically damped harmonic oscillator." It is important to note that though the physical phenomena considered are quite distinct, the underlying process in the language of impulse-response of the system in the time domain is a unique one. To the best of our knowledge we have not seen anywhere the above analysis of determining the unique function or its description as a transfer function in literature.