The missing piece of the jigsaw puzzle: probability conservation as a consequence of time-reversal symmetry

Abstract

Physicists believe that there is a corresponding conservation law for each principle of symmetry. In standard textbooks on quantum mechanics, however, it is asserted that the time-reversal symmetry does not lead to any conservation law. There seems to be a missing piece of the jigsaw puzzle of the connection between the principles of symmetry and the conservation laws. In this paper, we resolve this difficulty by considering the time-reversal invariance of the transition probability a basic physical requirement of the time-reversal symmetry. We show that this requirement, together with the time-reversal covariance of the Schrödinger equation, guarantees the hermiticity of the Hamiltonian which ensures the probability conservation. Thus we complete the missing piece by demonstrating that the time-reversal symmetry leads to the probability conservation. The present paper can help undergraduate students gain deeper insight into the connection between the principles of symmetry and the conservation laws in quantum mechanics.