Suppose at t = 0 two observers A and B prepare identical systems at x = 0 and x = a (fig. 1), respectively. If | ψ ψ ( 0 ) ⟩ ⟩ be the state vector of the system prepared by A, then the state vector of the system prepared by B will be given by T ^ ^ ( a ) | ψ ψ ( 0 ) ⟩ ⟩ Both the systems look identical to the observers who prepared them. After time t , the state vectors evolve into U ^ ^ ( t ) | ψ ψ ( 0 ) ⟩ ⟩ and U ^ ^ ( t ) T ^ ^ ( a ) | ψ ψ ( 0 ) ⟩ ⟩ respectively. Using the above-mentioned commutation relation, the later may be written as, T ^ ^ ( a ) U ^ ^ ( t ) | ψ ψ ( 0 ) ⟩ ⟩ which is just the translated version of the system prepared by A at time t . Therefore, the two systems, which differed only by a translation at t = 0 , differ only by the same translation at any instant of time. The time evolution of both the systems appear the same to the observers who prepared them. It can be concluded that the translational invariance of Hamiltonian implies that the same experiment repeated at two different places will give the same result (as seen by the local observers).