In a 1D quantum system the position and momentum operators 2, p satisfy the standard commutation relations [î, p] = ih. iba (a) Evaluate the commutator [p, e*] where b is any real number. (b) Use your result to prove that if (p) is the momentum ket such that p|p) = p|p) then ibâ the state ep) is the shifted momentum ket |p+b). (c) Assuming that b is infinitesimally small in p+b) and using the fundamental definition of the momentum-space wavefunction, namely (p') = (p'|v) for any p', prove that a (p||v) = ih- др = a ih. -v(p). др

In a 1D quantum system the position and momentum operators 2, p satisfy the standard commutation relations [î, p] = ih. iba (a) Evaluate the commutator [p, e*] where b is any real number. (b) Use your result to prove that if (p) is the momentum ket such that p|p) = p|p) then ibâ the state ep) is the shifted momentum ket |p+b). (c) Assuming that b is infinitesimally small in p+b) and using the fundamental definition of the momentum-space wavefunction, namely (p') = (p'|v) for any p', prove that a (p||v) = ih- др = a ih. -v(p). др