Question

2. Schrodinger equation In quantum mechanics, physical quantities cor- respond to Hermitian operators. In particular, the total energy of the system corresponds to the Hamiltonian operator H, which is a hermitian operator The 'state of the system' is a time dependent vector in an inner product space, l(t)). The state of the system obeys the Schrodinger equation We assume that there are no time-varying external forces on the system, so that the Hamiltonian operator H is not itself time-dependent a) Take the Hermitian conjugate of the Schrodinger equation, which puts it into bra form (this is a simple one-liner; use the fact that H is hermitian) b) Show that( (t)|v(t)) is time independent, i.e (hint: use the product rule). This means that we can normalize the state vector at one time, say t = 0, so that(o(0)lp(0)) = 1, and it will stay nor- malized at all times, (v(t)l>(t))-1 c) Define the time-evolution operator where the exponential is defined by a Taylor expansion in powers of H it Show that U(t) is unitary: U(t)tU(t) = U(t)0(t)t = 1 d) Let f(H) be any function of H that can be defined by a Taylor expansion in powers of H. Show that H and f(H) commute e) find an expression for U (t f) Show that lo(t))-U(t)|39(0)) is a solution of the Schrodinger equation Here lv() is the initial state of the system, which we assume is given

Answer 1