Question

1. Problems on unitary operators. For a function f(r) that can be expanded in a Taylor series, show that Here a is a constant, and pis the momentum operator. The exponential of an operator is defined as ea_ ??? i,O" Verify that the unitary operator elo/h can be constructed as follows (Hint: Notice that f(x +a) (al) and eohf())) e Prove that Here is the position operator. (Hint: You may work in the momentum space, in which p = p and = ih,, (see problem 2).) Show that is a solution of the Schrodinger equation i)) 2. The Hamiltonian of a 1D harmonic oscillator is given by H/2mmu/2. In the momentum! space (p-space), one has p = p and i = ih (Notie that plp) = PP), so p p: On the other hand, p,i, Define B-1/mhw. Show that the lowering and raising operators are given by 3 dp Find out the normalized ground state o(p) in the momentum space. . Let (x) be the ground state in the position space (see Lecture 2). Verify that The above equation is a standard fourier transformation. (In terms of dirac notation, the above equation reads (odprlpplo).)

Answer 1

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