Write the matrices which produce a rotation θ about the x axis, or that rotation combined with a reflection through the (y,z) plane. [Compare (7.18) and (7.19) for rotation about the z axis.]

Explain the difference between time-dependent and independent SchrÖdinger's equations.

Show that (x,t)=Asin(kxt) and (x,t)=Acos(kxt) do not obey Schrödinger's time-dependent equation.

Compute |(x,t)|2 for the function (x,t)=(x)sint, where is a real constant.

Sketch the potential energy function of an electron in a hydrogen atom, (a) What is the value of this function at r=0 ? in the limit that r=? (b) What is unreasonable or inconsistent with the former result?

Show that when 1(x,t) and 2(x,t) are solutions to the time-dependent Schrödinger equation and A, B are numbers then a function (x,t) that is a superposition of these functions is also a solution: (x,t)=A1(x,t)+B1(x,t).

Show that (a) (b) (c)

Consider the van der Waals potential U(r)=U0[( R 0 r)122( R 0 r)6] , used to model the potential energy function of two molecules, where the minimum potential is at r=R0 . Find the force as a function of r. Consider a small displacement R=R0+r and use the binomial theorem: (1+x)n=1+nx+n( n1)2!x2+n( n1)( n2)3!x3+ , to show that the force does approximate a Hooke’s law force.

Is it possible for to be smaller than unity? `

Suppose the primed and laboratory observers want to measure the length of a rod that rests on the ground horizontally in the space between the helicopter and the tower (Fig. 39.8B). To derive the length transformation L = L (Eq. 39.5), we had to assume that the positions of the two ends were determined simultaneously. What happens to the length transformation equation if both observers measure the end below the helicopter at one time t1 and the other end at a later time t2?