2.4.3 So­lu­tion math­ops-c

Ques­tion:

A less self-ev­i­dent op­er­a­tor than the above ex­am­ples is a trans­la­tion op­er­a­tor like ${\cal T}_{\pi /2}$ that trans­lates the graph of a func­tion to­wards the left by an amount $\pi$$\raisebox{.5pt}{$/$}$​2: ${\cal T}_{\pi /2}f(x)$ $\vphantom0\raisebox{1.5pt}{$=$}$ $f\left(x+\frac 12\pi\right)$. (Cu­ri­ously enough, trans­la­tion op­er­a­tors turn out to be re­spon­si­ble for the law of con­ser­va­tion of mo­men­tum.) Show that ${\cal T}_{\pi /2}$ turns $\sin(x)$ into $\cos(x)$.

An­swer:

Us­ing var­i­ous stan­dard trig ma­nip­u­la­tions, [1, pp. 43-44]:

$${\cal T}_{\pi /2}\sin(x)=\sin\left(x+{\textstyle\frac{1}{2}}\pi\right) = \cos(-x) = \cos(x).$$

Or just com­pare the graphs vi­su­ally, [1, p. 43].