From Ticcati's textbook, he asks to show that from the axioms of position operator we get that:
e−ia⋅P|x⟩=|x+a⟩
where the axioms are:
X=X†
If Δa is a space translation, then U(Δa)†XU(Δa)=X+a, where U is the representation of a unitary operator, we know that eia⋅PXe−ia⋅P=X+a.
If R is a space rotation then U(R)†XU(R)=RX.
Here's what I tried so far to do with this:
eia⋅PXe−ia⋅P|x+a⟩=(X+a)|x+a⟩=(x+a)|x+a⟩
X|y⟩:=Xe−ia⋅P|x+a⟩=(x+a)e−ia⋅P|x+a⟩=(x+a)|y⟩
Now I want to show somehow that |y⟩=|x+2a⟩, but I don't see how, any hints?
Thanks in advance.
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