Suppose you have a state la) and an infinitesimal translation dr, the translation operator is given by T(dx) = 1 - idr, where p is a momentum operator. Show that (a) T(dx)T(dx') = T(dx + dx'). (b) T(dx)T¹ (dx) = 1. (c) In the position representation, T(dr) |a) = dr' \r') va (r' - dx). (d) In the continuum limit, 7(dx) = exp(-idx).
Suppose you have a state la) and an infinitesimal translation dr, the translation operator is given by T(dx) = 1 - idr, where p is a momentum operator. Show that (a) T(dx)T(dx') = T(dx + dx'). (b) T(dx)T¹ (dx) = 1. (c) In the position representation, T(dr) |a) = dr' \r') va (r' - dx). (d) In the continuum limit, 7(dx) = exp(-idx).
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Transcribed Image Text:Suppose you have a state la) and an infinitesimal translation dr, the translation
operator is given by
T(dx)=1-idx,
where p is a momentum operator. Show that
(a) T(dr)T(dx') = T(dx + dx').
(b) T(dx)T¹ (dx) = 1.
(c) In the position representation, T(dr) |a) = f dr' \r') a (x' — dr).
exp(-ifdx).
(d) In the continuum limit, T(dr) = exp
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