a) Give a bra/ket expression of the operator N using outer products of its eigenkets |n> (hint: it has an infinite number of terms). b) Give a similar bra/ket expression (using outer products of eigenkets |n>) of the annihilation operator a (hint: think of the explicit general bra/ket form of the transformation operator U.) in which a does not explicitly appear. C++ map function. Maps are part of the C++ STL (Standard Template Library). Maps are the associative containers that store sorted key-value pair, in which each key is unique and it can be inserted or deleted but cannot be altered.

3D spatial variation, use the del (nabla) operator. •This is a vector operator •Del may be applied in three different ways •Del may operate on scalars, vectors, or tensors This is written in Cartesian ordinates Einstein notation for del Del Operator

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