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I have some basic questions concerning operators. What is actually meant by the following:

1) The dimensionality of an operator? E.g., what does it mean to say that the operator K has the dimension of 1/length (an example from Sakurai's book)? Operators act on abstract mathematical states to produce other states - how can you ascribe a dimension to such a quantity?

2) The derivative of an operator? Like dA/dt, where A is an operator. Can anyone offer an intuitive explanation?

3) An arbitrary function applied to an operator? Like exp(A) where A is an operator. In this case we can write exp(A) = 1 + A + A^2/2 + A^3/3 + ... - is this how you define a function of an operator in the general case, by using the taylor expansion?

1) The dimensionality of an operator? E.g., what does it mean to say that the operator K has the dimension of 1/length (an example from Sakurai's book)? Operators act on abstract mathematical states to produce other states - how can you ascribe a dimension to such a quantity?

2) The derivative of an operator? Like dA/dt, where A is an operator. Can anyone offer an intuitive explanation?

3) An arbitrary function applied to an operator? Like exp(A) where A is an operator. In this case we can write exp(A) = 1 + A + A^2/2 + A^3/3 + ... - is this how you define a function of an operator in the general case, by using the taylor expansion?

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