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3

A derivation is here: http://en.wikipedia.org/wiki/Lamb_shift#Derivation or in Landau-Lifshitz. Bethe's original derivation is found e.g. in Matt Schwartz's Harvard lecture here http://isites.harvard.edu/fs/docs/icb.topic792163.files/20-LambShift.pdf The leading contribution to the Lamb shift is the one-loop level (the first non-classical ...


2

This is incorrect. The Lamb shift makes the $2\,{}^2 P_{1/2}$ state lower than the $2\,{}^2 S_{1/2}$; as such, the former will not spontaneously decay into the latter. In any case, these are states within the $n=2$ shell, so that their main decay channel is to the ground state, $1\,{}^2S_{1/2}$; this can and will happen on a ~nanosecond timescale (i.e. slow ...


2

The scalar product is just a shortcut notation for multiplication and then addition: $$\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y + a_z b_z\tag{1}$$ It's commutative if the underlying multiplication is commutative, and otherwise it is not. Notation like $\vec{a}\cdot\nabla$ is not really a scalar product, but it takes the same form of (1) and applies it to ...


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This can be a little subtle the first time you see it, so I'll move through the rationale carefully: $\langle (\delta \mathbf{r} \cdot \nabla )^2 \rangle _{vac} = \langle (\delta x \ \partial_x + \delta y\ \partial_y + \delta {z}\ \partial_z )^2\rangle _{vac} $ On expanding you get squared and cross terms. The text mentions $\langle \delta \mathbf r ...



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