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MaestroBach

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- TL;DR Summary
- Reading the intro to a book on non equilibrium green's functions, where it defines the orthonormality relation of two kets in the continuum formulation. Not sure how to understand the dirac deltas.

I'm reading Stefanucci's Nonequilibrium Many Body Theory of Quantum Systems.

In the first chapter, where it goes over basic quantum mechanics, it first defines the usual orthonormality condition I'm familiar with,

$$\langle n' | n \rangle = \delta_{n, n'} $$

where $$ | n \rangle$$ is the ket describing a particle in the interval ##x_n \pm \Delta/2##

However, it then goes on to say that in the continuum formulation, the orthonormality relation becomes:

$$\langle n' | n \rangle = \lim_{\Delta\to0}\frac{\delta_{n, n'}}{\Delta} = \delta(x_{n'}-x_n) $$

I am not sure how to think about this, specifically in the case where ## x_{n'} = x_n ##

Obviously in all other cases, then the kronecker delta i'm used to is equivalent to the dirac delta, but in that one case, the book seems to be implying that instead of = 1, ##\langle n' | n \rangle = \infty##?

In the first chapter, where it goes over basic quantum mechanics, it first defines the usual orthonormality condition I'm familiar with,

$$\langle n' | n \rangle = \delta_{n, n'} $$

where $$ | n \rangle$$ is the ket describing a particle in the interval ##x_n \pm \Delta/2##

However, it then goes on to say that in the continuum formulation, the orthonormality relation becomes:

$$\langle n' | n \rangle = \lim_{\Delta\to0}\frac{\delta_{n, n'}}{\Delta} = \delta(x_{n'}-x_n) $$

I am not sure how to think about this, specifically in the case where ## x_{n'} = x_n ##

Obviously in all other cases, then the kronecker delta i'm used to is equivalent to the dirac delta, but in that one case, the book seems to be implying that instead of = 1, ##\langle n' | n \rangle = \infty##?