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## Main Question or Discussion Point

I am reading Tinkham's "introduction to superconductivity" 1975 by McGraw-Hill, Inc.

Tinkham derives the BCS theory by canonical transformation. At the beginning of the chapter he writes:

"We start with the observation that the characteristic BCS pair interaction Hamiltonian will lead to a ground state which is some phase-coherent superposition of many-body states with pairs of Bloch states $(k\uparrow, -k\downarrow)$ occupied or unoccupied as units. Because of the coherence, operators such as $c_{-k\downarrow}c_{k\uparrow}$ can have nonzero expectation values b_k in such a state, rather than averaging to zero as in a normal metal, where the phases are random. "

I have some questions:

1. Is the characteristic BCS pair interaction Hamiltonian the so-called "reduced Hamiltonian"?

2. The BCS ground state is a state with pairs of Bloch states $(k\uparrow, -k\downarrow)$ either occupied or unoccupied, but what does the phase-coherence mean?

3. Why does it follow from coherence, that operators such as $c_{-k\downarrow}c_{k\uparrow}$ can have nonzero expectation values? When he writes phase, he means the phase of what?

It is the first time posting here, so bear with me.^^

Tinkham derives the BCS theory by canonical transformation. At the beginning of the chapter he writes:

"We start with the observation that the characteristic BCS pair interaction Hamiltonian will lead to a ground state which is some phase-coherent superposition of many-body states with pairs of Bloch states $(k\uparrow, -k\downarrow)$ occupied or unoccupied as units. Because of the coherence, operators such as $c_{-k\downarrow}c_{k\uparrow}$ can have nonzero expectation values b_k in such a state, rather than averaging to zero as in a normal metal, where the phases are random. "

I have some questions:

1. Is the characteristic BCS pair interaction Hamiltonian the so-called "reduced Hamiltonian"?

2. The BCS ground state is a state with pairs of Bloch states $(k\uparrow, -k\downarrow)$ either occupied or unoccupied, but what does the phase-coherence mean?

3. Why does it follow from coherence, that operators such as $c_{-k\downarrow}c_{k\uparrow}$ can have nonzero expectation values? When he writes phase, he means the phase of what?

It is the first time posting here, so bear with me.^^