- #1
Niles
- 1,866
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Hi
I am, at the moment, reading about propagators and Green's functions in QM. An example of a Green's function in some γ-basis is
[tex]
G(\gamma, t, t') = - i\left\langle {c_\gamma \left( t \right),c_\gamma ^ \dagger\left( {t'} \right)} \right\rangle
[/tex]
Now, if I expand this in terms of eigenstates [itex]\left| n \right\rangle [/itex]of the Hamiltonian, we obtain
[tex]
G(\gamma, t, t') \propto \sum\limits_{n,n'} {\left\langle n \right|c_\gamma ^{} \left| {n'} \right\rangle \left\langle {n'} \right|c_\gamma ^\dag \left| {n'} \right\rangle f\left( {E_n ,E_{n'} ,t,t'} \right)}
[/tex]
where f is some function (the exact form of the Green's function is not that important). So this is the expression for the Green's function in some γ-basis written in terms of the Hamiltonians basis states (i.e., the Hamiltonian representation).
My question is: I cannot see what we mean when we distinguish between the basis and the representation of an operator (or state). I think the basis refers to the quantum numbers and the representation refers to the basis states (i.e., the functional form of the expression). But if this is correct, then any arbitrary state, say [itex]{\left| \varepsilon \right\rangle }[/itex], is written in ε-basis and the x-representation (e.g.) is given by [itex]{\left\langle x \right.\left| \varepsilon \right\rangle }[/itex]?Niles.
I am, at the moment, reading about propagators and Green's functions in QM. An example of a Green's function in some γ-basis is
[tex]
G(\gamma, t, t') = - i\left\langle {c_\gamma \left( t \right),c_\gamma ^ \dagger\left( {t'} \right)} \right\rangle
[/tex]
Now, if I expand this in terms of eigenstates [itex]\left| n \right\rangle [/itex]of the Hamiltonian, we obtain
[tex]
G(\gamma, t, t') \propto \sum\limits_{n,n'} {\left\langle n \right|c_\gamma ^{} \left| {n'} \right\rangle \left\langle {n'} \right|c_\gamma ^\dag \left| {n'} \right\rangle f\left( {E_n ,E_{n'} ,t,t'} \right)}
[/tex]
where f is some function (the exact form of the Green's function is not that important). So this is the expression for the Green's function in some γ-basis written in terms of the Hamiltonians basis states (i.e., the Hamiltonian representation).
My question is: I cannot see what we mean when we distinguish between the basis and the representation of an operator (or state). I think the basis refers to the quantum numbers and the representation refers to the basis states (i.e., the functional form of the expression). But if this is correct, then any arbitrary state, say [itex]{\left| \varepsilon \right\rangle }[/itex], is written in ε-basis and the x-representation (e.g.) is given by [itex]{\left\langle x \right.\left| \varepsilon \right\rangle }[/itex]?Niles.