Extracting ground state in path integral

[geoduck]

Suppose you have the transition amplitude in the presence of a source $<q''t''|q't'>_{f}$

To extract the ground state, we change the Hamiltonian to $H-i\epsilon$ , because we can write:
$$|q't'>=e^{iHt'} |n><n|q> \rightarrow e^{iE_0t'} |0><0|q>=<0|q>e^{iHt'} |0>=<0|q> |0 t'> $$

where only the ground state survives when $t' \rightarrow -\infty$ due to the imaginary term we added. So we have what we want: $|0 t'>$

But shouldn't the above really be $|q't'>_{f=0}=<0|q> |0 t'>_{f=0}$?

It seems what we really need is:

$$|q't'>_f=<0|q>e^{iHt'-if(t')xt'} |0>=<0|q>|0t'>_f $$

 

[Avodyne]

I can't follow the equations. Please learn the tex symbols \langle and \rangle.

 

[geoduck]

Absolutely. I also made some mistakes. I hope this is more clear:

Suppose you have the transition amplitude in the presence of a source $\langle q''t''|q't' \rangle_{f}$.

For example in terms of path integrals $\langle q''t''|q't' \rangle_{f} =\int [dx(t)]\, e^{i\{ L+f(t)x(t)\}dt}$

To extract the ground state, we change the Hamiltonian to $H-i\epsilon$ , because we can write:
$$|q't' \rangle=e^{iHt'} \sum_n |n \rangle \langle n|q' \rangle \rightarrow e^{iE_0t'} |0 \rangle \langle 0|q' \rangle= \langle 0|q' \rangle e^{iHt'} |0 \rangle= \langle 0|q' \rangle |0 t' \rangle $$

where only the ground state survives when $t' \rightarrow -\infty$ due to the imaginary term we added. So we have what we want: $|0 t' \rangle$ instead of $|q't' \rangle$, up to a coefficient of $\langle 0|q' \rangle$.

But shouldn't the relationship we just derived really be $|q't' \rangle_{f=0}= \langle 0|q' \rangle |0 t' \rangle_{f=0}$?

It seems what we really want is this relationship:

$$|q't' \rangle_f= \langle 0|q' \rangle e^{i \int \{H-f(t)x\} dt} |0 \rangle= \langle 0|q' \rangle |0t' \rangle_f $$

as it'll allow us to write:

$\langle q''t''|q't' \rangle_{f} =[\langle q''|0 \rangle \langle 0|q' \rangle ] *\, \,_f \langle 0 t'' |0t' \rangle_f$

Does the $H \rightarrow H-i\epsilon\$ work if your Hamiltonian includes a source?