Physics Asked by completeStranger on January 5, 2021
I’m reading "An introduction to quantum fields on a lattice" by Jan Smit.
In chapter 2, the transfer operator ?̂ is defined and shown to be equal to
$$
hat{T} = e^{-omega^2 hat{q}^2/4} e ^{-hat{p}^2/2}e^{-omega^2 hat{q}^2/4}
$$
(There is another question about this section: Eigenvalue spectrum of the transfer operator for the harmonic oscillator)
With the usual coordinate representation:
$$hat{q} to q , ~~~ hat{p} to -i partial / partial q$$
The coordinate representation of the ground state is given by:
$$
langle q|0rangle = e^{- frac{1}{2} sinh tilde{omega} , q²}
$$
$ tilde{omega}$ and $omega$ are related via
$$ cosh tilde{omega} = 1 + frac{1}{2} omega^2$$
Now the question:
I don’t know how to derive this equation:
$$
hat{T} |0rangle = e^{-frac{1}{2} tilde{omega}}|0rangle
$$
What I thought of so far:
To bring $hat{T}$ to the form $hat{T}=e^{-hat{H}}$ by using the Baker-Campbell-Hausdorff formula.
But the exact BCH, $$ e^{X}e^{Y}=e^{{X+Y+[X,Y]/2}} $$ is not applicable here, since $[X,[X,Y]] neq 0$, for $X = hat{p}^2$ and $Y = hat{q}^2$.
Use the coordinate representation of $hat{T} |0rangle $, e.g.
$$ e^{-omega^2 hat{q}^2/4} e ^{-hat{p}^2/2}e^{-omega^2 hat{q}^2/4} e^{- frac{1}{2} sinh tilde{omega} , q²}, $$
where the last part is the ground state.
Then I use $ e^{-hat{p}^2/2} = 1 – frac{p²}{2} + frac{p^4}{8} + …$ and then insert the coordinate representation of the momentum operator, giving a ladder of derivatives of zeroth, then second, then fourth order… But this also does not yield the desired result.
I am thankful for any ideas!
Inserting complete sets of coordinate eigenstates yields finally the result:
$$ langle q |hat{T} |0rangle = int dq^{prime prime} int dq^{prime} langle q | e^{-omega^2 hat{q}^2/4} |q^{prime} rangle langle q^{prime} | e^{-hat{p}^2/2} |q^{prime prime} rangle langle q^{prime prime} |e^{-omega^2 hat{q}^2/4} |0rangle $$
using $$ langle q | e^{-omega^2 hat{q}^2/4} |q^{prime} rangle= e^{-omega^2 q^{prime 2}/4} delta(q-q^{prime}) $$ $$ langle q^{primeprime} | e^{-omega^2 hat{q}^2/4} |0 rangle= e^{-omega^2 q^{prime prime 2}/4} e^{- frac{1}{2} sinh tilde{omega} , q^{primeprime 2}} $$ and $$ langle q^prime | e^{-hat{p}^2/2} |q^{prime prime} rangle= sqrt{ frac{1}{2 pi}} e^{-frac{(q^prime-q^{prime prime})^2}{2}} $$ (can be shown by inserting complete set of momentum eigenstates). One arrives at the desired result $$ langle q |hat{T} |0rangle= e^{-frac{1}{2} tilde{omega}} e^{- frac{1}{2} sinh tilde{omega} , q^{ 2}} = e^{-frac{1}{2} tilde{omega}} langle q |hat{T} |0rangle $$
The only thing needed are Gauss integral and the relation $cosh tilde{omega} = 1 + frac{1}{2} omega^2$
Correct answer by completeStranger on January 5, 2021
Get help from others!
Recent Answers
Recent Questions
© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP