Base on the Lecture note.[1]
Second Quantized States[edit]
Minimal Uncertainty States[edit]
Heisenberg uncertainty principle: for any Hermitian operator
and
and any state
, the following inequality holds
,
where
,
, and
.
The equality is achieved if and only if
is a solution of the minimal uncertainty equation
,
for any
. There is an one-to-one correspondence between the angle θ and the state
that minimize the uncertainty between
and
.
Coherent State[edit]
Displacement operator[edit]
Definition: for
,
.
Unitarity:
.
Action of displacement operator performs translation in the phase space
,
.
Applying to the vacuum state leads to the coherent state
, such that
.
Properties of Coherent State[edit]
Expansion in particle number representation
![{\displaystyle |\alpha \rangle =e^{-|\alpha |^{2}/2}\sum _{n=0}^{\infty }{\frac {\alpha ^{n}}{\sqrt {n!}}}|n\rangle }](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/dc6dca67c031073a16f845551f4f834b077d0045)
Overlap:
.
Completeness:
.
Squeezed State[edit]
Squeezing operator[edit]
Definition: for
,
.
Unitarity:
.
Action of squeezing operator performs the Bogoliubov transform
,
.
Applying to the vacuum state leads to the squeezed state
, such that
.
Reference[edit]