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
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]