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User:EverettYou/Second Quantization

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Base on the Lecture note.[1]

Second Quantized States

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Minimal Uncertainty States

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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

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Displacement operator

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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

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Expansion in particle number representation

Overlap:

.

Completeness:

.

Squeezed State

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Squeezing operator

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Definition: for ,

.

Unitarity: .

Action of squeezing operator performs the Bogoliubov transform

,
.

Applying to the vacuum state leads to the squeezed state , such that

.

Reference

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  1. ^ Leonid Levitov. "Strongly Correlated Systems in Condensed Matter Physics". MIT open course. Retrieved 2003. {{cite web}}: Check date values in: |accessdate= (help)