The coherent state |α > is defined as an eigenstate of the annihilation operator with an eigenvalue

| α> = |α >

Commutation relation between creation operator and annihilation operator [a, a^dagger] = 1 The Hermitian amplitude operators (essentially dimensionless position and momentum operators):

X1 = 1/2 (a^dagger + a), X2 = 1/(2) (a −a ^dagger)

Prove the uncertainties in coherent state

(∆X1)^2 =< |X1^2 | > −(< |X1 | >)^2 = 1/4

(∆X2)^2 =< |X2^2 | > −(< |X2 | >)^2 = 1/4