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Problems and Solutions in Quantum Mechanics
Substituting into the Heisenberg inequality, the latter then takes the form 1 |c − c∗ |2 ≼ 2(1 − |c|2 ) 16 Defining c = |c|eiβ , we arrive at sin2 β ≤
2 1 − |c|2
which is always true. (d) Since, as can be shown straightforwardly, a coherent state |z is an eigenstate of the annihilation operator a with eigenvalue z, we have on the one hand z|N |z = |z|2 ,
z|N 2 |z = |z|2 (|z|2 + 1)
and ( N )2 = |z|2 On the other hand, using the matrix element n|eiφ |n = δn,n +1 , we get z|eiφ |z = z e−|z|
2
∞ n=0
|z|2n √ n! n + 1
and z|e2iφ |z = z 2 e−|z|
2
∞ n=0
|z|2n √ n! (n + 1)(n + 2)
from which we obtain ∞ 1 |z|2n 2 z| cos φ|z = (z + z ∗ ) e−|z| √ 2 n=0 n! n + 1
and z| cos2 φ|z =
∞ |z|2n 1 1 2 2 + z + (z ∗ )2 e−|z| √ 2 4 n=0 n! (n + 1)(n + 2)
The resulting phase uncertainty is ( cos φ)2 =
∞ |z|2n 1 1 2 2 + z + (z ∗ )2 e−|z| √ 2 4 n=0 n! (n + 1)(n + 2) 2 ∞ 2n 1 |z| 2 − (z + z ∗ )2 e−2|z| √ 4 n=0 n! n + 1