Ground states and Annihilation operators

1. The equation a | 0 \rangle = 0 means that the eigenvalue of a on | 0 \rangle is 0:

a | 0 \rangle = 0 | 0 \rangle

2. The length of the vector a | 0 \rangle is 0:

\langle 0 | a^\dagger a | 0 \rangle = 0

3. The physical meaning is that the probability of the system being at state a | 0 \rangle is 0.
 
In other words, there is no state with an eigen-energy lower than the ground state one.
 
 
4. For the equation a | 0 \rangle = 0 | 0 \rangle, the 0 at the right is a scalar.
 
 
5. For the equation a | 0 \rangle = 0, the 0 at the right is a zero vector – a state vector with length zero.
 
6. | 0 \rangle is a state vector. However, it is NOT the zero vector.

Instead, it is the state vector of the ground state. Its length is 1 unit.

— Me@2015-11-03 03:26:58 PM
 
 
 
2015.11.04 Wednesday (c) All rights reserved by ACHK