# 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