Final answer:
In matrix notation, the three states corresponding to the possible measurement results of a quantum observable A are represented as column vectors |a1} = [1 0 0]¹, |a2} = [0 1 0]¹, and |a3} = [0 0 1]¹, with each vector indicating the state the system collapses to upon measurement.
Step-by-step explanation:
In the context of a quantum system with an observable A that has three possible measurement results in the states |a1}, |a2}, and |a3}, we can use the matrix notation to represent these states. Given that quantum states can be represented as vectors in a Hilbert space, the matrix representation for these states in a three-dimensional Hilbert space would be as follows:
- For the state corresponding to measurement result a1: |a1} = [1 0 0]¹ (Where the superscript ¹ symbolizes that this is a column vector.)
- For the state corresponding to measurement result a2: |a2} = [0 1 0]¹
- For the state corresponding to measurement result a3: |a3} = [0 0 1]¹
Each vector has a '1' in the position corresponding to the measurement outcome and '0's in all other positions. These vectors are often referred to as state vectors or kets in the Dirac or bra-ket notation. The values a1, a2, and a3 typically represent the eigenvalues associated with the observable A for each respective state. The act of measurement projects the quantum system into one of these states.