Question

Here is my problem: LetA^Aᵇe an operator A^=ϵ(|1⟩⟨1|−|2⟩⟨2|+|1⟩⟨2|+|2⟩⟨1|)A^=ϵ(|1⟩⟨1|−|2⟩⟨2|+|1⟩⟨2|+|2⟩⟨1|) where|1⟩|1⟩and|2⟩|2⟩are eigenstates which orthogonal to each other, andϵϵis a constant Find the matrix representation of the operator^A^ From my understanding,|1⟩|1⟩and|2⟩|2⟩can be written as:⎡⎣⎢⎢⎢⎢100⋮⎤⎦⎥⎥⎥⎥[100⋮]⎡⎣⎢⎢⎢⎢010⋮⎤⎦⎥⎥⎥⎥[010⋮]respectively. ⟨1|⟨1|and⟨2|⟨2|can be written as:[100⋯][100⋯][010⋯][010⋯]respectively. are my understanding correct?

Answer 1

The matrix representation of the operator A^ in the basis of the orthogonal eigenstates |1⟩ and |2⟩ is a 2x2 matrix with the elements multiplied by the constant ε. It results in a matrix with elements [ε ε; ε -ε].

Step-by-step explanation:

To find the matrix representation of the operator Î , we need to express the operator in terms of the basis vectors |1⟩ and |2⟩. These basis vectors are orthogonal and can be represented in matrix form as:

• |1⟩ = [1 0]
• |2⟩ = [0 1]

The dual vectors (bra vectors) of these basis vectors are represented as ⟩ 1| = [1 0] and ⟩ 2| = [0 1], which correspond to the rows of the identity matrix. Utilizing these vectors, we rewrite the operator Î as a matrix:

Î = ε(|1⟩⟩ 1| - |2⟩⟩ 2| + |1⟩⟩ 2| + |2⟩⟩ 1|)

Applying the matrix multiplication, we obtain:

Î = ε


[1 1]
[1 -1]

This is the matrix representation of the operator with respect to the basis 1⟩, .