Write down the expression for the commutator [σi, σj] of two Pauli matrices. Show that the antic

Write down the expression for the commutator [σi, σj] of two...

Write down the expression for the commutator $\left[\sigma_{i}, \sigma_{j}\right]$ of two Pauli matrices. Show that the anticommutator of two Pauli matrices is
$$
\left\{\sigma_{i}, \sigma_{j}\right\}=2 \delta_{i j}
$$

Key Concepts

Anticommutator

The anticommutator, defined as {A, B} = AB + BA, is used to explore properties of operators under a symmetric product. For Pauli matrices, the anticommutator relation is given by {?_i, ?_j} = 2?_ij, which encapsulates the orthogonality and normalization conditions of these matrices. This property indicates that the square of any Pauli matrix is proportional to the identity matrix.

Kronecker Delta

The Kronecker delta, ?_ij, is a function of two discrete indices that equals 1 when the indices are equal and 0 otherwise. It is commonly used in mathematical expressions to represent orthogonality and acts as an identity in summations. In the anticommutator of the Pauli matrices, ?_ij is employed to succinctly express the condition that the matrices are orthogonal in the sense of their algebraic structure.

Pauli Matrices

Pauli matrices are a set of three 2x2 complex Hermitian matrices that serve as the generators of the SU(2) group. They play a central role in quantum mechanics, particularly in representing the spin operators for spin-half particles. Their algebraic properties, including specific commutation and anticommutation relations, are crucial for understanding the structure of angular momentum in quantum systems.

Commutator

The commutator is an operation defined for any two operators by [A, B] = AB - BA. It measures the degree to which two operators do not commute and is fundamental in quantifying the non-classical behavior of observables in quantum mechanics. In the context of the Pauli matrices, the commutator reveals the Lie algebra structure of SU(2) through expressions like [?_i, ?_j] = 2i ?_ijk ?_k, where ?_ijk is the Levi-Civita symbol.

Transcript

00:01 All righty, so today we are looking at, or rather here, we are looking at three matrices which come from quantum physics.

00:08 So we have sigma 1 equals 0 110, sigma 2 equals 0 i negative i 0, sigma 3 equals 1 0 negative 1.

00:16 And we have three equations that we need to verify using these three matrices.

00:21 So we just go through and do our matrix multiplication.

00:26 So for the product of two matrices, the ijth element, it's going to be.

00:30 Going to be the dot product of row i from the left -hand matrix with column j from the right -hand matrix.

00:38 So element 1 -1 here is going to be 0 times 0 plus 1 times i, which is i.

00:45 Element 2 -1 is going to be 1 times 0 plus 0 times i, that's just 0.

00:52 Element 1 -2, 0 times negative i, which is still 0, plus 1 times 0, 0, 0.

01:00 Element 2 -2 is going to be 1 times negative i plus 0 times 0 so we have negative i there and we need this we need to see if that's equal to i times sigma 3 well we can see we have a common factor of i here we can just factor that out so we have i times 1 0 0 negative 1 which is i times sigma 3 next up element 1 1 is going to be 0 times 1 plus negative i times 0 so that's going to be 0.

01:42 Element 2 1 is i times 1 plus 0 0 times 0.

01:45 That's just going to be i.

01:47 Element 1 2 is going to be 0 times 0 plus negative i times negative 1.

01:52 We have double negative that's going to be a positive i there.

01:56 And element 2 2 is going to be i times 0 plus 0 times negative 1 so that's 0...

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