Proving Spin Commutation Relations: [Sx, Sy], [Sy, Sz], [Sz, Sx]

This problem involves proving the fundamental commutation relations for spin operators in quantum mechanics using Pauli matrices.

Answer

The problem asks to prove the commutation relations for spin operators:

These are the fundamental commutation relations for angular momentum operators, and spin is a form of angular momentum. To prove these relations, we typically use the Pauli matrices, which are representations of the spin-1/2 operators.

The spin operators

are related to the Pauli matrices

by:

The Pauli matrices are given by:

We need to calculate the commutators

1. Prove

First, let’s calculate the product

Now, calculate

So,

Next, let’s calculate the product

Now, calculate

So,

Now, calculate the commutator

Since

, we can write

This proves the first relation.

2. Prove

First, let’s calculate the product

Now, calculate

So,

Next, let’s calculate the product

Now, calculate

So,

Now, calculate the commutator

Since

, we can write

This proves the second relation.

3. Prove

First, let’s calculate the product

Now, calculate

So,

Next, let’s calculate the product

Now, calculate

So,

Now, calculate the commutator

We want to express this in terms of

.
Notice that

.
So,

Since

, we can write

This proves the third relation.

The final answer is

Proving Spin Commutation Relations: [Sx, Sy], [Sy, Sz], [Sz, Sx]

Question

Answer

Related math solutions

永久解锁网页版

Related math solutions