Schwarz inequality: ⟨ α | α ⟩ ⟨ β | β ⟩ ≥ | ⟨ α | β ⟩ | 2 We substitute: | α ⟩ = ( ˆ A − ⟨ A ⟩ ) | ψ ⟩ | β ⟩ = ( ˆ B − ⟨ B ⟩ ) | ψ ⟩ into the inequality where the expectation values ⟨ A ⟩ = ⟨ ψ | ˆ A | ψ ⟩ and ⟨ B ⟩ = ⟨ ψ | ˆ B | ψ ⟩ are real numbers because the operators are Hermitian. Notice that ⟨ α | α ⟩ = 〈 ψ ∣ ∣ ∣ ( ˆ A − ⟨ A ⟩ ) 2 ∣ ∣ ∣ ψ 〉 = ⟨ ( ˆ A − ⟨ A ⟩ ) 2 ⟩ = (∆ A ) 2 ⟨ β | β ⟩ = 〈 ψ ∣ ∣ ∣ ( ˆ B − ⟨ B ⟩ ) 2 ∣ ∣ ∣ ψ 〉 = ⟨ ( ˆ B − ⟨ B ⟩ ) 2 ⟩ = (∆ B ) 2 The right-hand side of the Schwarz inequality for the states becomes ⟨ α | β ⟩ = ⟨ ψ | ( ˆ A − ⟨ A ⟩ )( ˆ B − ⟨ B ⟩ ) | ψ ⟩ For any operator ˆ O , we may write ˆ O = ˆ O + ˆ O † 2 + ˆ O − ˆ O † 2 = ˆ F 2 + i ˆ G 2 where ˆ F = ˆ O + ˆ O † and ˆ G = − i ( ˆ O − ˆ O † ) are Hermitian operators. If we take the operator ˆ O to be ( ˆ A − ⟨ A ⟩ )( ˆ B − ⟨ B ⟩ ), we find ˆ O − ˆ O † = [ ˆ A, ˆ B ] = i ˆ C and therefore ˆ G = ˆ C . Thus |⟨ α | β ⟩| 2 = ∣ ∣ ∣ ∣ 1 2 ⟨ ψ | ˆ F | ψ ⟩ + i 2 ⟨ ψ | ˆ C | ψ ⟩ ∣ ∣ ∣ ∣ 2 = |⟨ ψ | ˆ F | ψ ⟩| 2 4 + |⟨ ψ | ˆ C | ψ ⟩| 2 4 ≥ |⟨ C ⟩| 2 4 where we have made use of the fact that the expectation values of the Hermitian operators ˆ F and ˆ C are real. Combining this and the left-hand side from above, we obtain: (∆ A ) 2 (∆ B ) 2 ≥ |⟨ C ⟩| 2 4 or simply ∆ A ∆ B ≥ |⟨ C ⟩| 2 which is a very important result. If we apply this uncertainty relation to the specific commutation relation between the spin matrix, we find: ∆ J x ∆ J y ≥ ℏ 2 |⟨ J z ⟩| 1