The Eigenvalues of any Observable $$\hat{L}$$ must be Real

An eigenvector of any operator $$\hat{M}$$ is defined as simply a special particular vector $$|λ_M⟩$$ such that the only effect of $$\hat{M}$$ acting on $$|λ_M⟩$$ is to span the vector by some constant $$λ_M$$ (called the eigenvalue of $$\hat{M}$$). Any eigenvector $$|λ_M⟩$$ and eigenvalue $$λ_M$$ of $$\hat{M}$$ is defined as

$$\hat{M}|λ_M⟩=λ_M|λ_M⟩.$$

Thus the eigenvectors and eigenvalues of any observable $$\hat{L}$$ is defined as

$$\hat{L}|L⟩=L|L⟩.\tag{9}$$

Let’s take the Hermitian conjugate of both sides of Equation (9) to get

$$⟨L|\hat{L}^†=⟨L|L^*.\tag{10}$$

According to Principle 1, any observable $$\hat{L}$$ is Hermitian. Since any Hermitian operator $$\hat{H}$$ satisfies the equation $$\hat{H}=\(\hat{H}^†$$, it follows that $$\hat{L}=\hat{L}^†$$ and we can rewrite Equation (10) as

$$⟨L|\hat{L}=⟨L|L^*.\tag{11}$$

Let’s multiply both sides of Equation (9) by the bra $$⟨L|$$ and both sides of equation (11) by the ket $$|L⟩$$ to get

$$⟨L|\hat{L}|L⟩=L⟨L|L⟩\tag{12}$$

and

$$⟨L|\hat{L}|L⟩=L^*⟨L|L⟩.\tag{13}$$

Subtracting Equation (12) from Equation (13), we get

$$⟨L|\hat{L}|L⟩-⟨L|\hat{L}|L⟩=0=(L^*-L)⟨L|L⟩.\tag{14}$$

In general, $$⟨L|L⟩≠0$$ and is equal to a number $$z$$. Therefore it follows that, in general, $$L^*-L=0$$ and $$L^*=L$$. In order for a number $$z$$ to equal its own complex conjugate, it must be real with no imaginary part. Thus, in general, the eigenvalue $$L$$ of any observable $$\hat{L}$$ is always real. This is a good thing; otherwise Principle 3 wouldn’t make any sense. Recall that principle 3 states that the possible measured values of any quantity (i.e. charge, position, electric field, etc.) are the eigenvalues $$L$$ of the corresponding observable $$\hat{L}$$. But the measured values obtained from an experiment are always real; in order for Principle 3 to be consistent with this fact, the eigenvalues $$L$$ better be real too.