Question

Show that ${\mathbf{L}}^2,{\mathbf{L}}_x,{\mathbf{L}}_y$, and ${\mathbf{L}}_z$ are hermitian operators in the space of square-integrable functions.

Short answer

The four operators ${\mathbf{L}}^2,{\mathbf{L}}_x,{\mathbf{L}}_y$, and ${\mathbf{L}}_z$ are all Hermitian because they satisfy the necessary condition for being Hermitian in the square-integrable function space.

Step-by-step solution

Definition of Hermitian Operator

For an operator $O$ to be classified as Hermitian, it should satisfy the following condition for all pairs of functions $f(x)$ and $g(x)$ in the square-integrable function space: $$\int f^{\ast }(x)O[g(x)]dx=\int [O^{†}f(x){]}^{\ast }g(x)dx$$ where $O^{†}$ represents the conjugate transpose of $O$, $f^{\ast }(x)$ is the complex conjugate of $f(x)$, and $g(x)$ is any function in the function space.

Verify ${\mathbf{L}}^{2}asHermitian$

Now, substitute the given operator ${\mathbf{L}}^{2}intotheaboveequation.Iftheresultingtwoequationsareequivalent,\text{(}{\mathbf{L}}^2$ is a Hermitian operator.

Verify ${\mathbf{L}}_x,{\mathbf{L}}_y$, and ${\mathbf{L}}_{z}asHermitian$

Repeat the same procedure in step 2 for the remaining three operators ${\mathbf{L}}_x$, ${\mathbf{L}}_y$, and ${\mathbf{L}}_z$.

Conclusion

If all four operators satisfy the condition for being Hermitian, then we can conclude they are Hermitian operators in the square-integrable function space.