For a one-dimensional quantum harmonic oscillator of mass $m$ and frequency $\omega$, which of the following statements is/are TRUE?
(A)
The energy eigenvalues are given by $E_n = \left(n + \frac{1}{2}\right)\hbar\omega$ for $n = 0, 1, 2, \dots$
(B)
The ground state energy is non-zero ($E_0 = \frac{1}{2}\hbar\omega$) due to the Heisenberg uncertainty principle
(C)
The ladder operators satisfy the commutation relation $[a, a^\dagger] = 1$
(D)
The ground state wavefunction has a node at $x = 0$