Explain the variational method for computing the ground state energy for a quantum Hamiltonian.

For the one-dimensional Hamiltonian

$$H=\frac{1}{2} p^{2}+\lambda x^{4},$$

obtain an approximate form for the ground state energy by considering as a trial state the state $|w\rangle$ defined by $a|w\rangle=0$, where $\langle w \mid w\rangle=1$ and $a=(w / 2 \hbar)^{\frac{1}{2}}(x+i p / w)$.

[It is useful to note that $\left\langle w\left|\left(a+a^{\dagger}\right)^{4}\right| w\right\rangle=\left\langle w\left|\left(a^{2} a^{\dagger 2}+a a^{\dagger} a a^{\dagger}\right)\right| w\right\rangle$.]

Explain why the states $a^{\dagger}|w\rangle$ may be used as trial states for calculating the first excited energy level.