Describe the variational method for estimating the ground state energy of a quantum system. Prove that an error of order $\epsilon$ in the wavefunction leads to an error of order $\epsilon^{2}$ in the energy.

Explain how the variational method can be generalized to give an estimate of the energy of the first excited state of a quantum system.

Using the variational method, estimate the energy of the first excited state of the anharmonic oscillator with Hamiltonian

$$H=-\frac{d^{2}}{d x^{2}}+x^{2}+x^{4}$$

How might you improve your estimate?

[Hint: If $I_{2 n}=\int_{-\infty}^{\infty} x^{2 n} e^{-a x^{2}} d x$ then

$$\left.I_{0}=\sqrt{\frac{\pi}{a}}, \quad I_{2}=\sqrt{\frac{\pi}{a}} \frac{1}{2 a}, \quad I_{4}=\sqrt{\frac{\pi}{a}} \frac{3}{4 a^{2}}, \quad I_{6}=\sqrt{\frac{\pi}{a}} \frac{15}{8 a^{3}}\right]$$