Give an account of the variational principle for establishing an upper bound on the ground state energy of a Hamiltonian $H$.

A particle of mass $m$ moves in one dimension and experiences the potential $V=A|x|^{n}$ with $n$ an integer. Use a variational argument to prove the virial theorem,

$$2\langle T\rangle_{0}=n\langle V\rangle_{0}$$

where $\langle\cdot\rangle_{0}$ denotes the expectation value in the true ground state. Deduce that there is no normalisable ground state for $n \leqslant-3$.

For the case $n=1$, use the ansatz $\psi(x) \propto e^{-\alpha^{2} x^{2}}$ to find an estimate for the energy of the ground state.