The Variational Principle

Also known as the Ritz theorem.

From $H | n ⟩ = E_{n} | n ⟩$ . We have $| ψ ⟩ = \sum_{n} c_{n} | n ⟩$ . Thus, $⟨ ψ | H | ψ ⟩ = \sum_{n} | c_{n} |^{2} E_{n}$ and $⟨ ψ | ψ ⟩ = \sum_{n} | c_{n} |^{2}$ . The weighted average is then, $⟨ H ⟩ = \frac{⟨ ψ | H | ψ ⟩}{⟨ ψ | ψ ⟩} = \frac{\sum_{n} | c_{n} |^{2} E_{n}}{\sum_{n} | c_{n} |^{2}}$ . Note that, $⟨ ψ | H | ψ ⟩ \geq E_{0} \sum_{n} | c_{n} |^{2}$ since that would assume all energies are the ground state at best and there exist larger energies at worst. Thus, $⟨ H ⟩ \geq E_{0}$ .

Choose a trial ket $| ψ (α) ⟩$ , $α$ is called a Ritz parameter
Compute $⟨ H ⟩ (α)$
Minimize with respect to $α$ , ${\frac{\partial ⟨ H ⟩}{\partial α} |}_{α = α_{0}} = 0$ .
Thus, $⟨ H ⟩ (α_{0})$ is the lowest upper bound for the energy of the ground state for that trial function.

Example: Simple Harmonic Oscillator

$H = - \frac{ℏ^{2}}{2 m} \frac{d^{2}}{d x^{2}} + \frac{1}{2} m ω^{2} x^{2}$

Let $ψ_{α} (x) = \exp (- α x^{2})$ .
Then, $⟨ ψ_{α} | H | ψ_{α} ⟩ = \int_{R} \exp (- α x^{2}) (- \frac{ℏ^{2}}{2 m} \frac{d^{2}}{d x^{2}} + \frac{m ω^{2} x^{2}}{2}) \exp (- α x^{2}) d x = (\frac{ℏ^{2}}{2 m} α + \frac{m ω^{2}}{8 α}) \sqrt{\frac{π}{2 α}}$ . $⟨ ψ_{α} | ψ_{α} ⟩ = \sqrt{\frac{π}{2 α}}$ .

Thus, $⟨ H ⟩ (α) = \frac{ℏ^{2}}{2 m} α + \frac{m ω^{2}}{8 α}$ .

$\frac{\partial ⟨ H ⟩}{\partial α} = \frac{ℏ^{2}}{2 m} - \frac{m ω^{2}}{8 α_{0}^{2}} = 0$ implies $α_{0}^{2} = \frac{m^{2} ω^{2}}{4 ℏ^{2}} = {(\frac{m ω}{2 ℏ})}^{2}$ .
The lowest upper bound is then, $⟨ H ⟩ = \frac{ℏ ω}{2}$ .

Example: SHO with a different Trial Function

$ψ_{α} = \frac{1}{x^{2} + α}$ with $α > 0$ .
Then, $⟨ ψ_{α} | H | ψ_{α} ⟩ = \frac{ℏ^{2}}{8 m} \frac{π}{α^{5 / 2}} + \frac{m ω^{2} π}{4 \sqrt{α}}$ . $⟨ ψ_{α} | ψ_{α} ⟩ = \frac{π}{2 α \sqrt{α}}$ . $⟨ H ⟩ (α) = \frac{ℏ^{2}}{4 m α} + \frac{m ω^{2} α}{2}$ .
$\partial_{α} ⟨ H ⟩ = - \frac{ℏ^{2}}{4 m α_{0}^{2}} + \frac{m ω^{2}}{2} = 0 \Rightarrow α_{0}^{2} = \frac{ℏ^{2}}{2 m ω}$ .
$\frac{ℏ ω}{\sqrt{2}}$ .