We have a spinor evolving according to the one-dimensional Hamiltonian , where


We want to find the Ground state Energy using the Variational Method.


We parameterise our Quantum states as where

  • .
  • describes the expected energy.
  • . We use this to simplify the calculations below.

We can write the denominator as

The 2nd derivative can be written as

The above allows us to calculate the numerator of via

Using results from the Table of integrals, we know that . This gives us the result

We write the Energy as

In order to find the minimal Energy we need to solve . We will use the quotient rule for differentiation in a specific form: . Applying this to the ratio above, where

We get from :

  • Solve the above for . We should get a connection to Hermite polynomials.