2.8. Variational theorem

Here, we derive the variational theorem, which states that the energy of any wavefunction \(\psi\) is always larger or equal to the true ground-state energy \(E_0\)

(2.144)\[E[\psi] \ge E_0\]

We start with a general wavefunction \(\psi\) that is not necessarily normalized. We write it as a linear combination of the complete orthonormal set of eigenfunctions \(\psi_i\) of the Hamiltonian (\(\hat{H}\psi_i = E_i\psi_i\) and \(\langle\psi_i|\psi_j\rangle=\delta_{i,j}\)):

(2.145)\[\psi = \sum_i c_i \psi_i\]

\(c_i\) are the linear-combination coefficients. The expectation value of the energy of \(\psi\) is

(2.146)\[E[\psi] = \frac{\langle\psi|\hat{H}|\psi\rangle}{\langle\psi|\psi\rangle} = \frac{ \sum_j\sum_i c_j^* c_i \langle\psi_j|\hat{H}|\psi_i\rangle} {\sum_j\sum_i c_j^*c_i \langle\psi_j|\psi_i\rangle}\]

\(\hat{H}\psi_i\) equals to \(E_i\psi_i\), so that we get

(2.147)\[E[\psi] = \frac{ \sum_j\sum_i c_j^* c_i \langle\psi_j|\psi_i\rangle E_i} {\sum_j\sum_i c_j^*c_i \langle\psi_j|\psi_i\rangle}\]

The set of \(\psi_i\) are orthonormal, i.e. \(\langle\psi_j|\psi_i\rangle = \delta_{j,i}\). With this, all terms in the two double sums except those with \(j=i\) vanish:

(2.148)\[E[\psi] = \frac{ \sum_j\sum_i c_j^* c_i \delta_{j,i} E_i} {\sum_j\sum_i c_j^*c_i \delta_{j,i} } = \frac{\sum_i|c_i|^2E_i}{\sum_i|c_i|^2}\]

Now comes the key step of the proof: The energy of any energy eigenstate is always larger or equal to that of the ground state, \(E_i\ge E_0\). Therefore, we can replace all \(E_i\) by \(E_0\) and, at the same time, replace the equal sign by \(\ge\). This yields

(2.149)\[E[\psi] \ge \frac{\sum_i|c_i|^2E_0}{\sum_i|c_i|^2}\]

After pulling \(E_0\) out of the sum, the sums in the enumerator and denominator cancel, so that the final result is

(2.150)\[E[\psi] \ge E_0 \frac{\sum_i|c_i|^2}{\sum_i|c_i|^2} = E_0\]

This proves the variational theorem.

(2.151)\[E[\psi] = \frac{ \sum_j\sum_i c_j^* c_i \delta_{j,i} E_i} {\sum_j\sum_i c_j^*c_i \delta_{j,i} } = \frac{\sum_i|c_i|^2E_i}{\sum_i|c_i|^2}\]

Now comes the key step of the proof: The energy of any energy eigenstate is always larger or equal to that of the ground state, \(E_i\ge E_0\). Therefore, we can replace all \(E_i\) by \(E_0\) and, at the same time, replace the equal sign by \(\ge\). This yields