Chapter 12, Problem 12.51E

Interpretation Introduction

Interpretation:

The variation theorem for the given wavefunction is to be proved. The value of $\langle E\rangle$, by using $\varphi$ as the trial wavefunction, is to be determined. The expression $\langle E\rangle \ge E_1$ is to be stated by equaling $E_1$ if $\varphi$ is identically equal to ${\Psi }_1$ and greater than $E_1$ if $\varphi$ is not identically equal to ${\Psi }_1$.

Concept introduction:

Variation theory states that the average energy of any wavefunction for a given system is equal to or greater than its ground-state energy. It provides more accurate value of energy for a particular system. It is used to find the approximate solutions to the Schrödinger equation.

The Schrödinger equation is used to find the allowed energy levels for electronic transitions in the quantum mechanics. It is generally expressed as follows.

$\stackrel{⌢}{\text{H}}\Psi =\text{E}\Psi$

Where,

• $\stackrel{⌢}{\text{H}}$ is the Hamiltonian operator.

• $\Psi$ is the wavefunction.

• $\text{E}$ is the energy