A trial function \(\psi\) differs from an eigenfunction \(u_{E}\) by a small amount, so that \(\psi=u_{B}+\epsilon \psi_{1}\), where \(u_{E}\) and \(\psi_{1}\) are normalized and \(\epsilon \ll 1 .\) Show that \(\langle H\rangle\) differs from \(E\) only by a term of order \(\epsilon^{2}\), and find this term.

In quantum mechanics, eigenfunctions are crucial for understanding systems at a fundamental level. An eigenfunction is a special type of wave function that satisfies the eigenvalue equation for an operator, usually the Hamiltonian operator. This means if you apply the Hamiltonian operator to an eigenfunction \(u_E\), you get the same function back, multiplied by a constant \(E\), which is the eigenvalue. Mathematically, this is written as: \[ H(u_E) = E u_E \].
Here, \(H\) is the Hamiltonian operator, representing the total energy of the system. \(u_E\) is the eigenfunction corresponding to an eigenvalue \(E\), which represents the quantized energy level of the system.
This concept is vital because quantum systems exhibit discrete energy levels, and understanding these levels helps in predicting how the system will behave. Eigenfunctions form the basis for this understanding.

The expectation value is a way to predict the average result of measurements on a quantum system. It is a weighted average and provides insight into the probable outcomes. The formula for the expectation value \(\langle A \rangle \) of an operator \(A\) when the system is in the state \(\psi\) is: \[ \langle A \rangle = \langle \psi | A | \psi \rangle \].
In simple terms, \(\langle A \rangle \) is the inner product of \(\psi\) and \(A \psi\). This tells us what we'd expect to measure for the observable associated with \(A\) if we made many identical measurements on identically prepared systems.
This concept helps bridge the gap between the abstract mathematical formulation of quantum mechanics and real-world measurements. For the exercise given, understanding the expectation value of the Hamiltonian is essential to see how small perturbations affect the system's energy.

Quantum mechanics often deals with complex systems that are not easily solvable. Perturbation theory helps approximate solutions by starting from a simple system with a known solution and adding a small 'perturbation'.
The idea in perturbation theory is to express the complex operator (like the Hamiltonian) as the sum of a solvable operator and a small perturbing term: \[ H = H_0 + \epsilon H' \]
where \(H_0\) is the simple, solvable Hamiltonian, and \(\epsilon H'\) is the small perturbation.
Usually, \(\epsilon\) is very small (\( \epsilon \ll 1 \)), allowing us to consider only the first-order perturbation. The total wave function and energy can also be expanded in a similar series.
For the provided exercise, we used a trial function \(\psi = u_E + \epsilon \psi_1\), where \(u_E\) and \(\psi_1\) are normalized. This approach shows how perturbation theory helps us understand small deviations from the eigenfunction and the resulting changes in expected energy.