Question

The eigenvalues of the Hamiltonian operator represent the:

• A. Momentum of the system
• B. Energy of the system
• C. Position of the particle
• D. Probability density

Hint:

Remember the operator-observable pairs:
Hamiltonian \( \rightarrow \) Energy.
Momentum operator \( \rightarrow \) Momentum.
Position operator \( \rightarrow \) Position.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
In Quantum Mechanics, physical observables are represented by linear operators.
When an operator acts on its eigenfunction, the resulting scalar value is called an eigenvalue, which represents the measurable value of that observable.
Step 2: Key Formula or Approach:
The time-independent Schrödinger equation is an eigenvalue equation for the Hamiltonian operator \( \hat{H} \):
\[ \hat{H} \psi = E \psi \]
Where:
\( \hat{H} \) is the Hamiltonian operator (Total Energy operator).
\( \psi \) is the wave function (eigenfunction).
\( E \) is the eigenvalue.
Step 3: Detailed Explanation:
The Hamiltonian operator \( \hat{H} \) corresponds to the total energy of the system, consisting of kinetic energy and potential energy components:
\[ \hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + \hat{V} \]
The solution to the eigenvalue equation yields a set of allowed energy levels \( E \).
Therefore, measuring the Hamiltonian of a system in a state \( \psi \) will return one of these energy eigenvalues.
Step 4: Final Answer:
The eigenvalues of the Hamiltonian operator represent the total energy of the system.