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: Recall how observables are treated in quantum mechanics. 
In quantum mechanics, every physical quantity that can be measured is described using a linear operator.
When such an operator acts on a special function called its eigenfunction, the outcome is a number known as the eigenvalue.
This eigenvalue corresponds to a possible result of a physical measurement.

Step 2: Identify the relevant eigenvalue equation.
The time-independent Schrödinger equation has the form of an eigenvalue problem involving the Hamiltonian operator:

\[ \hat{H}\psi = E\psi \]

Here:

• $\hat{H}$ denotes the Hamiltonian operator.
• $\psi$ is the wave function associated with the system.
• $E$ is the eigenvalue obtained from the equation.

Step 3: Interpret the Hamiltonian operator.
The Hamiltonian represents the total energy of the system.
It consists of a kinetic energy term and a potential energy term, and is commonly written as:

\[ \hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + \hat{V} \]

Solving the Schrödinger equation yields a discrete or continuous set of allowed values for $E$.
These values correspond to the energies that the system can physically possess.

Step 4: Final conclusion.
The eigenvalues associated with the Hamiltonian operator give the total energy levels of the quantum system.