Question

The time-independent Schrödinger equation is fundamentally an eigenvalue equation for which operator?

• A. Momentum Operator
• B. Hamiltonian Operator
• C. Position Operator
• D. Parity Operator

Hint:

Hamiltonian = Total Energy. Whenever you see the time-independent Schrödinger equation \( H\psi = E\psi \), it is finding the eigenvalues of the Hamiltonian operator, which correspond to the allowed energy levels of the system.

Answer 1

The Correct Option is B

Concept: In the mathematical framework of quantum mechanics, physical observables are represented by Hermitian operators that act on states within a Hilbert space. When an operator acts on a wavefunction and yields the same wavefunction multiplied by a scalar constant, the relation forms an eigenvalue equation: \[ \hat{O} \psi = o \psi \] Where \( \hat{O} \) is the operator and \( o \) is the scalar eigenvalue.

Step 1: Examine the structure of the time-independent Schrödinger equation.
The time-independent Schrödinger equation is written as: \[ \hat{H}\psi = E\psi \] Here, the operator under study is \( \hat{H} \), which represents the Hamiltonian Operator.

Step 2: Define the role of the Hamiltonian.
The Hamiltonian operator corresponds to the total energy of the system. In one dimension, it is written explicitly as: \[ \hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) \]
• The first term, \( -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} \), represents the kinetic energy operator (\( \frac{\hat{p}^2}{2m} \)).
• The second term, \( V(x) \), represents the potential energy operator. When the Hamiltonian operator \( \hat{H} \) acts on a stationary state wavefunction \( \psi \), it yields the total energy value \( E \) as its eigenvalue. Therefore, the Schrödinger equation is fundamentally an eigenvalue equation for the Hamiltonian operator.