Question

The quantum operator for energy is:

• A. \(i\hbar\nabla\)
• B. \(-i\hbar\nabla\)
• C. \(i\hbar\,\partial/\partial t\)
• D. \(-i\hbar\,\partial/\partial t\)

Hint:

From \(\psi \sim e^{-iEt/\hbar}\), the time derivative gives \(E\psi = i\hbar\,\partial\psi/\partial t\).

Answer 1

The Correct Option is C

Step 1: Start from a plane-wave solution \(\psi = e^{i(kx - \omega t)}\), where the energy is \(E = \hbar\omega\).

Step 2: Differentiate with respect to time:
\[\frac{\partial \psi}{\partial t} = -i\omega\, \psi = -\frac{iE}{\hbar}\psi.\]
Step 3: Rearrange to isolate \(E\):
\[E\psi = i\hbar \frac{\partial \psi}{\partial t}.\]
Step 4: Hence the energy observable corresponds to the operator
\[\hat{E} = i\hbar \frac{\partial}{\partial t}.\]
This is exactly the operator appearing on the left of the time-dependent Schrodinger equation.
\[\boxed{\hat{E} = i\hbar\,\frac{\partial}{\partial t}}\]