Question

The energy-momentum (E-K) relationship in a crystalline solid is usually obtained by solving :

• A. Maxwell equation
• B. Laplace equation
• C. Poisson equation
• D. Schrödinger equation

Hint:

Energy band diagrams in solids are obtained by solving the Schrödinger equation under periodic crystal potential conditions.

Answer 1

The Correct Option is D

Concept: In solid-state physics, the energy-momentum relationship: \[ E-k \] describes how electron energy varies with crystal momentum inside a periodic lattice. This relationship is obtained by solving the: \[ \text{Schrödinger equation} \] under periodic potential conditions.

Step 1: Understand the meaning of E-k relationship. The E-k diagram represents: \[ \text{Energy band structure of a crystal} \] where: \[ E = \text{electron energy} \] and \[ k = \text{wave vector} \] It explains:
• Allowed energy bands
• Forbidden energy gaps
• Electron motion inside solids

Step 2: Determine the governing equation. Electron motion at atomic scale is governed by: \[ \text{Quantum mechanics} \] The fundamental quantum mechanical equation is: \[ \text{Schrödinger equation} \] For electrons in periodic crystal potential: \[ -\frac{\hbar^2}{2m}\nabla^2\psi + V\psi = E\psi \] Solving this equation gives: \[ E-k \] relationship.

Step 3: Analyze other options. Option A: Maxwell equation Used for electromagnetic fields. Not used for electronic band structure. Option B: Laplace equation Used in electrostatics and potential theory. Not used for band structure calculations. Option C: Poisson equation Relates electric potential and charge density. Important in semiconductor electrostatics but not directly for E-k relationship.

Step 4: Write final answer. Thus the correct answer is: \[ \boxed{\text{Schrödinger equation}} \]