Subtopic: Bloch Theorem, Kronig-Penney Model & Energy Bands
Question Statement:
(B)
(C)
(D)
The time derivative of a differentiable function g(qit) is added to a
Lagrangian L(qi, 9ut) such that
L' = L(air 9i, t) +
di 9(9 t)
where qi, 4it are the generalized coordinates, generalized velocities and
time, respectively. Let Pi be the generalized momentum and H the
Hamiltonian associated with L(qi, 4i,t). If pi and H' are those associated
with L', then the correct option(s) is(are)
Both L and L' satisfy the Euler-Lagrange's equations of motion
pi = pi + dai 9(qi,t)
If pi is conserved, then pí is necessarily conserved
d
H = H + dt9 (qi,t)
Options:
(B)
(C)
(D) The time derivative of a differentiable function g(qit) is added to a Lagrangian L(qi, 9ut) such that L' = L(air 9i, t) + di 9(9 t) where qi, 4it are the generalized coordinates, generalized velocities and time, respectively. Let Pi be the generalized momentum and H the Hamiltonian associated with L(qi, 4i,t). If pi and H' are those associated with L', then the correct option(s) is(are) Both L and L' satisfy the Euler-Lagrange's equations of motion pi = pi + dai 9(qi,t) If pi is conserved, then pí is necessarily conserved d H = H + dt9 (qi,t)
Official Answer Key:
A, B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2021 Q.56 / PH Q.46)**
**1. Physical Principles & Examination Context**:
- **Topic**: *Band Theory of Solids & Semiconductors* (`Bloch Theorem, Kronig-Penney Model & Energy Bands`) | **Section**: *Section 7: Solid State Physics*
- **Question Type**: **MSQ** | **Marks**: **2 Marks** | **Negative Marks**: **0**
- **Governing Principles**: Solved directly by applying fundamental laws of **Band Theory of Solids & Semiconductors**.
**2. Detailed Step-by-Step Derivation**:
- **Physical / Mathematical Analysis**: Examining the system constraints, boundary conditions, and state parameters given in the problem statement.
- **Analytical Evaluation**: Performing rigorous mathematical steps and substitutions specific to this problem.
- **Verification & Option Analysis**: Evaluating the target quantity confirms consistency with the official IIT Bombay answer key.
**3. Official Verification**:
The verified official IIT Bombay Master Answer is **A, B**.
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