GATE PH 2021 • Q61 NAT • 2 Marks

Section 4: Quantum Mechanics

Angular momentum & Hydrogen Atom

Subtopic: Hydrogen Atom Spectrum & Radial Wavefunctions

Question Statement:

The normalized radial wave function of the second excited state ($n = 3, l = 0$) of hydrogen atom is $R(r) = \frac{2}{(3a)^{3/2}} \left( 1 - \frac{2r}{3a} + \frac{2r^2}{27a^2} \right) e^{-r/3a}$ (or $R(r) = \frac{1}{\sqrt{24}} a^{-3/2} \left(2 - \frac{r}{a}\right) e^{-r/2a}$ for $n=2$). The distance at which the electron is most likely to be found is $y \times a$. The value of $y$ (in integer) is _

Official Answer Key:

4 to 4

Step-by-Step Solution & Explanation

### **Official Solution & Step-by-Step Derivation (GATE PH 2021 Q.61 / PH Q.51)** **1. Physical Principles & Examination Context**: - **Topic**: *Semiconductor Devices & Diodes* (`p-n Junction Diode, Zener Diode & Voltage Regulation`) | **Section**: *Section 8: Electronics* - **Question Type**: **NAT** | **Marks**: **2 Marks** | **Negative Marks**: **0** - **Governing Principles**: Solved directly by applying fundamental laws of **Semiconductor Devices & Diodes**. **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 **4 to 4**.

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