The wavefunction of a particle in a one-dimensional infinite well of size 2 a at a
certain time is 𝜓(x) =
1
√8 √2sin (
\pi
) +√3cos (
\pi
2) + cos (
^3\pi
2 ). Probability of
finding the particle in n = 2 state at that time is ______% (Round off to the nearest
integer)
Official Answer Key:
33 to 34
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.34)**
**1. Quantum Mechanics Formulation (Infinite Square Well)**:
- Potential well of width $2a$ centered at $x = 0$ ($-a \le x \le a$).
- Stationary state wavefunctions:
- Even states ($n = 1, 3, \dots$): $\psi_n(x) = \frac{1}{\sqrt{a}} \cos\left(\frac{n\pi x}{2a}\right)$
- Odd states ($n = 2, 4, \dots$): $\psi_n(x) = \frac{1}{\sqrt{a}} \sin\left(\frac{n\pi x}{2a}\right)$
- For the first excited state ($n = 2$):
$\psi_2(x) = \frac{1}{\sqrt{a}} \sin\left(\frac{2\pi x}{2a}\right) = \frac{1}{\sqrt{a}} \sin\left(\frac{\pi x}{a}\right)$
**2. Detailed Step-by-Step Derivation**:
- The given state at time $t$ is:
$\psi(x) = \frac{1}{\sqrt{6a}} \left[ \sqrt{2}\sin\left(\frac{\pi x}{a}\right) + \sqrt{3}\cos\left(\frac{\pi x}{2a}\right) + \cos\left(\frac{3\pi x}{2a}\right) \right]$
- Express $\psi(x)$ in terms of the orthonormal basis $\{\psi_n(x)\}$:
$\psi(x) = \frac{\sqrt{2}}{\sqrt{6}} \left[\frac{1}{\sqrt{a}}\sin\left(\frac{\pi x}{a}\right)\right] + \frac{\sqrt{3}}{\sqrt{6}} \left[\frac{1}{\sqrt{a}}\cos\left(\frac{\pi x}{2a}\right)\right] + \frac{1}{\sqrt{6}} \left[\frac{1}{\sqrt{a}}\cos\left(\frac{3\pi x}{2a}\right)\right]$
$\psi(x) = \frac{1}{\sqrt{3}} \psi_2(x) + \frac{1}{\sqrt{2}} \psi_1(x) + \frac{1}{\sqrt{6}} \psi_3(x)$
- Check normalization:
$\sum |c_n|^2 = \left|\frac{1}{\sqrt{3}}\right|^2 + \left|\frac{1}{\sqrt{2}}\right|^2 + \left|\frac{1}{\sqrt{6}}\right|^2 = \frac{1}{3} + \frac{1}{2} + \frac{1}{6} = \frac{2 + 3 + 1}{6} = 1$
- Probability of finding the particle in the $n = 2$ state:
$P(n=2) = |c_2|^2 = \left|\frac{1}{\sqrt{3}}\right|^2 = \frac{1}{3} \approx 33.33\%$
- Rounding to the nearest integer yields **33%**.
**3. Official Verification**:
The verified official IIT Master Answer range is **33 to 34**.
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