GATE 2022 PH – Question 34

Quantum Mechanics · one dimensional potentials: step potential, finite rectangular well, tunnelling from a potential barrier, particle in 1,2,3-dimensional box, particle in single and double delta function potentials, 1,2,3 dimensional harmonic oscillator: concept of degeneracy · 1 mark · Numerical answer

In a width-2a infinite well, $$\psi(x)=\frac{\sqrt2\sin(\pi x/a)+\sqrt3\cos(\pi x/(2a))+\cos(3\pi x/(2a))}{\sqrt{6a}}.$$ Find probability in n=2 as a percentage, nearest integer.

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Correct answer: 33

Explanation

For an infinite well of width $2a$ the energy eigenfunctions are $\psi_n(x)=\dfrac{1}{\sqrt a}\sin\dfrac{n\pi x}{2a}$ for $0<x<2a$.

$n=2$: $\psi_2=\dfrac{1}{\sqrt a}\sin\dfrac{\pi x}{a}$. This is the term in the given wave function.

**Write the given state with this eigenfunction.**
$$\psi=\frac{\sqrt2\sin(\pi x/a)+\cdots}{\sqrt{6a}}=\frac{\sqrt2}{\sqrt6}\cdot\frac{\sin(\pi x/a)}{\sqrt a}+\cdots=\frac{1}{\sqrt3}\,\psi_2+\cdots$$

The other terms, $\sqrt3\cos(\pi x/2a)$ and $\cos(3\pi x/2a)$, are combinations of the other eigenfunctions (odd $n$) and are orthogonal to $\psi_2$.

**Probability of finding the particle in $n=2$:**
$$P_2=\left|\frac1{\sqrt3}\right|^2=\frac13=33.3\%\approx\mathbf{33\%}.$$