What is the probability of finding a particle in a well of width ∝ at a position $$\frac{\alpha }{4}$$ from the wall if n = 1, if n = 2, if n = 3. Use the normalized wavefunction ψ(x,t) =$$\left ( \frac{2}{a} \right )^{\frac{1}{2}}$$ sin$$\left ( \frac{n\pi x}{a} \right )e^{-\frac{iEt}{h cut}}$$.

Solution :

Here is given, width of well = a

Position (x) = $$\frac {a}{4}$$

Normalized wave function $$\Psi$$ (x,t) = $$\sqrt \frac {2}{a}$$sin$$\frac {n\pi x}{a}$$ $$e^{- \frac {iEt}{ℏ}}$$

We know that, probability of finding a particle, P = $$\Psi\times\Psi$$

Then, at x = $$\frac {a}{4}$$

P = $$\left ( \sqrt \frac {2}{a} sin\frac{n\pi}{a}.\frac{a}{4}e^{\frac{iEt}{ℏ}} \right )$$$$\left ( \sqrt \frac{2}{a}sin\frac{n\pi}{a}.\frac{a}{4}e^{-\frac{iEt}{ℏ}} \right )$$

P = $$\frac{2}{a}sin^{2}\frac{n \pi}{4}$$…………………. (1)

If n = 1, P1 = $$\frac{2}{a}sin^{2}\frac{\pi}{4} = \frac {1}{a}$$

For n = 2, P2 = $$\frac{2}{a} sin^{2}\frac{2\pi}{4} = \frac{2}{a}$$

And for n = 3, P3 = $$\frac{2}{a}sin^{2}\frac{32\pi}{4} = \frac{2}{a}sin^{2}135^{\circ} = \frac{2}{a}\left ( \frac{1}{\sqrt2} \right )^{2}$$

∴ P3 = $$\frac{1}{a}$$

Hence, the probability of finding a particle in a well of width of a position x = $$\frac{a}{4}$$ form the wall for n =1, n = 2 and n = 3 are $$\frac{1}{a},\frac{2}{a} \enspace and \enspace \frac{1}{a}$$ respectively.