Answer to Question #118511 in Quantum Mechanics for Moel Tariburu

Question #118511
For the infinite square well potential, calculate the probability that a particle in its second excited state is in each third of the one-dimensional box:
0≪x≪L/3, L/3≪x≪2L/3,2L/3≪x≪L,
Check to see that the sum of the probabilities is one.
1
Expert's answer
2020-05-28T13:38:32-0400

Wave function of particle in infinite square well potential is ψn(x)=2Lsin(πnLx)\psi_n(x) = \sqrt{\frac{2}{L}} \sin \left( \frac{ \pi n}{L} x\right).

Hence, for n=2n=2:

p(0xL/3)=0L/3ψ2(x)2dx=2L0L/3sin22πLxdx=13+38πp(0 \leq x \leq L/3) = \int_0^{L/3} |\psi_2(x)|^2 dx = \frac{2}{L}\int_0^{L/3} \sin^2 {\frac{2 \pi}{L} x} dx = \frac{1}{3} + \frac{\sqrt{3}}{8 \pi},

p(L/3x2L/3)=L/32L/3ψ2(x)2dx=2LL/32L/3sin22πLxdx=1334πp(L/3 \leq x \leq 2L/3) = \int_{L/3}^{2L/3} |\psi_2(x)|^2 dx = \frac{2}{L}\int_{L/3}^{2L/3} \sin^2 {\frac{2 \pi}{L} x} dx = \frac{1}{3} - \frac{\sqrt{3}}{4 \pi},

p(2L/3xL)=2L/3Lψ2(x)2dx=2L2L/3Lsin22πLxdx=13+38πp(2L/3 \leq x \leq L) = \int_{2L/3}^L |\psi_2(x)|^2 dx = \frac{2}{L}\int_{2 L/3}^{L} \sin^2 {\frac{2 \pi}{L} x} dx = \frac{1}{3} + \frac{\sqrt{3}}{8 \pi}.

The sum of probabilities is one, because terms with 3\sqrt{3} cancel out, and one thirds sum to one.


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