Question

) The wavefunction for a particle is defined by:
ψ (x)={ N cos(2πx/L) for - L/4 ≤x≤ L/4
0 otherwise


Determine
i) the normalization constant N, and
ii) the probability that the particle will be found between x = 0 and x = L / 8. (5+5)

Answer 1

Answer on Question #85010, Physics / Quantum Mechanics

Question:

The wavefunction for a particle is defined by:

$$\psi (x) = \left\{N \cos \left(2 \pi x / L\right) \text{ for } - L / 4 \leq x \leq L / 4 \right.$$

0 otherwise

Determine

i) the normalization constant $N$, and

ii) the probability that the particle will be found between $x = 0$ and $x = L / 8$. (5+5)

Solution:

By entering $t = 2x + 0.5L$, we get its range $[0;L]$ and the wavefunction $\psi (t) = N\sin (\pi t / L)$. In this case

$$N = \sqrt{\frac{2}{L}} \quad \text{and} \quad \text{the probability}$$ $$p = \frac{0.75L - 0.5L}{L} - \frac{\sin 2\pi \cdot 0.75 - \sin 2\pi \cdot 0.5}{6.28} = 0.25 + 0.16 = 0.41$$

The answer:

The normalization constant $N = \sqrt{\frac{2}{L}}$

The probability $p = 0.41$

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