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schrodinger equation help?
What is the probability of locating a particle of mass m between x = L/4 and x = L/2 in a 1-D box of length L? Assume the particle is in the n=1 energy state.
with steps plz
1 Answer
- 8 years agoFavorite Answer
The wavefunction for a particle in the n = 1 energy eigenstate is:
psi(x) = sqrt(2/L) sin(pi x/L)
The probability density is:
|psi|^2 = (2/L) sin^2 (pi x/L)
Thus, the probability that the particle is between x = L/4 and x = L/2 is:
P = integral (2/L) sin^2 (pi x/L) dx from x = L/4 to x = L/2
Let u = pi x/L. Thus:
P = (2/pi) integral sin^2 u from u = pi/4 to u = pi/2
P = (2/pi) [u/2 - (1/4)sin(2u)] from u = pi/4 to u = pi/2
P = (2/pi) [(1/2)(pi/4) - (1/4)(-1)]
P = 1/4 + 1/(2 pi)
