Consider an electron trapped in a box of side L, in one dimension. After making a measurement of the particle position, what is the probability of finding it within L/4 of either side of the box, if the particle is initially in:
a) the ground stationary state?
b) the first excited stationary state?
To determine the probability of finding the electron within a specific range after measuring its position, we need to consider the wavefunction of the particle and calculate the probability density.
a) For the ground stationary state, the wavefunction is given by:
ψ(x) = √(2/L) * sin(πx/L)
The probability density, P(x), is the square of the absolute value of the wavefunction:
P(x) = |ψ(x)|^2 = |√(2/L) * sin(πx/L)|^2
Simplifying this, we have:
P(x) = (2/L) * sin^2(πx/L)
To find the probability of finding the electron within L/4 of either side of the box, we integrate the probability density over that range and normalize it:
P(L/4 ≤ x ≤ 3L/4) = ∫[L/4,3L/4] [(2/L) * sin^2(πx/L)] dx
This integral evaluates to 1/2, so the probability of finding the electron within L/4 of either side of the box, given it is initially in the ground stationary state, is 1/2.
b) For the first excited stationary state, the wavefunction is given by:
ψ(x) = √(2/L) * sin(2πx/L)
Following the same steps as above, we find that the probability of finding the electron within L/4 of either side of the box, given it is initially in the first excited stationary state, is also 1/2.
Note:
The probability density function considers the normalized wavefunction, which ensures that the total probability over the entire range is equal to 1. By integrating the probability density function over a specific range, we can determine the probability of finding the particle within that range.