The wave function for a quantum particle is given byψ(x)=Aexp(−|x|a) where A and a = 1 are constants and −∞ < x < +∞.Hint: It will be useful to break any integration into 2 parts.Find the value of the normalisation constant A.Answer for part 1 Find the probability that the particle will be found in the interval −a < x < a.
Question
The wave function for a quantum particle is given byψ(x)=Aexp(−|x|a) where A and a = 1 are constants and −∞ < x < +∞.Hint: It will be useful to break any integration into 2 parts.Find the value of the normalisation constant A.Answer for part 1 Find the probability that the particle will be found in the interval −a < x < a.
Solution
To find the normalization constant A, we need to set the integral of the absolute square of the wave function over all space equal to 1. This is because the total probability of finding the particle somewhere in space must be 1.
The absolute square of the wave function is |ψ(x)|² = |Aexp(−|x|/a)|² = A²exp(−2|x|/a).
We break the integration into two parts, from -∞ to 0 and from 0 to +∞, because the absolute value function is discontinuous at x = 0.
∫ from -∞ to +∞ |ψ(x)|² dx = ∫ from -∞ to 0 A²exp(2x/a) dx + ∫ from 0 to +∞ A²exp(-2x/a) dx = 1.
Both integrals are equal, so we can write
2 ∫ from 0 to +∞ A²exp(-2x/a) dx = 1.
This integral can be solved by substituting u = 2x/a, du = 2 dx/a, dx = a du/2.
2 ∫ from 0 to +∞ A²exp(-u) (a/2) du = 1.
This is a standard integral, ∫ from 0 to +∞ exp(-u) du = 1, so
A² a = 1/2.
Therefore, A = sqrt(1/(2a)).
For the second part, the probability that the particle will be found in the interval -a < x < a is given by the integral of |ψ(x)|² from -a to a.
P(-a < x < a) = ∫ from -a to a |ψ(x)|² dx = ∫ from -a to a A²exp(-2|x|/a) dx.
Again, we break the integral into two parts, from -a to 0 and from 0 to a.
P(-a < x < a) = 2 ∫ from 0 to a A²exp(-2x/a) dx.
Substituting the value of A from the first part, we get
P(-a < x < a) = 2 ∫ from 0 to a (1/(2a))exp(-2x/a) dx = 1 - exp(-2).
So, the probability that the particle will be found in the interval -a < x < a is 1 - exp(-2).
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