Question

We have a particle with the following wave function:

Ψ(x) = √k * e^(-|x| / 2a)
where a = 3. Calculate the probability of finding the particle between x = 1 and x = 3. First, you have to normalize the function.

Answer 1

To calculate the probability of finding the particle between x = 1 and x = 3, we first need to normalize the given wave function Ψ(x). The normalization condition is ∫|Ψ(x)|² dx = 1. After calculating the normalization constant, we can integrate the square of the wave function over the specified interval to find the probability.

Step-by-step explanation:

To calculate the probability of finding the particle between x = 1 and x = 3, we first need to normalize the given wave function Ψ(x).

The normalization condition is given by: ∫|Ψ(x)|² dx = 1, where the integral is taken over all space.

Using the given wave function, we calculate the normalized constant k as:

∫|√k * e^(-|x| / 2a)|² dx = 1

∫k * e^(-|x| / a) dx = 1

k * ∫e^(-|x| / a) dx = 1

k * [∫e^(-x / a) dx + ∫e^(x / a) dx]

k * [(∫e^(-x / a) dx) + (-∫e^(x / a) dx)] = 1

k * [(-a * e^(-x / a)) + (a * e^(x / a))] = 1

2k * a * (e^(x / a) - e^(-x / a)) = 1

2k * a * sinh(x / a) = 1

k = (1 / (2a * sinh(x / a)))

Now that we have the normalized constant, we can calculate the probability of finding the particle between x = 1 and x = 3.

The probability is given by: ∫|Ψ(x)|² dx, where the integral is taken over the specified interval.

∫|(√k * e^(-|x| / 2a))|² dx

∫(k * e^(-|x| / a))² dx

∫(k^2 * e^(-2|x| / a)) dx

(k^2 / 2) * ∫e^(-2|x| / a) dx

(k^2 / 2) * {-0.5 * a * (e^(-2 * x / a)) - 0.5 * a * (e^(2 * x / a))}

(k^2 / 2) * {-0.5 * a * (e^(-2 * (3 / a) / a)) - 0.5 * a * (e^(2 * (3 / a) / a)) - (-0.5 * a * (e^(-2 * (1 / a) / a)) - 0.5 * a * (e^(2 * (1 / a) / a)))}

(k^2 / 2) * {-0.5 * a * (e^(-6 / a) - e^(-2 / a)) - (-0.5 * a * (e^(-2 / a) - e^(6 / a)))}

(k^2 / 2) * {-0.5 * a * (e^(-6 / a) - e^(-2 / a)) + 0.5 * a * (e^(-2 / a) - e^(6 / a))}

(k^2 / 2) * (-a * (e^(-6 / a) - e^(-2 / a)) + a * (e^(-2 / a) - e^(6 / a)))

(k^2 / 2) * (a * (-e^(-6 / a) + e^(-2 / a)) + a * (e^(-2 / a) - e^(6 / a)))

(k^2 / 2) * (a * (e^(-2 / a) - e^(-6 / a)) + a * (e^(-2 / a) - e^(6 / a)))

(k^2 / 2) * (2a * (e^(-2 / a) - e^(-6 / a)))

(2a * sinh(2 / a))^2 / (2 * 2a * (e^(-2 / a) - e^(-6 / a)))

(sinh(2 / a))^2 / (2 * (e^(-2 / a) - e^(-6 / a)))