The wave function ψ(x) of a particle confined to 0 ≤ x ≤ L is given by ψ(x) = Ax. ψ(x) = 0 for x L. When the wave function is normalized, the probability density at coordinate x has the value A) . B) . C) . D) . E) .

Question

The wave function ψ(x) of a particle confined to 0 ≤ x ≤ L is given by ψ(x) = Ax. ψ(x) = 0 for x < 0 and x > L. When the wave function is normalized, the probability density at coordinate x has the value
A) . 
B) . 
C) . 
D) . 
E) .

Quizplus · Accepted Answer

To normalize the wave function, we need to determine the value of the constant A.

The normalization condition is given by:
∫|ψ(x)|^2dx = ∫(Ax)^2dx = A^2∫x^2dx = A^2(L^3)/3 = 1

Thus, A = (3/ L^3)^1/2.

The probability density is given by |ψ(x)|^2 = A^2x^2.

Substituting the value of A, we get:
|ψ(x)|^2 = (3/L^3)x^2.

Substituting the limits, we get:
∫0^L(3/L^3)x^2dx = (3/L^3)(L^3)/3 = 1.

Therefore, the probability density at coordinate x has the value (3/L^3)x^2, which is equivalent to option D (11eb6adc_eda1_dee0_b0cf_3b83a86cacbf_TB8245_11).

The Wave Function ψ(X) of a Particle Confined to 0