A particle in a box (infinite square well) has the following stationary-state wave functions: -{VE sta where n = Yn(x) = 1, 2, 3, . . . . You will find the following trigonometric identities useful: sin²0 = (sin 0₁) (sin 0₂) = L sin (²) if 0 < x < L, otherwise, 1 (1 - cos 20), 2 1 [cos(0₁ − 0₂) − cos(01 + 02)]. Also notice that Eq. (3) is just a special case of Eq. (4). (a) Consider a particle assigned the wave function y(x) = ₁(x) for its position. Cal- culate the probability that the particle will be detected in the leftmost quarter of the box (i.e., 0 < x < ). Repeat this calculation for the three other quarters of the box (i.e., ¼ < x < ½, etc.). Express your answers in decimal form. Verify that the four probabilities you have calculated satisfy the laws of probability theory (see tutorials). (b) Repeat part (a) with the assigned wave function y(x) = √2₁(x) + ₂(x) instead. [Caution! Remember that |z1+z2|² ‡ |21|² + |z2|².]

A particle in a box (infinite square well) has the following stationary-state wave functions: -{VE sta where n = Yn(x) = 1, 2, 3, . . . . You will find the following trigonometric identities useful: sin²0 = (sin 0₁) (sin 0₂) = L sin (²) if 0 < x < L, otherwise, 1 (1 - cos 20), 2 1 [cos(0₁ − 0₂) − cos(01 + 02)]. Also notice that Eq. (3) is just a special case of Eq. (4). (a) Consider a particle assigned the wave function y(x) = ₁(x) for its position. Cal- culate the probability that the particle will be detected in the leftmost quarter of the box (i.e., 0 < x < ). Repeat this calculation for the three other quarters of the box (i.e., ¼ < x < ½, etc.). Express your answers in decimal form. Verify that the four probabilities you have calculated satisfy the laws of probability theory (see tutorials). (b) Repeat part (a) with the assigned wave function y(x) = √2₁(x) + ₂(x) instead. [Caution! Remember that |z1+z2|² ‡ |21|² + |z2|².]