1. A particle moving in one dimension has a wave function (A is a constant) v(x) = 1 (2πA2)1/4 exp(-x²/4A²) (a) Show that the wave function is correctly normalised. (b) Using the normalised momentum wave function, show that the probability that the particle has linear momentum in the range p to p +dp is P(p) dp where P(p) = = 1/2 Δ h exp(-2p²²/h²) (c) Show that the product of the uncertainties in the position and momentum has the the minimum value allowed by the uncertainty principle. By definition, the standard deviation of the square modulus of the position wave function is A. Hints: [*exp(-ar²)dr πT = a [*exp(-x²/4^²)exp(-ipx/h)dr = exp(−p²^²/h²) c{p) = {ſp|¥) = A [∞ e¯ipz/ħy(x,t)dx = Þ(p,t) L е

1. A particle moving in one dimension has a wave function (A is a constant) v(x) = 1 (2πA2)1/4 exp(-x²/4A²) (a) Show that the wave function is correctly normalised. (b) Using the normalised momentum wave function, show that the probability that the particle has linear momentum in the range p to p +dp is P(p) dp where P(p) = = 1/2 Δ h exp(-2p²²/h²) (c) Show that the product of the uncertainties in the position and momentum has the the minimum value allowed by the uncertainty principle. By definition, the standard deviation of the square modulus of the position wave function is A. Hints: [exp(-ar²)dr πT = a [exp(-x²/4^²)exp(-ipx/h)dr = exp(−p²^²/h²) c{p) = {ſp|¥) = A [∞ e¯ipz/ħy(x,t)dx = Þ(p,t) L е