Question

part (e),(f) and (g)

3. Wave functions (40 marks) Consider particle described by wave function (x) = Ce-x/a for x > 0, and otherwise (C is a real and non-negative). (a) Normalise (x) and plot it (you can use a computer to plotit). (b) Calculate the probability that the particle is located within distance a from the origin. (e) Find mean value of position measurement. (d) Find mean value of momentum measurement. Hint: use the fact that (x) is purely real, what can you deduce about the mean of momentum in such a case? (e) Calculate position variance. (f) Calculate momentum variance. (g) Is the Heisenberg uncertainty relation saturated?

Answer 1

for any quantity'n': Variance is <12) - <272, 4 della for Ondo and o otheumise on Normalization (C= 1/5a); (e) Position valuiance a) <h²> - <^7²; <by Turgydyn lei" jom za dom, hutting an at; draadt; cha) la Ligy egte 914x2! un Using Gamma La function 1e22"-1.nl; <w- le fazer em kvalificeret Position variance a² - (a)

(1) Momentum variance a <$2>- <$T; pritha ; 84 h 22h Cole, the year h2 cena: 42 X o Yo with a cola, ita con las (pas. Juk ne dhe ich faz malo da S T 05-4:3)[- t2 Qa² <p) - Wro de 11² ita cale de Momentun Vamance t te 23 2a2 ua² ? 3 t² 4 a² 1g) using Hupia on op at: x = los variance op- mann. Variance An ap > t =) / 307 x 32 = 3h which is the N 64 16 yaz Hup is valid / Hauce