A Particle in a box In order to answer the following questions you will need toconsideran infinite potential well of length L, in which a particle has been confined.
1. For principal quantum number n=7, write code to calculatethenormalization constant. Then, plot the probability density function for this state (usingL= 10). 2. Using the properly normalized wave equation from Question1,write code to determine the expectation values <x> and <x2> . Is <x>oneof the most probable positions? Explain why. 3. Using the properly normalized wave equation from Question1,write code to determine the expectation values <p> and <p2> . Show the valueyouobtain for <p2> is equal to 2mE7 (where E7 is the energy of the n = 7 level). Remember that pisa differential operator so you will have to use the diff command. 4. The uncertainty of a quantum measurement is defined on page222,Equation 6.41 (3rd edition SMM). Using this equation calculate the uncertainties oftheposition and momentum of a particle in the n = 5 state. Compute the productofthese uncertainties and compare this with the prediction of the Heisenberguncertaintyprinciple. Quantum oscillator 5. Repeat calculations in the previous question for the groundstatewave function of the quantum oscillator. Comment on the obtained result.
(x) = A Sin( n pi x/L) En = n^2 pi^2 h^2/2mL^2 1. For principal quantum number n=7, write code to calculate thenormalization constant. Then, plot the probability density function for this state (using L= 10). 2. Using the properly normalized wave equation from Question 1,write code to determine the expectation values and . Is oneof the most probable positions? Explain why. 3. Using the properly normalized wave equation from Question 1,write code to determine the expectation values
and . Show the valueyou obtain for is equal to 2mE7 (where E7 is the energy of the n = 7 level). Remember that pis a differential operator so you will have to use the diff command. 4. The uncertainty of a quantum measurement is defined on page 222,Equation 6.41 (3rd edition SMM). Using this equation calculate the uncertainties ofthe position and momentum of a particle in the n = 5 state. Compute the product ofthese uncertainties and compare this with the prediction of the Heisenberg uncertaintyprinciple. Quantum oscillator 5. Repeat calculations in the previous question for the groundstate wave function of the quantum oscillator. Comment on the obtained result.
