Reference solution of the particle in a one-dimensional box model. For a box of length L, positioned between x=0 and x=L, we found a family of solutions to the time-independent Schrödinger Equation as follows:
ϕₙ(x)= 2/L sin(nπx/L) with n=1,2,3,4⋯ for 0≤x≤L Eₙ= n² h²/8mL² with n=1,2,3,4⋯For state n,ϕₙ (x) is the wavefunction and Eₙ is the energy (where h is Planck's constant and m is the mass of the particle). 1a. Calculate Eₙ in units of electron Volts for an electron in a box of length L=3.0 nm for n= 1,2,3, and 4 .
Using any convenient software (e.g., Excel or Matlab), plot Eₙ in units of electron Volts versus n for these four states and insert the plot into your assignment. 1b. Using any convenient software plot the functions ϕₙ(x) versus x for 0≤x≤L and L=3.0 nm and values of n=1,2,3, and 4 . The best option is to place them on the same graph, with either vertical offsets or different colors for clarity, but you can also make four separate plots if necessary. Be sure your plot(s) are clearly labeled and/or have legends so that it's clear which function is which. Be sure to specify the units and values on both axes of the plots. 1c. The probability density of the electron in state n is given by the function Pₙ(x)=∣ϕₙ(x)∣². Repeat part b, now showing the probability density of the particle between x=0 and x=L for L=3.0 nm and n=1,2,3, and 4 . Be sure to specify the units on both axes of the pl 1d. The probability density functions Pₙ(x) found in part c tell us the probability of finding the particle at any given position. ¹. What is the probability for the electron to be located at the position x=1 nm ) for an electron in the state ϕ3(x) ?