Orthonormality

The orthonormality of the eigenfunctions of the Schroedinger equation are an essential attribute of the mathematical hardware of quantum mechanics. Here we consider the functions

                                            (Subscript[ϕ, k])(x) = (1/(Sqrt[a]))(e^ikx)
                           
defined over the interval -a/2 ≤  x ≤  a/2 that form an orthogonal set in the limit as ‘a’ goes to infinity. In this laboratory we study the attributes of continuous wave functions in quantum mechanics and apply our results to an example.

1. Three Dimensional Plotting

The command for making three dimensional plots is exhibited below. The option PlotPoints determines the density of points that are used to calculate the surface. If the function changes rapidly over the region of the function you may have to increase this number to get a reasonable picture.

f[x_,y_] := Sin[x*y/10]
Plot3D[f[x,y], {x, -7, 7},
               {y, -7, 7},
               PlotPoints->20,
               AxesLabel->{"x","y","f(x,y)"}
            ]

2. Questions and Calculations

As mentioned above we consider the functions

                                            (Subscript[ϕ, k])(x) = (1/(Sqrt[a]))(e^ikx)
                           
defined over the interval -a/2 ≤  x ≤  a/2 that form an orthogonal set in the limit as ‘a’ goes to infinity.

1. What are the conditions of orthonormality?

2. Calculate the inner product of these wave functions ⟨(Subscript[ϕ, k'])(x)|(Subscript[ϕ, k])(x)⟩ over the interval -a/2 ≤ x ≤ a/2. 

3. How is the previous result related to the orthonormality of the wave functions? (Hint: Express your result in terms of Δk = k - k’ and consider the cases (a) when Δk=0 with a→∞ and (b) when Δk≠0 with a→∞.

3. Letting the Size of the Box Go To Infinity

If we let the width of the box containing the particle go to infinity, then we should recover the orthonormality condition for the free particle. Why? Show why this occurs graphically. For the graphical part make three dimensional plots of the result from part 2.2 above as a function of ‘a’ and Δk and discuss your results in terms of this figure. In other words, what features of the function correspond to the conditions of orthonormality?

4. The Uncertainty Principle.

Now consider a case where V=0 and a particle is initially confined to the region  -a/2 ≤ x ≤ a/2 where the size of the box remains finite. For the wave packet defined by the following function.

                                            ψ(x) = (1/(Sqrt[a]))(e^((Subscript[ik, 0])x))      -a/2 ≤ x ≤ a/2
                                            ψ(x) = 0                   otherwise

1. What is the spectral distribution (i.e. the spectrum of wave numbers) necessary to produce such a wave packet? Use the standard eigenfunctions of the free particle, ϕ(x) = (e^ikx)/(Sqrt[2π]).

2. Plot the spectral distribution for a fixed value of ‘a’. What are the values of k of the minima on either side of the central peak? What is the separation between these minima? Treat this result as the width Δk of the distribution.

3. Get the width Δx of the probability distribution ((|ψ(x)|)^2). Recall that, in general, the width Δx is the standard deviation so

                                       ⟨Δ(x^2)⟩ = ⟨(x^2)⟩ -((⟨x⟩)^2)

where the brackets imply averaging over all x.

4. Generate an uncertainty principle appropriate for this wave packet.

Include purpose, calculations, figures,  and a discussion in your report.