Particle in a box waves

Find the formulas for the matrix elements of the matrix representatives of x and px in the particle-in-a-box energy representation. (This is the representation with particle-in-a-box wave functions as basis functions.) Use the general formulas to write down the first several elements in the northwest corner of each matrix. 

The property (2.27) of the wave functions is c ed orthonormality. We proved ortho-normality only for the particle-in-a-box wave functions. We shall prove it more generally in Section 7.2. 

Integrals involving trigonometric functions can often be evaluated using the identities of Problem 1.26. Use the complex-exponential form of the sine function to verify Eq. (2.27) for the particle-in-a-box wave functions. 

Expansion of a Function Using Particle-in-a-Box Wave Functions. Let us consider expanding a function in terms of the particle-in-a-box stationary-state wave functions, which are [Eq. (2.23)]... 

This is the desired expression for the expansion of an arbitrary well-behaved function f x) (0 /) as a linear combination of the particle-in-a-box wave functions... 

FIGURE 7.2 Plots of (a) the error and (b) the percent error in the expansion of the function of Fig. 7.1 in terms of particle-in-a-box wave functions when 1 and 5 terms are taken in the expansion. 

The very high probability of finding the = 1 energy is related to the fact that the parabolic state function — x) closely resembles the n = 1 particle-in-a-box wave function... 

Since / dr = 1 30, the normalized form of (8.11) is (30// ) x(/ - x). Figure 7.3 shows that this function rather closely resembles the true ground-state particle-in-a-box wave function. 

FIGURE 8.1 Percent deviation of the linear variation function (8.75) from the true ground-state particle-in-a-box wave function. 

As the depth and width of the internal potential is dialed up, these particle-in-a-box wave functions are modified - increasing the frequency in the region of the potential well and can even localize some of them entirely within the potential well. The latter, having no probability far-removed from the potential well, are the bound-states. ... 

Expansion of a Function Using Particle-in-a-Box Wave Functions... 

Numerically integrate the 1-D particle-in-a-box wave-function product over all space and show that the two functions are orthogonal. 

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