Particle in a box energy

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. 

PROBLEM 5.2.8. Prove Eq. (5.2.17), assuming that a perfect gas PV = nRT = NfikBT is treated as a fiCE of CB with quantum-mechanical particle-in-a-box energies. 

Expansion of a Function in Terms of Eigenfimctions. We have seen an example of the expansion of a function in terms of a set of functions— the particle-in-a-box energy eigenfunctions. Many different sets of functions can be used to expand an arbitrary function. A set of functions g, g2. > is said to be a complete set if any well-behaved function / that obeys the same boundary conditions as the g/s can be expanded as a linear combination of the g, s according to... 

Evaluate fm H f ) if (a) H is the harmonic-oscillator Hamiltonian operator and f and f are harmonic-oscillator stationary-state wave functions with vibrational quantum numbers m and n (b) H is the particle-in-a-box H and / and f are particle-in-a-box energy eigenfunctions with quantum numbers m and n. 

EXERCISE Let G(x) = 1 for 0 x Z and G(x) = 0 elsewhere. Expand G in terms of the particle-in-a-box energy eigenfunctions. Since G is not zero at 0 and at /, the expansion will not represent G at these points but will represent G elsewhere. Use the first 7 nonzero terms of the expansion to calculate G at x = /. Repeat this using the first 70 nonzero terms (use a programmable calculator). (Answers 0.1219,0.9977.)... 

In an extremely crude model of anthracene, C14H10, the n electrons are considered to be confined to a rectangular box of dimensions 4 A x 7 A. Using the appropriate particle-in-a-box energy expression, calculate the wavelength (in A) of the transition from the ground state to the first excited state. 

Next, using this value for the change in energy for the transition and the expression for the particle-in-a-box energy values, we can set up the following relationship ... 

The particle-in-a-box energy levels can be used to predict the qualitative behavior of an electron trapped in a spherical cavity of radius r. The relevant equation from Section 1.7 is now... 

Show that the particle-in-a-box energy eigenfunction given in Eq. (16.4-10) is normalized. Solution... 

We know the particle-in-a-box energy levels are very close together when the dimension L of the... 

The zero-point energy is the lowest possible energy in a quantum mechanical system, such as the "particle-in-a-box" energy corresjxjnding to w = 1 (page 326). 

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