CHEM 101 Lecture Notes - Lecture 13: Degenerate Energy Levels, Product Rule, New York Yankees

It has been claimed that as the quantum number n increases, the motion of a particle in a 1-dimensional box becomes more classical. In this problem you will have the opportunity to convince yourself of this fact. For a particle of mass m moving in a 1-D box of length L, with ends at x = 0 and x = L, the classical probability density function can be shown to be independent of x. Formally, it is given by P(x)dx = dx/L regardless of the energy of the particle. Evaluate the probability that the classical particle will be found within the interval x = 0 to x = L/4. Now consider the quantum mechanical system. Evaluate the probability of finding the particle in the interval x = 0 to x = L/4 for the system in its n^th quantum state. For what value of n is there the largest probability of finding the particle between x = 0 and x = L/4? Justify this conclusion mathematically. Take the limit obtained in (b) as n rightarrow infinity. How does this compare to the classical result in (a)?

esources except othe peopie. Sing what you know the x direction. Show that the about the momentum operator along ion e A is an eigenfunction of the momentum operator in the xdirection with an eig pw etermine whether the eand Dz -th commute. In other words, operators please evaluate the commutator e nsider the 1-dimensional time dependent Schrodinger equation, ih 2m ax2 where y is a function at above, function only of x). Use the of separation of variables technique to recover the time independent Schroinger equation. Show 4. and justify your steps where appropriate. with the Consider a oscillator with and force constant k, and a particle same simple harmonic mass m mass in a one-dimensional box of length L a. What is the relationship between kandLsuch that the zero-point (ground state) energies of these two systems will be the same? o. If m is equal to the mass of a 1H atom, what is the value of k (in N/m) corrsesponding to L 1.4 nm? 5. If a Ha molecule rotates in the plane of a crystalline surface (in a chemisorption situation), it can approximated as a two-dimensional rigid rotor calculate (in k/mo) the lowest energy rotation hydrogen atom with a value of requal to the bond length of H2, o 7416 angrstoms. in the far infrared spectrum of HysBr, there is a series of lines separated by 16.72 cm 1. Calculate the values of the moment of inertia and the internuclear separation in H79Br. The Jao and J-1 transition for carbon monoxide (12cieo) occurs at 1.153x10 MHz. Calculate the value of the bond length in carbon monoxide. Show that the harmonic oscillator wave functions are alternately even and odd functions of x because the Hamiltonian operator obeys the relation A(x) A(-x). Define a reflection operator R by Ru(x) show that R is linear and that it commutes with A. Show also that the eigenvalues of R are t1. What are its eigenfunctions? Show that the harmonic-oscillator wave functions are eigenfunctions of R. Note that they are eigenfunctions of both H and R. What does this observation say about A and R