A Primer on Quantum Mechanics and Orbitals



The Dreaded Harmonic Oscillator

A Few Hints

1. Force constants, k, used here are in units of Newtons per meter, N/m, so masses of particles must be in kg and distances between them in terms of m. For several problems you'll need to convert atomic weights ( in g/mol) into atomic masses ( kg/atom). I trust you can do this. For our purposes, unless explicitly stated, just use the atomic weights as the masses of the atoms rather than the masses of specific isotopes of those atoms.

2. Remember that when you are dealing with a situation when one of the things that the force constant "spring" is attached to does not have a "infinitely" larger mass than the other, that you must use the reduced mass of the system

recall for a system of two masses m1 and m2, the reduced mass μ is

[pic]

Again, if m1 is much much greater than m2, then the sum of m1 and m2 is pretty much just m1. So the reduced mass simply approaches the mass of the smaller particle.

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Question 1

What is the ultimate source of quantization for the quantum mechanical harmonic oscillator? What are the allowed values of the quantum number v

Question 2

What is the meaning of a zero-point energy, and how does it apply to the quantum mechanical harmonic oscillator? Why do we have to have a zero point energy? How does the zero point energy of a harmonic oscillator change with the force constant, k? How does it change with the mass of the particle?

Question 3

What is the zero point energy for the vibration of the amino acid, glycine, tethered by a hydrogen bond to a cell surface? Assume that the force constant, k, for this hydrogen bond is 106 N/m. Assume also that the mass of the cell (and its surface) can be regarded as infinite for this problem.

Question 4

Estimate the energy required to excite the vibration of the tethered amino acid in question 3 to its first excited state. What is the wavelength of the photon required to effect this transition? What is this energy in terms of cm-1?

Question 5

What is the energy difference between successive energy levels of the quantum mechanical harmonic oscillator having a mass m and a force constant k. (ΔE=Ev+1 -Ev )Does this energy difference, ΔE, change with changing v?

Question 6

Sketch the ground state, first excited state, and second excited state wavefunctions of the quantum mechanical harmonic oscillator. How many nodes are in each? How does the energy change with the number of nodes? How does the quantum number v relate to the number of nodes?

Question 7

In question 6, you sketched the ground, first excited, and second excited state wavefunctions for the harmonic oscillator. If you did it correctly, then the 'tail' of each wavefunction extends beyond the potential energy 'surface' defined by the parabola V(x)=-1/2 kx2. What is the meaning of this tail in each case.

Question 8

You'll notice that the expression for the expectation value of the position, , for the harmonic oscillator has the same value for all energy levels. What is this value and what does it mean that it does not change with increasing energy.

Question 9

You'll also notice that the expectation value for the square of the position, does not equal zero. How does change with increasing energy of the harmonic oscillator (i.e. with increasing values of v). How does this result correspond to the case of the classical harmonic oscillator?

Question 10

Calculate the zero point vibrations for each of the following diatomic gasses having their respective force constants:

N2(k=2293 N/m), NO(k=995 N/m), O2(k=1177 N/m).

Question 11

The value of the force constant between the atoms in 1HCl is the same as that in the deuterated molecule, 2HCl. Why would this be the case?

Question 12

The of the force constant in both 1HCl and 2HCl is 1375 N/m. What are the zero point vibrational energies of each? What wavelengths of light are needed to excite each to their respective first excited states?

Question 13

The energy required to excite F2 from its ground vibrational state to its first excited vibrational state is 2.207 x 10 -26 J. What wavelength of light does this correspond to? What is the force constant for F2?

Question 14

What would be the effect of doubling the force constant on the zero point vibration of a molecule? What about doubling the mass of the particle.

Question 15

Write the general expression for a harmonic oscillator wavefunction and describe the components.

What is the function of the gaussian part? What does the Hermite polynomial introduce into the wavefunction?

Question 16

What do you suppose would be the value of the 'overlap' S between two different harmonic oscillator wavefunctions for the same particle? The overlap between two wavefunctions ψv and ψv' is deffined as:

[pic]

What about the overlap of a wavefunction with itself?

Question 17

What is anharmonicity and what effect does it have on the spacing between successive vibrational energy levels in 'real' oscillators (such as actual molecules?

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