### Around and around in circles: the rigid rotor

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Physical Chemistry I: Quantum Chemistry

Bryn Mawr College

This semester we will be covering quantum chemistry, including an introduction to quantum mechanics, molecular quantum mechanics, and spectroscopy.

We consider one more model problem, this one concerning the rigid rotation of a diatomic molecule. Though the problem is simple compared to most molecular systems chemists are interested in, it yielded our first example of a wavefunction that was complex and not real valued. It will also provide a basis for an atomic model problem - the hydrogen atom.

MP3 podcast

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MP3 podcast

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You can pick any "particle" you like for the problem. Pertinent data are in the notebook we used in class and in your text.

Problem 5-22 refers you to Problem 5-20 for the rms displacement. Note that the formula in 5-20 is for v=2 and question 5-22 asks about v=0! You can derive the formula for v=0 by following 5-22, but I hadn't intended that. The formula for v=0 is the same as for v=2 except it's 1/2 instead of 5/2 as the factor.

What happens when you shoot an electron at a wall? We consider the case of a particle impinging on a rectangular potential barrier to discover that under many conditions - the particle goes right through! An electron with an energy of 4 eV has about a 70% chance of going "through" a 5eV wall and coming out the other side.

MP3 podcast

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Mathematica notebook

Worksheet

MP3 podcast

screencast

Mathematica notebook

Worksheet

How can parity work for you in quantum mechanics? We see that the parity of the harmonic oscillator solutions can make some problems trivial (the average value of the position, for example). We computed the probability of a harmonic oscillator being stretched or compressed further than classical mechanics would permit, to discover that in the ground state of HCl the probability was 15% that the classical limits would be exceeded.

MP3 podcast

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MP3 podcast

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Read about how the particle in the box quantum mechanical model and quantum dots are connected in a short article from the Strouse group at UC Santa Barbara. The use of quantum dots as biosensors is being explored at Livermore National Labs. Such dyes are now commercially available!

Photo of the dyes from the gallery at theONR

Photo of the dyes from the gallery at theONR

The boundary conditions on z in the MicroTest should be 20 to 200 - not 200 to 200 as stated.

With apologies!

mmf

With apologies!

mmf

Moving beyond the particle in the box, a model for simple molecular vibrations is constructed. The solutions depend on the Hermite polynomials and exhibit parity. Wavefunctions for states with an even quantum number have even parity (are symmetric about the y axis) and those with odd quantum numbers are odd. Parity considerations can simplify quantum chemical calculations.

MP3 podcast

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Mathematica notebook

MP3 podcast

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Mathematica notebook

We show that the product of one-dimensional functions is indeed a solution to the 3-dimensional problem. We find the energy and show that the energy and wavefunction now depend on 3 independent quantum numbers. When the box is symmetric, for example, a cube, some energy levels are degenerate. We noted that symmetry generally leads to degeneracy, though not all degeneracy is a result of symmetry (accidental symmetry). We extended the concept of the product wavefunction to systems of more than one particle. We drew a quick concept map of where we have been in the course so far.

MP3 podcast

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Mathematica notebook - same as previous lecture

Trout Fishing in America - the band

Six

and on...to

MP3 podcast

screencast

Mathematica notebook - same as previous lecture

Trout Fishing in America - the band

Six

What do you get when you add three plus three?

I believe the answer is six.

And how ‘bout seven when take away one now?

I believe the answer is six.

Well, how do you do that in your head?

I would need a pencil all filled with lead,

A huge piece of paper ‘bout the size of my bed.

Now…You must be a mathematician.....

and on...to

What is the dimension of the field of complex numbers over

the real numbers, times the order of the alternating group on

three elements divided by the definite integral from zero to pi

over two of sine of X D X?

If you have an idea for a class shirt - let me know in the next couple of weeks and I'll get things organized over fall break.

Monday night, 7-9 pm in Park 354. Improve your Mathematica skills, whatever level you are!

Thanks for all your feedback on the course so far! Overall, you'd like me to continue using the tablet instead of the white board (even if you don't listen to the podcasts or watch the screencasts). About 1/4 of you are using the recorded materials, and are happy to have them.

The pace seems to be alright for some to slightly fast for others. If you lose the thread in lecture, be sure to stop me and ask. It's very likely that if you have lost the thread or missed a point, two other people have as well.

Interesting questions you raise:

Can I connect all the different equations?

Why if there is a node in a 2s, is it still classifed as "s"? Does the node have any significance?

Why an eigenvalue?

What does ψ measure?

Mathematica is confusing!

Why are Cheetos orange?

Where does Schrodinger's equation come from and why should I just accept it?

How to find uncertainty and probablity?

How to deal with electrons in rings in the free particle model?

What's the point of quantum mechanics?

What math should we review?

How can you predict the color of something using PIB?

What does orthonormality have to do with anything?

Watch this space for the answers!

The pace seems to be alright for some to slightly fast for others. If you lose the thread in lecture, be sure to stop me and ask. It's very likely that if you have lost the thread or missed a point, two other people have as well.

Interesting questions you raise:

Can I connect all the different equations?

Why if there is a node in a 2s, is it still classifed as "s"? Does the node have any significance?

Why an eigenvalue?

What does ψ measure?

Mathematica is confusing!

Why are Cheetos orange?

Where does Schrodinger's equation come from and why should I just accept it?

How to find uncertainty and probablity?

How to deal with electrons in rings in the free particle model?

What's the point of quantum mechanics?

What math should we review?

How can you predict the color of something using PIB?

What does orthonormality have to do with anything?

Watch this space for the answers!

A "real" quantum mechanical box - that is, one that is 3-dimensional. We consider the case of a particle confined to a rectangular parallelepiped. The Schrodinger equation for this system separates neatly into 3 one dimensional cases and we propose that the solutions to these problems are the 1-D particle in a box wavefunctions. We will verify this in the next lecture.

MP3 podcast

screencast

Mathematica notebook - same as previous lecture

MP3 podcast

screencast

Mathematica notebook - same as previous lecture

Our first postulate of quantum mechanics is that the wavefunction is a complete description of the system. This is great in principle, but what are the practical details? How do we use the wavefunction to produce information that is useful to chemists? The difference between average (or expectation) values and the probability density are explored. We consider the radial distribution of electron density in 1s and 3s orbitals using Mathematica.

MP3 podcast

screencast

Mathematica notebook

MP3 podcast

screencast

Mathematica notebook

An in-class Mathematica exercise using the particle in the box wavefunctions. Expectation values, the free-electron model and electronic transitions.

PIBWavefns.nb: Mathematica notebook to accompany the lesson.

PIBWavefnsWorksheet.pdf: Worksheet

PIBWavefns.nb: Mathematica notebook to accompany the lesson.

PIBWavefnsWorksheet.pdf: Worksheet

How can we use the framework of quantum mechanics to tell us something useful to chemists? The expectation value and the probability density are the keys. The eigenfunctions of the Hamiltonian are an orthonormal set. We graphed the probability density and wavefunctions for the particle in the box. We noticed that as n increased, the probabilty profile appeared more classical - a manifestation of the Bohr Correspondance Principle.

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*Mathematica* notebook to plot the wavefunctions and probality densities

MP3 podcast

webcast

Open hours in Park 354 are Tuesdays 7-9pm and Wednesdays 6-8pm, beginning next Tuesday, September 13th.

If you are looking for practice problems, a selection is posted at the course blackboard site. Practice problems for credit are due one week after they are posted. I will post solutions one week after the problem is posted. If you would like problems of a particular sort, please let me know and I'd be happy to create some to order!

If Hψ=eψ, why can't you just cancel the ψs? Operators, rules that change one function into another, play a key role in quantum chemistry. Each measurable quantity has a corresponding operator, and the operator, used in tandem with the wavefunction, can be used to calculate expected values of these quantities. We introduced the notion of the wavefunction as a vector in a function space and used Dirac's bra and ket notation to express the wavefunction and the expectation value.

MP3 podcast

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MP3 podcast

screencast

Schrodinger's equation provides a way to describe the wave nature of matter, most important for the small bits of matter that concern chemists. We solve Schrodinger's equation for a model problem of a particle in a 1-D universe, confined to a small line segment, to get the wave functions and the energy. The solution requires using the boundary conditions for the problem, including the condition that the total probability of finding the particle somewhere in the universe is 1. We notice that not every wavefunction of the proper form is allowed, nor is every energy. The solutions are characterized by a quantum number "n".

MP3 podcast

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MP3 podcast

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Office hours are W 9-10, 12-1 and Th 9-10.

Virtual hours are S-Th 8:30 to 9:30 by IM (quantophrenic)

Anyone interested in a Mathematica boot camp later this week, please send me an e-mail with some good times.

Virtual hours are S-Th 8:30 to 9:30 by IM (quantophrenic)

Anyone interested in a Mathematica boot camp later this week, please send me an e-mail with some good times.

In late 1925 Erwin Schrodinger, prompted by a question asked by Sommerfeld in a seminar Schrodinger had given, developed the wave equation. This lecture discusses the 1-dimensional, time independent Schrodinger equation for a single particle. We set up a sample problem, for a single particle trapped in an infinitely deep potential energy well and found solutions (by inspection) for the wave equation in 3 different regions (outside the box, where it is zero, and inside the box). We discussed the basic form of the Hamiltonian operator.

MP3 podcast

screencast

MP3 podcast

screencast