Difference between revisions of "PCh7 Lec 1 2"
(Created page with "Chapter 7, Lectures 1 and 2 (previously delivered) Quantum mechanics seeks to address all modes motion/energy. As discussed at the beginning of Chapter 7, there are three typ...") |
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Chapter 7, Lectures 1 and 2 (previously delivered) | Chapter 7, Lectures 1 and 2 (previously delivered) | ||
| − | Quantum mechanics seeks to address all modes motion/energy. As discussed at the beginning of Chapter 7, there are three types of motion: | + | Quantum mechanics seeks to address all modes of motion/energy. As discussed at the beginning of Chapter 7, there are three types of motion: |
:- Translational | :- Translational | ||
:- Vibrational | :- Vibrational | ||
:- Roational | :- Roational | ||
| − | Translational motion in QM is described by the particle in a box (Chap 4). Vibrational motion is described by the harmonic oscillator (sec 7.1). Rotational motion is described by a particle on a ring (2D - Sec 7.2) or sphere (3D - Sec 7.3). | + | Translational motion in QM is described by the particle in a box (PIB - Chap 4). Vibrational motion is described by the harmonic oscillator (sec 7.1). Rotational motion is described by a particle on a ring (2D - Sec 7.2) or sphere (3D - Sec 7.3). |
It is convenient for us to discuss all of these modes of motion/energy as separate things, but in the end, we cannot separate the motions from each other and so the final "solution" is a bit messy. | It is convenient for us to discuss all of these modes of motion/energy as separate things, but in the end, we cannot separate the motions from each other and so the final "solution" is a bit messy. | ||
I can draw an analogy to understanding 3D space by using the dimensions of x, y, and z. I could explain the concept of an x-axis (1D), or position along the x-axis and then tell you that a y-axis and z-axis follow all of the same concepts as the x-axis. We then put them all together in order to describe 3D space. In this case, x, y, and z fully describe all 3D space and we need to know nothing more than the x, y, and z values to know where an object is in this 3D space; we sometimes refer to x, y, and z as a "complete set." | I can draw an analogy to understanding 3D space by using the dimensions of x, y, and z. I could explain the concept of an x-axis (1D), or position along the x-axis and then tell you that a y-axis and z-axis follow all of the same concepts as the x-axis. We then put them all together in order to describe 3D space. In this case, x, y, and z fully describe all 3D space and we need to know nothing more than the x, y, and z values to know where an object is in this 3D space; we sometimes refer to x, y, and z as a "complete set." | ||
| + | |||
| + | As you have already learned from the PIB example, all quantum mechanical models have an operators, eigenfunctions, and eigenvalues. So i hope you have your "Hamiltonian Table" i passed out in class. I will link it here shortly. At this point you should have the following information in the table: | ||
| + | :- Translational (1D) | ||
| + | :- Translational (3D) | ||
| + | :- Vibrational | ||
| + | ...and by the end of this Chapter the entire table should be complete. | ||
===Vibrational Motion=== | ===Vibrational Motion=== | ||
| + | Points of noteworthiness | ||
| + | :- The Schrodinger Equation now includes a potential of 1/2kx^2 | ||
| + | :- The wavefunctions are related to the Hermite polynomials (eq. 7.5) | ||
| + | :- The energies are fairly straightforward (eq. 7.6) | ||
| + | :-...and this can all be pictured as shown in Fig 7.2 and 7.3 which you have already made in Mathematica. | ||
| + | |||
| + | To have a slightly different (physics) approach to this problem, i suggest you watch the following videos: | ||
| + | :[https://youtu.be/pVpqjVuN8M0 Quantum Harmonic Oscillator Part 1 (5:45 mins)] | ||
| + | :[https://youtu.be/Z7XHYLfAzV4 Quantum Harmonic Oscillator Part 2 (7:38 mins)] | ||
| + | ::he goes into LOTS of detail about solving...the answer is Hermite polynomials! | ||
Latest revision as of 16:43, 23 March 2020
Chapter 7, Lectures 1 and 2 (previously delivered)
Quantum mechanics seeks to address all modes of motion/energy. As discussed at the beginning of Chapter 7, there are three types of motion:
- - Translational
- - Vibrational
- - Roational
Translational motion in QM is described by the particle in a box (PIB - Chap 4). Vibrational motion is described by the harmonic oscillator (sec 7.1). Rotational motion is described by a particle on a ring (2D - Sec 7.2) or sphere (3D - Sec 7.3).
It is convenient for us to discuss all of these modes of motion/energy as separate things, but in the end, we cannot separate the motions from each other and so the final "solution" is a bit messy.
I can draw an analogy to understanding 3D space by using the dimensions of x, y, and z. I could explain the concept of an x-axis (1D), or position along the x-axis and then tell you that a y-axis and z-axis follow all of the same concepts as the x-axis. We then put them all together in order to describe 3D space. In this case, x, y, and z fully describe all 3D space and we need to know nothing more than the x, y, and z values to know where an object is in this 3D space; we sometimes refer to x, y, and z as a "complete set."
As you have already learned from the PIB example, all quantum mechanical models have an operators, eigenfunctions, and eigenvalues. So i hope you have your "Hamiltonian Table" i passed out in class. I will link it here shortly. At this point you should have the following information in the table:
- - Translational (1D)
- - Translational (3D)
- - Vibrational
...and by the end of this Chapter the entire table should be complete.
Vibrational Motion
Points of noteworthiness
- - The Schrodinger Equation now includes a potential of 1/2kx^2
- - The wavefunctions are related to the Hermite polynomials (eq. 7.5)
- - The energies are fairly straightforward (eq. 7.6)
- -...and this can all be pictured as shown in Fig 7.2 and 7.3 which you have already made in Mathematica.
To have a slightly different (physics) approach to this problem, i suggest you watch the following videos:
- Quantum Harmonic Oscillator Part 1 (5:45 mins)
- Quantum Harmonic Oscillator Part 2 (7:38 mins)
- he goes into LOTS of detail about solving...the answer is Hermite polynomials!