PCh7 Lec4 - Revision history

Bes at 21:53, 25 April 2020

2020-04-25T21:53:06Z

Bes: /* Sec 7.5: Visualizing the Spherical Harmonics */

2020-03-25T12:59:26Z

Sec 7.5: Visualizing the Spherical Harmonics

Bes: /* Sec 7.5: Visualizing the Spherical Harmonics */

2020-03-25T02:21:26Z

Sec 7.5: Visualizing the Spherical Harmonics

Bes: /* Sec 7.5: Visualizing the Spherical Harmonics */

2020-03-25T02:15:41Z

Sec 7.5: Visualizing the Spherical Harmonics

Bes: /* Sec 7.5: Visualizing the Spherical Harmonics */

2020-03-25T02:12:24Z

Sec 7.5: Visualizing the Spherical Harmonics

Bes: /* Sec 7.5: Visualizing the Spherical Harmonics */

2020-03-25T02:10:50Z

Sec 7.5: Visualizing the Spherical Harmonics

Bes: /* Sec 7.5: Visualizing the Spherical Harmonics */

2020-03-25T02:09:22Z

Sec 7.5: Visualizing the Spherical Harmonics

Bes: /* Sec 7.5: Visualizing the Spherical Harmonics */

2020-03-24T23:21:52Z

Sec 7.5: Visualizing the Spherical Harmonics

Bes: /* Sec 7.4 Quantization of Angular Momentum */

2020-03-24T22:54:24Z

Sec 7.4 Quantization of Angular Momentum

Bes: /* Sec 7.3: 3D Rotation */

2020-03-24T22:31:10Z

Sec 7.3: 3D Rotation

← Older revision		Revision as of 21:53, 25 April 2020
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		+	(3/25/20, bes)
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	===Sec 7.3: 3D Rotation===		===Sec 7.3: 3D Rotation===
	'''''The following expects that you have your textbook open to section 7.3 and are reading long...'''''		'''''The following expects that you have your textbook open to section 7.3 and are reading long...'''''

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 :[[File:Screen Shot 2020-03-24 at 9.20.34 PM.png|800px]]
-But wait, the book shows nice plots that look like p-orbitals and d-orbitals...the spherical harmonics do not give you the nice looking orbitals shown in Figure 7.7...these are a linear combination of the spherical harmonics...see eq. 7.34...can you plot one or two of these in Mathematica?
+But wait, the book shows nice plots that look like p-orbitals and d-orbitals...the spherical harmonics do not give you the nice looking orbitals shown in Figure 7.7...these are a linear combination of the spherical harmonics...see eq. 7.34...can you plot one or two of these in Mathematica..like this one?
 '''''End of Lecture 4.'''''

← Older revision		Revision as of 02:21, 25 March 2020
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	:[[File:Screen Shot 2020-03-24 at 9.05.37 PM.png\|800px]]		:[[File:Screen Shot 2020-03-24 at 9.05.37 PM.png\|800px]]
		+	:[[File:Screen Shot 2020-03-24 at 9.20.34 PM.png\|800px]]
		+
		+	But wait, the book shows nice plots that look like p-orbitals and d-orbitals...the spherical harmonics do not give you the nice looking orbitals shown in Figure 7.7...these are a linear combination of the spherical harmonics...see eq. 7.34...can you plot one or two of these in Mathematica?

−	But wait, the book shows nice plots that look like p-orbitals and d-orbitals...the spherical harmonics do not give you the nice looking orbitals shown in Figure 7.7...these are a linear combintion of the spherical harmonics...see eq. 7.34...can you plot one or two of these in Mathematica?

	'''''End of Lecture 4.'''''		'''''End of Lecture 4.'''''

← Older revision		Revision as of 02:15, 25 March 2020
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	:[[File:Screen Shot 2020-03-24 at 9.05.37 PM.png\|800px]]		:[[File:Screen Shot 2020-03-24 at 9.05.37 PM.png\|800px]]
		+
		+	But wait, the book shows nice plots that look like p-orbitals and d-orbitals...the spherical harmonics do not give you the nice looking orbitals shown in Figure 7.7...these are a linear combintion of the spherical harmonics...see eq. 7.34...can you plot one or two of these in Mathematica?
		+
		+	'''''End of Lecture 4.'''''

@@ Line 36: / Line 36: @@
 Either way, you should notice that some of these functions are imaginary, but that is okay since we know that the square of the wavefunction has physical meaning not the wavefunction itself. Now some of you want to run over to Mathematica and use the SphericalPlot3D function, like this:
-In the Mathematica document below, notice that the wavefunction i plotted with the Y(2,1) or also written: [[File:Screen Shot 2020-03-24 at 9.09.10 PM.png|100px]]
+In the Mathematica document below, notice that the wavefunction i plotted was the Y(2,1) or also written: [[File:Screen Shot 2020-03-24 at 9.09.10 PM.png|100px]]. Also notice that i plotted not only the square but the complex conjugate (change i to -i):
 :[[File:Screen Shot 2020-03-24 at 9.05.37 PM.png|800px]]

← Older revision		Revision as of 02:09, 25 March 2020
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	:If you just want to see what the functions are, then look in your book (eq. 7.33) or [https://en.wikipedia.org/wiki/Table_of_spherical_harmonics click here].		:If you just want to see what the functions are, then look in your book (eq. 7.33) or [https://en.wikipedia.org/wiki/Table_of_spherical_harmonics click here].

−	Either way, you should notice that some of these functions are imaginary, but that is okay since we know that the square of the wavefunction has physical meaning not the wavefunction itself. Now some of you want to run over to Mathematica and ~~do some~~, ~~SphericalPlot3D~~	+	Either way, you should notice that some of these functions are imaginary, but that is okay since we know that the square of the wavefunction has physical meaning not the wavefunction itself. Now some of you want to run over to Mathematica and use the SphericalPlot3D function, like this:
		+
		+	In the Mathematica document below, notice that the wavefunction i plotted with the Y(2,1) or also written: [[File:Screen Shot 2020-03-24 at 9.05.37 PM.png\|100px]]
		+	:[[File:Screen Shot 2020-03-24 at 9.05.37 PM.png\|600px]]

← Older revision		Revision as of 23:21, 24 March 2020
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	===Sec 7.5: Visualizing the Spherical Harmonics===		===Sec 7.5: Visualizing the Spherical Harmonics===
	Now onto the eigenfunctions, or the spherical harmonics...		Now onto the eigenfunctions, or the spherical harmonics...
		+
		+	:If you want to know everything there is to know about spherical harmonics, then visit the the [https://en.wikipedia.org/wiki/Spherical_harmonics wikipedia page]...
		+	:If you just want to see what the functions are, then look in your book (eq. 7.33) or [https://en.wikipedia.org/wiki/Table_of_spherical_harmonics click here].
		+
		+	Either way, you should notice that some of these functions are imaginary, but that is okay since we know that the square of the wavefunction has physical meaning not the wavefunction itself. Now some of you want to run over to Mathematica and do some, SphericalPlot3D

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 ===Sec 7.4 Quantization of Angular Momentum===
 ===Sec 7.5: Visualizing the Spherical Harmonics===
 Now onto the eigenfunctions, or the spherical harmonics...

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 Some important issues come out of this discussion:
 :1) The 3D rotation solution is kind of like the 2D + 1D. In the 2D solution, the integer ''m<sub>l</sub>'' = 0, ±1, ±2, ±3, ..., so the 3D solution contains this integer as well as one more, ''l'' = 0, 1, 2, 3, ... This is not a whole lot different than when we discussed the 1D box --> ''"n"'', 3D box --> ""''n<sub>x</sub>, n<sub>y</sub>, n<sub>z</sub>,''" one integer for each dimension or as your text notes, one integer for each set of boundary conditions.
-:2) Even though there are two integers, the total energy is only dependent on one, ''l''; [[File:Screen Shot 2020-03-24 at 4.49.34 PM.png|100px]], so once we take into account the relationship between the integers, ie. ''l'' = 0, 1, 2, 3, ..., but ''m<sub>l</sub>'' = 0, thru ±''l'', this leads to energy levels that are degenerate. Recall, ''Ι'' = moment of inertia and is equal to μr<sup>2</sup>.
+:2) Even though there are two integers, the total energy is only dependent on one, ''l''; [[File:Screen Shot 2020-03-24 at 4.49.34 PM.png|100px]], so once we take into account the relationship between the integers, ie. ''l'' = 0, 1, 2, 3, ..., but ''m<sub>l</sub>'' = 0, thru ±''l'', this leads to energy levels that are degenerate. Recall, ''I'' (eye) = moment of inertia and is equal to μr<sup>2</sup>, not to be confused with ''l'' (el).
 :::''l'' = 0 has only 1 energy level (''m<sub>l</sub>'' = 0)
 :::''l'' = 1 has only 3 energy level (''m<sub>l</sub>'' = -1, 0, +1)