'hh-atom fix'

Author: DPotoyan

ch05/note01.md

@@ -1,13 +1,15 @@
 ## Hydrogenlike atoms
 
-```{admonition} What you need to know
+:::{admonition} **What you need to know**
+
 :class: note
-- Hydrogen atom is the simplest atom for which Schrodinger equation can be solved exactly. He atom which only has one more electron already proves to be impossible to solve exactly. 
-- The solution of the Schrödinger equation for H atom uses the fact that the Coulomb potential produced by the nucleus is isotropic: it is radially symmetric in space and only depends on the distance to the nucleus. This symmetry gives rise to degenercaies in energy levels. 
-- Although the resulting energy eigenfunctions (the orbitals) are not necessarily isotropic themselves, their dependence on the angular coordinates follows completely generally from this isotropy of the underlying potential.
-- The angular momentum is conserved, therefore, the energy eigenvalues may be classified by two angular momentum quantum numbers, ℓ and m (both are integers) and quantize mangitude and projection of angular momentum. 
-- Atomic orbitals are introduced which are used for also describing multi-electron atoms and molecules. 
-```
+
+- The **hydrogen atom** is the simplest atom for which the Schrödinger equation can be solved exactly. In contrast, helium—though only having one additional electron—cannot be solved exactly due to the complexity introduced by electron-electron interactions.
+- Solving the Schrödinger equation for the hydrogen atom leverages the fact that the **Coulomb potential from the nucleus is isotropic**: it is radially symmetric and depends solely on the distance from the nucleus. This **symmetry leads to degeneracies in the energy levels**.
+- While the resulting energy eigenfunctions (atomic orbitals) are not necessarily isotropic, their dependence on angular coordinates arises fundamentally from the isotropy of the underlying potential.
+- Because angular momentum is conserved, energy eigenvalues can be classified by two angular momentum quantum numbers, $l$ and $m$ (both integers), which quantize the magnitude and projection of angular momentum.
+- **Atomic orbitals** derived for hydrogen are also foundational in describing the structure of multi-electron atoms and molecules.
+:::
 
 
  ### Schrodinger equation for hydrogenlike atoms
@@ -18,20 +20,23 @@
 
 - We have a porblem of one particle moving in a symmetric potential field in 3D. Expect to get 3 quantum numbers, anticipate some degenreacies due to this raidal symmetry. 
 
-- **Potential energy** operator consists of one coulomb term encoding the electrostatic attraction between nucleus and electron: 
+- **Kinetic energy operator** in 3D is the same as in the particle in a box in 3D: 
+
+$$\frac{\hbar^2}{2m_e}\nabla^2$$
+
+- **Potential energy operator** operator consists of one coulomb term encoding the electrostatic attraction between nucleus and electron: 
 
 $$V= - \frac{Ze^2}{4\pi\epsilon_0 r}$$
 
 - Where $m_e$ is the electron mass,  $\epsilon_0$ is the [vacuum permitivity](http://en.wikipedia.org/wiki/Vacuum_permittivity).
 
-- **Kinetic energy operator** in 3D is the same as in the particle in a box in 3D: 
-$$\frac{\hbar^2}{2m_e}\nabla^2$$
+
 
 
 
 ### H-atom in spherical coordinates system
 
-- Because of the spherical symmetry of the [Coulomb potential](http://en.wikipedia.org/wiki/Coulomb's_law)it is convenient to work in spherical coordinates:
+- Because of the spherical symmetry of the [Coulomb potential](http://en.wikipedia.org/wiki/Coulomb's_law) it is convenient to work in spherical coordinates:
 
 
 $${\left[ -\frac{\hbar^2}{2m_e}\Delta - \frac{Ze^2}{4\pi\epsilon_0 r}\right]\psi_i(r,\theta,\phi) = E_i\psi(r,\theta,\phi)}$$
@@ -78,14 +83,16 @@ $$V_{eff} = - \frac{Ze^2}{4\pi\epsilon_0r}+ \frac{l(l+1)\hbar^2}{2m_er^2} $$
 
 - The eigenvalues $E_{nl}$ and and the radial eigenfunctions $R_{nl}$ can be written as (derivations are lengthy but standard math):
 
-$${E_{nl} = -\frac{m_ee^4Z^2}{32\pi^2\epsilon_0^2\hbar^2n^2}{ with }n = 1,2,3...{ (independent of }l,l<n{)}}$$
+$${E_{nl} = -\frac{m_ee^4Z^2}{32\pi^2\epsilon_0^2\hbar^2n^2}{ \,\,\, }n = 1,2,3...{ (independent\, of\, }l,\,\,\,l<n{)}}$$
+
+$$R_{nl}(r) = \rho^le^{-\rho/2}{L_{n-l-1}^{2l+1}(\rho)}$$
 
-$${R_{nl}(r) = \rho^lL^{2l+1}_{n+l}(\rho){exp}\left(-\frac{\rho}{2}\right){ with }\rho = \frac{2Zr}{na_0}{ and }
-a_0 = \frac{4\pi\epsilon_0\hbar^2}{m_ee^2}}$$
 
-- where $L_{n+l}^{2l+1}(\rho)$ are [Laguerre polynomials](http://en.wikipedia.org/wiki/Laguerre_polynomials). The constant $a_0$ is called the [Bohr radius](http://en.wikipedia.org/wiki/Bohr_radius). Some of the first radial wavefunctions are listed on the next page.Some of the electronic energy levels of hydrogen atom are shown below.
+- **Bohr radius:** $a_0 = \frac{4\pi\epsilon_0\hbar^2}{m_ee^2}$
 
+- **Dimensionless distance defined via ratio of Bohr radius:** $\rho = \frac{2Zr}{na_0}$
 
+- **[Laguerre polynomials](http://en.wikipedia.org/wiki/Laguerre_polynomials) $L_{n+l}^{2l+1}(\rho)$** 
 
 #### Examples of the radial wavefunctions for hydrogenlike atoms
 
@@ -121,33 +128,47 @@ $${E_i = R_HZ^2\left(\frac{1}{1^2} - \frac{1}{\infty}\right)}$$
 
 ### Quantum numbers $n$, $l$ and $m$
 
-The quantum numbers in hydrogenlike atoms take on the following values dicated by the solution of Schrodinger equation with boundary conditions imposed respective radial and anuglar parts: 
+The quantum numbers in hydrogenlike atoms take on the following values dicated by the solution of Schrodinger equation with boundary conditions imposed respective radial and anuglar parts
+
+:::{admonition} **Quantum numbers of Hydrogen Atom**
 
 $${n = 1, 2, 3, ...}$$
 $${l = 0, 1, 2, ..., n-1}$$
 $${m = 0, \pm 1, \pm 2,...,\pm l}$$
+$${m_s = \pm 1/2}$$
+:::
 
-- For a given value of $n$, the level is $n^2$ times degenerate. 
 
 - For historical reasons, the following letters are used to express the value of $l$:
 
 $${\phantom{{symbo}}l = 0, 1, 2, 3, ...}{{symbol} = s, p, d, f, ...}$$
 
 
-- Recall that the wavefunctions for hydrogenlike atoms are $R_{nl}(r)Y_l^m(\theta,\phi)$ with $l < n$. For the first shell we have only one wavefunction: $R_{10}(r)Y_0^0(\theta,\phi)$. This state is usually labeled as $1s$, where 1 indicates the [shell number](http://en.wikipedia.org/wiki/Electron_shell) ($n$) and $s$ corresponds to orbital angular momentum $l$ being zero. For $n = 2$, we have several possibilities: $l = 0$ or $l = 1$. The former is labeled as $2s$. The latter is $2p$ state and consists of three degenerate states: (for example, $2p_x$, $2p_y$, $2p_z$ or $2p_{+1}$, $2p_0$, $2p_{-1}$). In the latter notation the values for $m$ have been indicated as subscripts.
+- For the $n=1$ we have only one wavefunction: $R_{10}(r)Y_0^0(\theta,\phi)$. This state is usually labeled as $1s$, where 1 indicates the [shell number](http://en.wikipedia.org/wiki/Electron_shell) ($n$) and $s$ corresponds to orbital angular momentum $l$ being zero. 
+- For $n = 2$, we have several possibilities: $l = 0$ or $l = 1$. The former is labeled as $2s$. The latter is $2p$ state and consists of three degenerate states: (for example, $2p_x$, $2p_y$, $2p_z$ or $2p_{+1}$, $2p_0$, $2p_{-1}$). In the latter notation the values for $m$ have been indicated as subscripts.
 
-- There is one more quantum number that has not been discussed yet: [Spin quantum number](http://en.wikipedia.org/wiki/Spin_quantum_number) For one-electron systems this can have values $\pm\frac{1}{2}$ (will be discussed in more detail later). In absence of magnetic fields the spin levels are degenerate and therefore the total degeneracy of the levels is $2n^2$.
+- There is one more quantum number that has not been discussed yet: [Spin quantum number](http://en.wikipedia.org/wiki/Spin_quantum_number) $m_s=\pm 1/2$
+
+- For one-electron systems this can have values $\pm\frac{1}{2}$ (will be discussed in more detail later). In absence of magnetic fields the spin levels are degenerate and therefore the total degeneracy of the levels is $2n^2$.
 
 ### Total wave function
 
 The total wavefunction for a hydrogenlike atom is ($m$ is usually denoted by $m_l$):
 
+:::{admonition}
+:class: important:
+
+$${\psi_{n,l,m_l}(r,\theta,\phi) = N_{nl}\cdot R_{nl}(r)\cdot Y_{l, m_l}(\theta,\phi)}$$
 
-$${\psi_{n,l,m_l}(r,\theta,\phi) = N_{nl}R_{nl}(r)Y_l^{m_l}(\theta,\phi)}$$
+$$|n, l, m\rangle$$
+
+:::
+
+Where the normalization factor is:
 
 $${N_{nl} = \sqrt{\left(\frac{2Z}{na_0}\right)^3\frac{(n - l - 1)!}{2n\left[(n + l)!\right]}}}$$
 
-$$R_{nl}(r) = \rho^le^{-\rho/2}{L_{n-l-1}^{2l+1}(\rho)}$$
+
 
 ### Table of Wavefunctions in cartesian coordinates
 

ch05/note03.md

@@ -4,48 +4,50 @@
 :::{admonition} What you need to know
 :class: note
 
-- A wavefunction for a one-electron system is called an orbital. For an atomic system such as H (hydrogen atom), it is called an [atomic](http://en.wikipedia.org/wiki/Atomic_orbital).
+- **A wavefunction for a one-electron system is called an orbital**. For an atomic system such as H (hydrogen atom), it is called an [atomic](http://en.wikipedia.org/wiki/Atomic_orbital).
 - The orbital plots demonstrated the shapes of the orbitals but this does not tell us anything about the radial extent (i.e., how far the orbital reaches).
 -  As the value of $Z$ is increased, the radial extent decreases. This indicates that for higher nuclear charge, the electrons will reside closer to the nucleus.
-- The radial functions have $n - l - 1$ zero values (nodes) between distances from zero to infinity.
-- The existence of the nodes makes the wavefunctions orthogonal. For example, $\psi_{1s}$ and $\psi_{2s}$ in hydrogenlike atoms are orthogonal.
+- **The radial wavefunctions have $n - l - 1$ nodes** between distances from zero to infinity.
+- **The angular part of wavefunctions ahve $l$ nodes**
+- **The total number of nodes is $n-1$**
 :::
 
 
 ### Radial profiles of atomic orbitals
 
-When visualizing the radial probabilities, it is possible to do directly plot the square of the radial wavefunction ($R_{nl}^2$) or the radial probability density ($P_{nl}$):
+- When visualizing the radial probabilities, it is possible to do directly plot the square of the radial wavefunction ($R_{nl}^2$) or the radial probability density ($P_{nl}$):
 
 $${P_{nl}(r) = r^2N_{nl}^2R_{nl}^2(r)}$$
 
 ![](images/AO_0.png)
 
-According to this expression, the most probable radius for an electron on hydrogen atom $1s$ orbital is $a_0$ (the Bohr radius). Previous figures showed examples of $R_{nl}$ and $P_{nl}$. Probability densities are useful, for example, in understanding charge distributions in atoms and molecules.
+- According to this expression, the most probable radius for an electron on hydrogen atom $1s$ orbital is $a_0$ (the Bohr radius). Previous figures showed examples of $R_{nl}$ and $P_{nl}$. Probability densities are useful, for example, in understanding charge distributions in atoms and molecules.
 
 ![](images/AO2.jpg)
-As the principal quantum number $n$ increases, the electron moves out to greater distances from the nucleus. The average distance for an electron in a given orbital (with quantum numbers $n$ and $l$) is given by (this is \textit{not} the expectation value):
+
+- As the principal quantum number $n$ increases, the electron moves out to greater distances from the nucleus. The average distance for an electron in a given orbital (with quantum numbers $n$ and $l$) is given by (this is \textit{not} the expectation value):
 
 $${\langle r\rangle_{nl} = \int_0^\infty r\times P_{nl}(r)dr}{= \frac{n^2a_0}{Z}\lbrace 1 + \frac{1}{2}\lbrack  1 - \frac{l(l+1)}{n^2}\rbrack\rbrace}$$
 
-Note that the expectation value of $r$ and the most probable value for $r$ are not equal. The expectation value can be thought of like *an average* and the most probable value like a *maximum value*.
+- Note that the expectation value of $r$ and the most probable value for $r$ are not equal. The expectation value can be thought of like *an average* and the most probable value like a *maximum value*.
 
 
-The probability density (including the angular variables) for the electron in a hydrogenlike atom is given by:
+- The probability density (including the angular variables) for the electron in a hydrogenlike atom is given by:
 
 $${\psi^*_{nlm}(r,\theta,\phi)\psi_{nlm}(r,\theta,\phi) = |N_{nl}R_{nl}(r)Y_l^m(\theta,\phi)|^2}$$
 
-This function depends on three variables and is difficult to plot directly. Previously, we have seen that it is convenient to plot contour levels, which contain the electron with, for example, 90\% probability.
+- This function depends on three variables and is difficult to plot directly. Previously, we have seen that it is convenient to plot contour levels, which contain the electron with, for example, 90\% probability.
 
 
 
 ### 3D shapes of orbitals
 
 
-For degenerate states with $l > 0$, we have an additional degree of freedom in choosing how to represent the orbitals. In fact, any linear combination of given $3l$ orthogonal eigenfunctions corresponding to a degenerate set with orbital angular momentum $l$, is also a solution to the Schr\"odinger equation.
+- For degenerate states with $l > 0$, we have an additional degree of freedom in choosing how to represent the orbitals. In fact, any linear combination of given $3l$ orthogonal eigenfunctions corresponding to a degenerate set with orbital angular momentum $l$, is also a solution to the Schr\"odinger equation.
 
 ![](images/AO.png)
 
-Two commonly used representations are the Cartesian form, which are real valued functions and have been, in the case of $l = 1$, denoted by $p_x$, $p_y$ and $p_z$, and the eigenfunctions of the angular momentum ($L^2$ and $L_z$), which are complex valued and are denoted by $p_{-1}$, $p_0$ and $p_{+1}$. The relation between the representations is:
+- Two commonly used representations are the Cartesian form, which are real valued functions and have been, in the case of $l = 1$, denoted by $p_x$, $p_y$ and $p_z$, and the eigenfunctions of the angular momentum ($L^2$ and $L_z$), which are complex valued and are denoted by $p_{-1}$, $p_0$ and $p_{+1}$. The relation between the representations is:
 
 
 
@@ -90,12 +92,13 @@ $${d_{x^2 - y^2} = \frac{1}{\sqrt{2}} \left(d_{+2} + d_{-2}\right)\textnormal{,
 - What are the energies of these states?
 - Consider several values for n, and show that the number of orbitals for each $n$ is $n^2$
 
-:::{dropdown} **Solution**
+:::{admonition} **Solution**
+:class: dropdown
 
 If  n=3 the allowed values of l are 0, 1, and 2. 
-- If  l=0; m=0 (1 state)
-- If  l=1; m=−1,0,1 (3 states);
-- If  l=2; m=−2,−1,0,1,2 (5 states). 
+- If  $l=0;\, m=0$ (1 state)
+- If  $l=1;\, m=−1,0,1$ (3 states);
+- If  $l=2;\, m=−2,−1,0,1,2$ (5 states). 
 
 In total, there are 1 + 3 + 5 = 9 allowed states. This confirms that number of orbitals for H-atom is $n^2$
 
@@ -115,7 +118,8 @@ $$
 - What are the possible orientations for the angular momentum vector?
 
 
-:::{dropdown} **Hint**
+:::{admonition} **Hint**
+:class: dropdown
 
 - 3d orbital corresponds to n=1 and l=2. A number of $m_l$ values are possible
 - Orientation of $L_z$. with respect to L is defined by $cos\theta = L_z/L$
@@ -130,7 +134,8 @@ $$
 \psi_{3p_z} = \frac{1}{81}\sqrt{\frac{2}{\pi}}\left(\frac{1}{a_0}\right)^{3/2}\left(6\frac{r}{a_0} - \frac{r^2}{a^2_0}\right) e^{-r/3a_0}	{cos}(\theta)
 $$
 
-:::{dropdown} **Solution**
+:::{admonition} **Solution**
+:class: dropdown
 
 - $3p_z$ corresponds to $n=3$, $l=1$ and $m_l=0$
 
@@ -147,7 +152,8 @@ $$
 - Calculate probabiltiy to find the electron within first Bohr raidus $a_0.$
 - calcualte the most probable value of r to find the electron. 
 
-:::{dropdown} **Solution**
+:::{admonition} **Solution**
+:class: dropdown
 
 - Ground state is 1s;
 
@@ -192,7 +198,8 @@ Carrying out the derivative we find that $r_max=a_0$
 
 Show that $|210\rangle$ is normalized and orthogoanal to $|200\rangle$
 
-:::{dropdown} Hint
+:::{admonition} Hint
+:class: dropdown
 
 - This problem is solved in Chapter 7 of McQuarrie's book. Page 326
 :::
@@ -201,7 +208,8 @@ Show that $|210\rangle$ is normalized and orthogoanal to $|200\rangle$
 
 Calculate average potential energy of H-atom in its ground and first excited states.
 
-:::{dropdown} Hint
+:::{admonition} Hint
+:class: dropdown
 
 - This problem is solved in Chapter 7 of McQuarrie's book. Page 3332
 :::