Atomic physics
The quantum description of atoms
Atomic physics
The quantum description of atoms
Recall: Bohr's Model of the Hydrogen Atom
Atomic models
Bohr's model
Quantization of electronic orbits
r_n=\frac{n^2\hbar^2}{kme^2}=n^2a_0
\text{where} \quad n=1,2,3,...
\text{and} \quad a_0=\frac{\hbar^2}{kme^2}=0.053\text{nm}
E_n=-\frac{k e^2 }{2r_n}=-\frac{1}{n^2}\frac{k e^2 }{2a_0}
=-\frac{13.606\ \text{eV }}{n^2}
\text{where} \quad n=1,2,3,...
Atomic Physics
Introduction
Recall: Bohr's Model of the Hydrogen Atom
Atomic physics
The quantum description of atoms
A quick word about Harmonics
Atomic Physics
Harmonics
What are Harmonics?
Waves are correlated amplitudes across space and time.
Atomic Physics
Harmonics
One Dimensional Harmonics
Classical: Standing waves on a string
Quantum: infinite well
1D ring in 2D
Atomic Physics
Harmonics
One Dimensional Harmonics: time dependence
\sin(kx-\omega t)+\sin(kx+\omega t)
The time evolution can be understood as an interference between oppositely traveling waves (with opposite but equal momenta)
\sin(kx) \times \cos(\omega t)
=
The number of nodes and antinodes depends on the momentum
\Psi(x,t)= e^{-i\omega t} \sin(kx)
The time evolution of quantum states is in the form of a time dependent phase.
Atomic Physics
Harmonics
Two Dimensional Harmonics
2D-harmonics
Cymatics
Atomic Physics
Harmonics
Two Dimensional Harmonics
2D-harmonics
Cymatics
Quantum infinite well
\psi(x,y)=A\ \sin (k_x x)\ \sin (k_y y)
Atomic Physics
Harmonics
Two Dimensional Harmonics
2D-harmonics
2D surface in 3D
spherical Harmonics
Atomic physics
The quantum description of atoms
Schrodinger's Wave Equation in 3D - Cartesian Coodinates
Atomic Physics
Introduction
Schrodinger's wave equation in 3-D
-\frac{\hbar^2}{2m}\frac{d^2 \psi
(x)}{d x^2}+V(x)\ \psi(x)=E\ \psi(x)
Recall the time-independent Schrodinger wave equation in one dimension
-\frac{\hbar^2}{2m}
\left[
\frac{\partial^2 }{\partial x^2}
+\frac{\partial^2 }{\partial y^2}
+\frac{\partial^2 }{\partial z^2}
\right]
\psi(x,y,z)
+V(x,y,z)\ \psi(x,y,z)=E\ \psi(x,y,z)
generalizing to 3-dimensions, this equation takes the form:
Atomic Physics
Introduction
3-D infinite box
Consider a 3-D box, where the potential vanishes inside, but is infinite outside, containing an otherwise free particle.
\psi(x)=A\ \sin (k_x x)\ \sin (k_y y)\ \sin (k_z z)
It is reasonable to assume a solution of the form
Satisfying the boundary conditions
\psi (x=0)=\psi (x=L_1)=0 \implies k_x L_1= n_1\pi
\psi (y=0)=\psi (y=L_2)=0 \implies k_y L_2= n_2\pi
\psi (z=0)=\psi (z=L_3)=0 \implies k_z L_3= n_3\pi
E=\frac{\pi^2\hbar^2}{2m}\left( \frac{n_1^2}{L_1^2}+\frac{n_2^2}{L_2^2}+\frac{n_3^2}{L_3^2}\right)
Which is a solution to Schrodinger's equation, with energies
E=\frac{\hbar^2}{2m}\left( k_x^2+k_y^2+k_z^2\right)
E
Atomic Physics
Introduction
3-D infinite box
If all the lengths are equal, cubic potential, then
E=\frac{\pi^2\hbar^2}{2m L^2}\left( {n_1^2}+{n_2^2}+{n_3^2}\right)
n_1
n_2
n_3
1
1
1
E=\frac{3\pi^2\hbar^2}{2m L^2}
E=\frac{3\pi^2\hbar^2}{m L^2}
\}
\}
Ground state is unique (nondegenerate)
First excited state is degenerate, with degeneracy = 3
2
1
1
1
2
1
1
1
2
Atomic Physics
Introduction
The Schrodinger Wave Equation in spherical coordinates
-\frac{\hbar^2}{2m}
\left[
\frac{1}{r^2}\frac{\partial }{\partial r} \left(r^2 \frac{\partial }{\partial r}\right)
+\frac{1}{r^2 \sin\theta}\frac{\partial }{\partial \theta} \left(\sin\theta \frac{\partial }{\partial \theta}\right)
+\frac{1}{r^2 \sin^2 \theta}\frac{\partial }{\partial \phi} \left(\frac{\partial }{\partial \phi}\right)
\right]
\psi(r,\theta,\phi)\\
\qquad \qquad
\qquad \qquad
\qquad \qquad
+V(r,\theta,\phi)\ \psi(r,\theta,\phi)=E\ \psi(r,\theta,\phi)
The Schrodinger wave equation in spherical coordinates:
Atomic physics
The quantum description of atoms
Schrodinger's Wave Equation in 3D - Spherical Coordinates
Atomic Physics
The Hydrogen Atom
The Schrodinger Wave Equation in spherical coordinates
To a good approximation the potential energy of the electron-proton system in the hydrogen atom is spherically symmetric:
V(r)=-\frac{k e^2}{r}
... to the time-independent Schrodinger wave equation:
\frac{1}{r^2}\frac{\partial }{\partial r} \left(r^2 \frac{\partial \psi}{\partial r}\right)
+\frac{1}{r^2 \sin\theta}\frac{\partial }{\partial \theta} \left(\sin\theta \frac{\partial \psi}{\partial \theta}\right)
+\frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 \psi }{\partial \phi^2}
+ \frac{2\mu}{\hbar^2}
\left[E-V(r)\right] \psi = 0\
So we can assume a separable solution
\psi(r,\theta,\phi)=R(r)f(\theta)g(\phi)
Atomic Physics
The Hydrogen Atom
The Schrodinger Wave Equation in spherical coordinates
... to the time-independent Schrodinger wave equation:
\frac{1}{r^2}\frac{\partial }{\partial r} \left(r^2 \frac{\partial \psi}{\partial r}\right)
+\frac{1}{r^2 \sin\theta}\frac{\partial }{\partial \theta} \left(\sin\theta \frac{\partial \psi}{\partial \theta}\right)
+\frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 \psi }{\partial \phi^2}
+ \frac{2\mu}{\hbar^2}
\left[E-V(r)\right] \psi = 0\
R(r)=A e^{-r/a_0}
f(\theta) g(\phi)= Y_{l,m_l}(\theta,\phi)
which, after many efforts by many people and quite a bit of math, leads to the solution:
E_n=-\frac{1}{n^2}\frac{ke^2}{2a_0}
\psi_{n,l,m_l}(r,\theta,\phi)=R_{n,l}(r) Y_{l,m_l}(\theta,\phi)
Atomic Physics
The Hydrogen Atom
Quantum Numbers
E_n=-\frac{1}{n^2}\frac{ke^2}{2a_0}
\psi_{n,l,m_l}(r,\theta,\phi)=R_{n,l}(r) Y_{l,m_l}(\theta,\phi)
n
Principal quantum number
1,2,3,....,\infty
l
Orbital angular momentum quantum number
0,1,2,....,n-1
m_l
Magnetic quantum number
-l,...-1,0,1,....,l
L=\sqrt{l(l+1)\hbar}
L_\alpha=m_l\hbar
Atomic Physics
The Hydrogen Atom
Quantum Numbers
Enumerate the quantum numbers for states with and count the degeneracy of each set.
n\le3
Atomic Physics
The Hydrogen Atom
Probability density
dP(r,\theta,\phi)=\psi^*\psi d\tau
The probability of finding the particle in an infinitesimal volume element
d\tau=r^2\sin\theta \ dr\ d\theta \ d\phi
is given by:
P(r)=r^2 |R_{n,l}(r)|^2
which, when integrated over
gives a radial probability density of
\theta , \phi
Atomic Physics
The Hydrogen Atom
Probability density
Atomic Physics
The Hydrogen Atom
Probability density
Atomic physics
The quantum description of atoms
Other Atoms
Atomic Physics
Other atoms
Shells, subshells, and the periodic table
Adding more protons and electrons to atoms is mathematically challenging.
Accurate results are obtained numerically.
Atomic Physics
Other atoms
Shells, subshells, and the periodic table
Two more important pieces to building atoms
m_s
Spin quantum number
-\tfrac{1}{2},\tfrac{1}{2}
1
Pauli's exclusion principle
2
No two electrons can have exactly the same set of quantum numbers
Atomic Physics
Other atoms
Shells, subshells, and the periodic table
Atomic Physics
Other atoms
Shells, subshells, and the periodic table
Atomic Physics
Other atoms
Shells, subshells, and the periodic table
Atomic Physics
Other atoms
Other forms of the periodic ...
Atomic Physics
Other atoms
Summary
Atomic Physics
Other atoms
Summary
Quantum description of Atoms
By omoussa
