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

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

2

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