← Part 1: Atomic Structure & Quantum Mechanics

Part 1, Topic 1 | Lectures 1–4

Atomic Structure & Models

From Bohr's quantized orbits to wave-particle duality

1.1 Early Atomic Models

The quest to understand the atom spans over two millennia. The ancient Greeks proposed that matter was composed of indivisible units (atomos), but it was not until the early 20th century that a quantitative model emerged. Key milestones include:

●Dalton (1808): Atoms are indivisible, elements consist of identical atoms
●Thomson (1897): Discovery of the electron; "plum pudding" model with electrons embedded in positive charge
●Rutherford (1911): Gold foil experiment reveals a dense, positively charged nucleus surrounded by electrons
●Bohr (1913): Quantized electron orbits explain the hydrogen emission spectrum

Rutherford's nuclear model posed a critical problem: according to classical electrodynamics, an orbiting electron should continuously radiate energy and spiral into the nucleus. Bohr resolved this by postulating that electrons occupy only certain allowed orbits where they do not radiate.

1.2 The Bohr Model of Hydrogen

Bohr's model assumes the electron orbits the proton in circular paths. The key postulates are:

Electrons move in circular orbits around the nucleus under Coulomb attraction
Only orbits with quantized angular momentum $L = n\hbar$ are allowed ($n = 1, 2, 3, \ldots$)
Electrons in allowed orbits do not radiate; radiation occurs only during transitions between orbits
The frequency of emitted radiation satisfies $E_i - E_f = h\nu$

Energy Levels

Balancing the Coulomb force with centripetal acceleration and applying the angular momentum quantization condition, we obtain the energy of the $n$-th level for a hydrogen-like atom with nuclear charge $Z$:

$$E_n = -\frac{13.6 \text{ eV} \cdot Z^2}{n^2}$$

For hydrogen ($Z = 1$): $E_1 = -13.6$ eV, $E_2 = -3.4$ eV, $E_3 = -1.51$ eV, ...

Bohr Radii

The radius of the $n$-th Bohr orbit is:

$$r_n = \frac{n^2 a_0}{Z}$$

where $a_0 = 0.529$ Angstrom ($52.9$ pm) is the Bohr radius

The ground state ($n = 1$) of hydrogen has $r_1 = a_0 = 52.9$ pm. As $n$ increases, the orbital radius grows as $n^2$, meaning excited states are much larger than the ground state. For hydrogen-like ions with $Z > 1$, the orbits are $Z$ times smaller due to the stronger nuclear attraction.

1.3 The Hydrogen Emission Spectrum

When an electron transitions from a higher energy level $n_2$ to a lower level $n_1$, it emits a photon. The wavelength of this photon is given by the Rydberg formula:

$$\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$$

where $R_H = 1.097 \times 10^7$ m$^{-1}$ is the Rydberg constant

Spectral Series

Lyman Series (UV)

Transitions to $n_1 = 1$

91.2 – 121.6 nm

Balmer Series (Visible)

Transitions to $n_1 = 2$

364.6 – 656.3 nm

Paschen Series (IR)

Transitions to $n_1 = 3$

820.4 – 1875 nm

The Balmer series is historically important because it falls in the visible range and was the first to be observed. The H-alpha line at 656.3 nm (red) is one of the most recognizable spectral lines in all of spectroscopy.

1.4 Wave-Particle Duality

de Broglie Wavelength

In 1924, Louis de Broglie proposed that all matter exhibits wave-like behavior. The wavelength associated with a particle of momentum $p$ is:

$$\lambda = \frac{h}{p} = \frac{h}{mv}$$

For macroscopic objects, the de Broglie wavelength is negligibly small. But for electrons, it is comparable to atomic dimensions, which is why quantum effects are essential at the atomic scale. An electron accelerated through 100 V has $\lambda \approx 0.12$ nm — similar to atomic spacings, enabling electron diffraction.

Photoelectric Effect

Einstein explained the photoelectric effect (1905) by proposing that light consists of discrete packets of energy (photons), each carrying energy $E = h\nu$. When a photon strikes a metal surface, it can eject an electron if its energy exceeds the work function $\phi$:

$$E_k = h\nu - \phi$$

$E_k$ = kinetic energy of ejected electron, $\phi$ = work function of the metal

Key observations: (1) below the threshold frequency $\nu_0 = \phi / h$, no electrons are emitted regardless of light intensity; (2) above the threshold, the kinetic energy of ejected electrons increases linearly with frequency; (3) increasing intensity increases the number of ejected electrons but not their kinetic energy.

1.5 Planck's Quantum Hypothesis

The quantum revolution began with Planck's resolution of the blackbody radiation problem (1900). Classical physics predicted that a heated object should radiate infinite energy at high frequencies (the "ultraviolet catastrophe"). Planck resolved this by proposing that energy is emitted and absorbed in discrete quanta:

$$E = nh\nu, \quad n = 0, 1, 2, \ldots$$

$h = 6.626 \times 10^{-34}$ J$\cdot$s is Planck's constant

This seemingly small assumption — that energy comes in discrete packages rather than being continuously variable — had revolutionary consequences. It led directly to Einstein's photon hypothesis, Bohr's atomic model, and ultimately the full framework of quantum mechanics.

The combination of the photoelectric effect ($E = h\nu$), the Bohr model ($E_n = -13.6/n^2$ eV), and de Broglie's matter waves ($\lambda = h/p$) established the foundation upon which Schrodinger and Heisenberg would build the complete theory of quantum mechanics in 1925–1926.

Interactive Simulation: Hydrogen Spectrum

This simulation plots the hydrogen energy levels and computes the wavelengths for the Lyman, Balmer, and Paschen spectral series using the Rydberg formula.

Hydrogen Energy Levels & Spectral Series

Python

Plots Bohr energy levels with transition arrows and computes wavelengths for the three main spectral series of hydrogen.

script.py87 lines

import numpy as np
import matplotlib.pyplot as plt
# Constants
R_H = 1.097e7  # Rydberg constant (1/m)
eV_per_level = 13.6  # eV for hydrogen

# --- Left panel: Energy level diagram ---
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 8))
fig.patch.set_facecolor('#0f172a')
for ax in (ax1, ax2):
    ax.set_facecolor('#1e293b')
    ax.tick_params(colors='#cbd5e1')
    ax.xaxis.label.set_color('#cbd5e1')
    ax.yaxis.label.set_color('#cbd5e1')
    ax.title.set_color('#cbd5e1')
    for sp in ax.spines.values():
        sp.set_color('#334155')

n_levels = 7
energies = [-13.6 / n**2 for n in range(1, n_levels + 1)]
colors_n = ['#f472b6', '#fb923c', '#facc15', '#4ade80', '#38bdf8', '#a78bfa', '#e879f9']

for i, (E, c) in enumerate(zip(energies, colors_n)):
    n = i + 1
    ax1.hlines(E, 0.2, 0.8, colors=c, linewidth=3)
    ax1.text(0.85, E, f'n={n}  ({E:.2f} eV)', color=c, fontsize=9, va='center')

# Draw Lyman transitions (to n=1)
for n2 in range(2, 5):
    E1 = energies[0]
    E2 = energies[n2 - 1]
    ax1.annotate('', xy=(0.35, E1), xytext=(0.35, E2),
                 arrowprops=dict(arrowstyle='->', color='#f472b6', lw=1.5))

# Draw Balmer transitions (to n=2)
for n2 in range(3, 7):
    E1 = energies[1]
    E2 = energies[n2 - 1]
    ax1.annotate('', xy=(0.5, E1), xytext=(0.5, E2),
                 arrowprops=dict(arrowstyle='->', color='#38bdf8', lw=1.5))

ax1.set_xlim(0, 1.2)
ax1.set_ylim(-15, 1)
ax1.set_ylabel('Energy (eV)', fontsize=12)
ax1.set_title('Hydrogen Energy Levels', fontsize=14, fontweight='bold')
ax1.set_xticks([])
ax1.text(0.35, 0.5, 'Lyman', color='#f472b6', fontsize=9, ha='center')
ax1.text(0.5, 0.5, 'Balmer', color='#38bdf8', fontsize=9, ha='center')

# --- Right panel: Spectral series wavelengths ---
series_data = {
    'Lyman (UV)': {'n1': 1, 'n2_range': range(2, 8), 'color': '#f472b6'},
    'Balmer (Vis)': {'n1': 2, 'n2_range': range(3, 8), 'color': '#38bdf8'},
    'Paschen (IR)': {'n1': 3, 'n2_range': range(4, 10), 'color': '#4ade80'},
}

y_offset = 0
for name, info in series_data.items():
    wavelengths = []
    for n2 in info['n2_range']:
        inv_lam = R_H * (1/info['n1']**2 - 1/n2**2)
        lam_nm = 1e9 / inv_lam
        wavelengths.append(lam_nm)
    ax2.barh([y_offset]*len(wavelengths), wavelengths, height=0.3,
             color=info['color'], alpha=0.7, edgecolor=info['color'])
    for w in wavelengths:
        ax2.plot(w, y_offset, 'o', color=info['color'], markersize=6)
        ax2.text(w, y_offset + 0.2, f'{w:.0f}', color=info['color'],
                fontsize=7, ha='center', va='bottom')
    ax2.text(10, y_offset, name, color=info['color'], fontsize=10,
             fontweight='bold', va='center')
    y_offset -= 1

ax2.set_xlabel('Wavelength (nm)', fontsize=12)
ax2.set_title('Hydrogen Spectral Series', fontsize=14, fontweight='bold')
ax2.set_yticks([])
ax2.set_xlim(0, 2200)
ax2.grid(True, color='#334155', alpha=0.5, axis='x')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0f172a')
plt.close()
print("Hydrogen energy levels and spectral series plotted successfully.")
print(f"Ground state energy: {energies[0]:.2f} eV")
print(f"Lyman alpha: {1e9/R_H/(1 - 1/4):.1f} nm")
print(f"Balmer alpha (H-alpha): {1e9/R_H/(1/4 - 1/9):.1f} nm")
print(f"Paschen alpha: {1e9/R_H/(1/9 - 1/16):.1f} nm")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Bohr Radii & Energy Levels

This Fortran program computes Bohr radii and energy levels for hydrogen-like atoms (H, He$^+$, Li$^{2+}$), demonstrating how nuclear charge $Z$ affects atomic properties.

Bohr Radii & Energies for Hydrogen-Like Atoms

Fortran

Computes quantized energy levels and orbital radii for H, He+, and Li2+ using the Bohr model.

bohr_model.f9064 lines

program bohr_radii_energies
  implicit none
  integer :: n, Z
  double precision :: a0, E_n, r_n, eV_conv
  double precision :: hbar, me, e_charge, k_e, pi

! Physical constants (SI)
  hbar = 1.0545718d-34
  me = 9.1093837d-31
  e_charge = 1.6021766d-19
  k_e = 8.9875518d9
  pi = 3.14159265358979d0
  a0 = 5.29177d-11  ! Bohr radius in meters
  eV_conv = 13.6d0  ! Energy in eV for hydrogen

write(*,'(A)') '=============================================='
  write(*,'(A)') '  Bohr Model: Hydrogen-Like Atoms'
  write(*,'(A)') '=============================================='
  write(*,*)

! Hydrogen (Z=1)
  Z = 1
  write(*,'(A,I2,A)') '--- Hydrogen (Z = ', Z, ') ---'
  write(*,'(A6, A18, A18)') 'n', 'r_n (pm)', 'E_n (eV)'
  write(*,'(A)') '----------------------------------------------'
  do n = 1, 7
    r_n = a0 * dble(n*n) / dble(Z) * 1.0d12  ! convert to pm
    E_n = -eV_conv * dble(Z*Z) / dble(n*n)
    write(*,'(I6, F18.2, F18.4)') n, r_n, E_n
  end do

write(*,*)

! He+ (Z=2)
  Z = 2
  write(*,'(A,I2,A)') '--- Helium Ion He+ (Z = ', Z, ') ---'
  write(*,'(A6, A18, A18)') 'n', 'r_n (pm)', 'E_n (eV)'
  write(*,'(A)') '----------------------------------------------'
  do n = 1, 7
    r_n = a0 * dble(n*n) / dble(Z) * 1.0d12
    E_n = -eV_conv * dble(Z*Z) / dble(n*n)
    write(*,'(I6, F18.2, F18.4)') n, r_n, E_n
  end do

write(*,*)

! Li2+ (Z=3)
  Z = 3
  write(*,'(A,I2,A)') '--- Lithium Ion Li2+ (Z = ', Z, ') ---'
  write(*,'(A6, A18, A18)') 'n', 'r_n (pm)', 'E_n (eV)'
  write(*,'(A)') '----------------------------------------------'
  do n = 1, 7
    r_n = a0 * dble(n*n) / dble(Z) * 1.0d12
    E_n = -eV_conv * dble(Z*Z) / dble(n*n)
    write(*,'(I6, F18.2, F18.4)') n, r_n, E_n
  end do

write(*,*)
  write(*,'(A)') '--- Key Relationships ---'
  write(*,'(A)') 'r_n = n^2 * a0 / Z    (a0 = 52.9 pm)'
  write(*,'(A)') 'E_n = -13.6 * Z^2 / n^2  eV'
  write(*,'(A)') 'As Z increases: orbits shrink, energies become more negative'

end program bohr_radii_energies

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Video Lectures

Lecture 1: Introduction to Solid-State Chemistry

Goodie Bag 1: Atoms and Reactions

Lecture 2: The Periodic Table

Lecture 3: Atomic Models

Lecture 4: Atomic Spectra

Key Takeaways

●The Bohr model correctly predicts hydrogen energy levels: $E_n = -13.6/n^2$ eV
●Bohr radii scale as $n^2$ and inversely with $Z$: $r_n = n^2 a_0 / Z$
●The Rydberg formula predicts all hydrogen spectral lines from a single constant
●Wave-particle duality: $\lambda = h/p$ connects wavelength to momentum for all matter
●The photoelectric effect proves light is quantized: $E_k = h\nu - \phi$
●Planck's quantum hypothesis ($E = nh\nu$) launched the quantum revolution

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