Bohr's model and physics of the atom

• Energy in the nth state: $E_n = -\frac{mZ^2e^4}{8\varepsilon_0^2h^2n^2}$

• For hydrogen (Z=1): $E_n = \frac{E_1}{n^2} = -\frac{13.6\text{ eV}}{n^2}$

• Derivation steps:

  1. Kinetic energy: $K = \frac{1}{2}mv^2 = \frac{mZ^2e^4}{8\varepsilon_0^2h^2n^2}$
  2. Potential energy: $V = -\frac{Ze^2}{4\pi\varepsilon_0r} = -\frac{mZ^2e^4}{4\varepsilon_0^2h^2n^2}$
  3. Total energy: $E = K + V = -\frac{mZ^2e^4}{8\varepsilon_0^2h^2n^2}$

• Energy scales with $Z^2$ and inversely with $n^2$

• All energy levels are negative (bound states)

• E = 0 corresponds to ionization (electron completely removed)

• Contradiction: According to Maxwell's theory, an accelerating electron must continuously emit electromagnetic radiation, but this doesn't happen in stable atoms.
• Problems this created:
  1. A revolving electron should continuously lose energy through radiation
  2. This energy loss should cause the electron to spiral into the nucleus
  3. The frequency of radiation should continuously change as the orbit shrinks
  4. Atoms should be inherently unstable (they aren't)
  5. Atoms should emit continuous spectra (they emit discrete line spectra)

This contradiction required Bohr's quantum approach to resolve.

Limitations of Bohr's model:

1. Works only for hydrogen and hydrogen-like ions (single electron systems)

   • Failed for multi-electron atoms like helium

2. Cannot explain fine structure of spectral lines

   • Higher precision measurements showed multiple components in lines

3. Doesn't account for the Zeeman effect (line splitting in magnetic fields)

4. Violates the uncertainty principle (assumes precise orbit with known position and momentum)

5. Arbitrary postulates lack theoretical foundation

   • Inconsistent with Maxwell's theory without proper justification

6. Fails to explain molecular bonding or chemical properties

Quantum mechanics replaced it by providing a comprehensive wave-based theory that overcomes all these limitations.

• Rydberg constant: $R = \frac{me^4}{8\varepsilon_0^2h^3c} \approx 1.097 \times 10^7 \text{ m}^{-1}$

• Significance:

  • It appears in the formula for all spectral lines: $\frac{1}{\lambda} = RZ^2(\frac{1}{n^2} - \frac{1}{m^2})$
  • Connects observable spectra to fundamental constants
  • Provides a way to experimentally verify atomic theory
  • Its value is the same for all hydrogen-like atoms after accounting for Z

• Relationship to energy:

  • Ground state energy: $E_1 = -hcR = -13.6$ eV
  • One Rydberg of energy = 13.6 eV

• It represents one of the most precisely measured fundamental constants in physics

Some alpha particles (approximately 1 in 8000) were deflected at very large angles (>90°), with some even bouncing back toward the source.

Key implications:

• Indicated concentrated positive charge and mass in a tiny central region (nucleus)
• Contradicted Thomson's "plum pudding" model where charge was uniformly distributed
• Showed atoms must be mostly empty space with a dense core
• Demonstrated that the positive charge must be associated with a very massive particle

The extreme deflection could only occur if the alpha particles encountered a concentrated, massive, positively charged entity within the atom.

The three main spectral series of hydrogen are:

1. Lyman series:

   • Wavelength range: ~90-121.6 nm
   • Ultraviolet region
   • Corresponds to electron transitions to the ground state (n=1)
   • Series limit at 91.2 nm (n=∞ to n=1 transition)

2. Balmer series:

   • Wavelength range: ~365-656.3 nm
   • Visible light region
   • Corresponds to electron transitions to the second energy level (n=2)
   • First line at 656.3 nm (n=3 to n=2 transition)

3. Paschen series:

   • Wavelength range: ~820-1875 nm
   • Infrared region
   • Corresponds to electron transitions to the third energy level (n=3)

To identify and calculate hydrogen spectral lines:

1. Series identification: Determine by wavelength region and corresponding transition:

   • Ultraviolet region (90-121.6 nm): Lyman series (transitions to n=1)
   • Visible region (365-656.3 nm): Balmer series (transitions to n=2)
   • Infrared region (820-1875 nm): Paschen series (transitions to n=3)
   • Longer infrared: Brackett (n=4), Pfund (n=5), etc.

2. Wavelength calculation using Rydberg formula: $\frac{1}{\lambda} = R\left(\frac{1}{n^2} - \frac{1}{m^2}\right)$

   Where:

   • R = 1.097 × 10⁷ m⁻¹ (Rydberg constant)
   • n = lower energy level (fixed for a series)
   • m = higher energy level (m > n)

3. Line identification within a series:

   • First line (m = n+1): Longest wavelength in series
   • Series limit (m = ∞): Shortest wavelength in series
   • Spacing between lines decreases as wavelength decreases

Example: For Balmer series first line (n=2, m=3): $\frac{1}{\lambda} = 1.097 × 10⁷ \left(\frac{1}{4} - \frac{1}{9}\right) = 1.097 × 10⁷ × \frac{5}{36} = 1.52 × 10⁶$ $\lambda = 6.56 × 10^{-7} \text{ m} = 656 \text{ nm}$

A metastable state is an excited energy state with:

• Unusually long lifetime (milliseconds vs. nanoseconds for typical excited states)
• Restricted transitions to lower states due to quantum selection rules
• ~10⁵ times longer lifetime than regular excited states

Importance for lasers:

• Serves as an "energy storage" level that accumulates excited atoms
• Allows population buildup required for achieving population inversion
• Provides time for stimulated emission to occur before spontaneous decay
• Enables continuous pumping to maintain inversion conditions

In a He-Ne laser, helium's metastable state (lifetime ~1 ms) accumulates energy that transfers to neon's upper laser level, facilitating the population inversion between neon's energy levels E₂ and E₁.

Relationship between coherence length and monochromaticity:

• Coherence length (L_c) is the distance over which light maintains a predictable phase relationship
• Monochromaticity is measured by spectral width (Δλ or Δν)

Mathematical relationship:

• L_c = c/Δν = λ²/Δλ
• Where c is speed of light, λ is wavelength

Implications:

• Perfect monochromaticity (Δλ=0) would give infinite coherence length
• More monochromatic light (smaller Δλ) has longer coherence length
• He-Ne laser with Δλ≈10⁻⁶ nm has coherence length of several hundred meters
• Ordinary light sources (Δλ≈1 nm) have coherence lengths of micrometers

Example: He-Ne laser with wavelength spread 1000× smaller than conventional sources has 1000× greater coherence length, enabling applications like holography and interferometry that depend on long-distance phase relationships.

Three fundamental atomic processes in laser physics:

1. Stimulated Absorption:

   • An atom in lower energy state E₁ absorbs an incident photon
   • The atom jumps to higher energy state E₂
   • Energy of absorbed photon equals E₂-E₁

2. Spontaneous Emission:

   • An atom in higher energy state E₂ spontaneously drops to lower state E₁
   • A photon with energy E₂-E₁ is emitted in a random direction
   • Occurs naturally without external stimulus
   • Average time before occurrence defines the state's lifetime

3. Stimulated Emission:

   • An atom in higher energy state E₂ interacts with an incident photon
   • The atom drops to lower state E₁, emitting a second photon
   • The emitted photon has identical energy, phase, and direction as the incident photon
   • This process is the fundamental mechanism enabling laser action