Spontaneous emission formula

Explore the spontaneous emission formula, its implications in quantum systems, and an example calculation for a hydrogen atom.

Understanding the Spontaneous Emission Formula

Spontaneous emission is a fundamental process in quantum physics, in which an excited atom, ion or molecule undergoes a transition to a lower energy state, emitting a photon in the process. This phenomenon is essential to the behavior of various systems, including lasers, atomic clocks, and quantum information processing. In this article, we will explore the spontaneous emission formula and its implications in these systems.

The Formula for Spontaneous Emission

The spontaneous emission rate, often denoted as A₂₁, describes the probability per unit time that an excited state (state 2) will decay to a lower energy state (state 1), emitting a photon in the process. The formula for spontaneous emission rate is given by:

1. A₂₁ = (e²ω³ |μ₂₁|²)/(3πε₀ħc³)

In this formula:

• e is the elementary charge,
• ω is the angular frequency of the emitted photon,
• μ₂₁ is the transition dipole moment between state 2 and state 1,
• ε₀ is the vacuum permittivity,
• ħ is the reduced Planck’s constant, and
• c is the speed of light in a vacuum.

Implications of the Spontaneous Emission Formula

The spontaneous emission formula offers valuable insights into the behavior of atoms and other quantum systems. Below are some of the key implications:

• Lifetime of Excited States: The inverse of the spontaneous emission rate, 1/A₂₁, provides an estimate for the lifetime of the excited state. This lifetime is crucial for understanding the behavior of atomic clocks and lasers, which rely on the precise control of atomic energy levels.
• Emission Spectrum: The spontaneous emission rate is directly related to the intensity of the emitted radiation, allowing us to calculate the emission spectrum of the system. This spectrum is essential for understanding the properties of light sources, such as LEDs and gas discharge lamps.
• Quantum Information Processing: Spontaneous emission is a source of decoherence in quantum systems, as it introduces randomness in the evolution of quantum states. Understanding the spontaneous emission rate is essential for developing strategies to counteract decoherence and improve the performance of quantum information processing devices.

In conclusion, the spontaneous emission formula is a powerful tool for understanding the behavior of various quantum systems. By providing insights into the decay of excited states, emission spectra, and decoherence in quantum information processing, the spontaneous emission rate allows us to design and analyze the performance of numerous devices and technologies that rely on the control of atomic and molecular energy levels.

Example of Spontaneous Emission Rate Calculation

Let’s consider a simple two-level system, such as a hydrogen atom, with an electron transitioning from the n=2 to the n=1 energy level. In this example, we will calculate the spontaneous emission rate for this transition.

1. First, we need to find the angular frequency (ω) of the emitted photon. We can obtain this from the energy difference (ΔE) between the two energy levels using the relation ω = ΔE/ħ. For a hydrogen atom, the energy difference between n=2 and n=1 levels is approximately 10.2 eV. We convert this to joules by multiplying with the elementary charge (e), resulting in ΔE ≈ 1.63 × 10^-18 J. Using the reduced Planck’s constant (ħ ≈ 1.05 × 10^-34 Js), we find ω ≈ 1.55 × 10¹⁵ rad/s.
2. Next, we need to determine the transition dipole moment (|μ₂₁|) for the hydrogen atom. For simplicity, we will use an approximate value of |μ₂₁| ≈ 1 × 10^-29 Cm, which is a reasonable estimate for hydrogen-like systems.
3. Now we have all the necessary values to calculate the spontaneous emission rate (A₂₁) using the formula:

A₂₁ = (e²ω³ |μ₂₁|²)/(3πε₀ħc³)

Plugging in the known values:

• e ≈ 1.6 × 10^-19 C
• ω ≈ 1.55 × 10¹⁵ rad/s
• |μ₂₁| ≈ 1 × 10^-29 Cm
• ε₀ ≈ 8.85 × 10^-12 C²/N·m²
• ħ ≈ 1.05 × 10^-34 Js
• c ≈ 3 × 10⁸ m/s

After substituting these values and calculating, we find:

A₂₁ ≈ 6.24 × 10⁶ s^-1

This means that the spontaneous emission rate for an electron transitioning from the n=2 to the n=1 energy level in a hydrogen atom is approximately 6.24 × 10⁶ s^-1. Thus, the lifetime of the excited state (1/A₂₁) is approximately 1.6 × 10^-7 s or 160 ns.