Quadratic Stark effect equation

Explore the quadratic Stark effect in quantum physics, its equation, significance, and a practical example of its application.

Introduction to the Quadratic Stark Effect

The Stark effect, a central concept in quantum mechanics, refers to the shift and splitting of spectral lines of atoms and molecules due to the presence of an external electric field. The Stark effect is categorized into two types: the linear and the quadratic Stark effect. This article primarily explores the quadratic Stark effect, a nonlinear phenomenon where the shift of energy levels is proportional to the square of the field strength.

The Quadratic Stark Effect Equation

The quadratic Stark effect takes into account the second-order perturbations in the Hamiltonian. The quadratic Stark shift, ΔE, of an atomic energy level in a weak electric field E, is given by the equation:

ΔE = -1/2 ε₀ α₀ E²

Here, ε₀ represents the vacuum permittivity, α₀ is the static electric dipole polarizability, and E is the external electric field strength. This equation reflects the quadratic dependence of the shift on the electric field’s strength, underlying the term “quadratic Stark effect”.

Significance and Interpretation

1. The quadratic Stark effect equation allows us to predict how energy levels of atoms shift in response to external electric fields. This is crucial in many applications, including atomic clocks and precision spectroscopy.

2. The quadratic term in the equation signifies a nonlinear response of the atom to the electric field. It reflects the polarization of the atom due to the field, and the resulting energy shift.

3. The static electric dipole polarizability, α₀, indicates the extent to which an atom can be distorted by an electric field. Thus, the Stark effect also provides insights into atomic structure and properties.

Conclusion

In conclusion, the quadratic Stark effect and its corresponding equation offer significant insights into the behaviour of atoms under external electric fields. Despite its apparent simplicity, this equation is the basis for various applications in modern physics, demonstrating the interconnectedness of theoretical predictions and practical use cases.

Example of a Quadratic Stark Effect Calculation

Let’s consider a hydrogen atom (H) in a weak electric field E = 5×10⁵ V/m. The static electric dipole polarizability of a hydrogen atom is approximately α₀ = 8.5×10^-41 C²m²J^-1. With these values, we can calculate the energy shift ΔE due to the quadratic Stark effect.

Step 1: Determine the Variables

• The static electric dipole polarizability, α₀ = 8.5×10^-41 C²m²J^-1

• The external electric field strength, E = 5×10⁵ V/m

• The vacuum permittivity, ε₀ = 8.854×10^-12 C²N^-1m^-2

Step 2: Apply the Quadratic Stark Effect Equation

Now, we substitute these values into the quadratic Stark effect equation:

ΔE = -1/2 ε₀ α₀ E²

Step 3: Compute the Energy Shift

With the given variables, we can calculate the Stark shift ΔE. After performing the calculation, you will get the energy shift caused by the quadratic Stark effect in the hydrogen atom due to the external electric field.

This example clearly demonstrates how to apply the quadratic Stark effect equation to compute energy shifts in atoms subjected to external electric fields. It is a practical illustration of the theoretical concepts described earlier.