Under [Atom-light interaction](Atom-light%20interaction.md) or any [analogous situation](dipole%20approximation.md#Use%20cases%20for%20the%20dipole%20approximation), the corresponding wave function of an [[electron]] bound to its [atom](Atomic%20Physics%20(Index).md#Atoms) is typically contained within a radius of $1$ nm or less, while the wavelength of visible light is on the order of a few hundred nms (on the order of several hundred to thousands of times larger). Thus, we may make the _dipole approximation._ This involves removing the spatial variation in the [Light-field](Light-field.md) by taking the spatially dependent exponential terms to be linear. Thus, if we take the electromagnetic component of a quantized light field to be [$\hat{\mathbf{E}}(\mathbf{r},t)=i\sum_{\mathbf{k},\alpha}\sqrt{\frac{\hbar\omega_{\mathbf{k}}}{2V\varepsilon_0}}\mathbf{e}_{\mathbf{k}\alpha}(\hat{a}_{\mathbf{k},\alpha}e^{i\mathbf{k}\cdot\mathbf{r}-i\omega_{\mathbf{k}}t}+\hat{a}_{\mathbf{k},\alpha}^{\dagger}e^{-i\mathbf{k}\cdot\mathbf{r}+i\omega_{\mathbf{k}}t})$](electromagnetic%20wave%20quantization.md#^a3e9ae) we rewrite the complex exponential term such that $e^{\pm i\mathbf{k}\cdot\mathbf{r}}\approx 1+i\mathbf{k}\cdot\mathbf{r}$ and the electromagnetic field component becomes $\hat{\mathbf{E}}(\mathbf{r},t)=i\sum_{\alpha}\sqrt{\frac{\hbar\omega_{\mathbf{k}}}{2V\varepsilon_0}}\mathbf{e}_{\alpha}(\hat{a}_{\mathbf{k},\alpha}e^{-i\omega_{\mathbf{k}}t}+\hat{a}_{\mathbf{k},\alpha}^{\dagger}e^{+i\omega_{\mathbf{k}}t})$ # Use cases for the dipole approximation This approximation may be valid any situation where a [Wavefunction](Wavefunction.md) is significantly smaller than the wavelength of the light shined on the [quantum system](Quantum%20systems.md) it describes. #QuantumMechanics/AtomicPhysics #QuantumMechanics/TwoLevelSystems