Derivation of Level Transitions and Transition Probabilities in Quantum Mechanics

In the framework of quantum mechanics, the motion of electrons within an atom is fundamentally different from the continuous trajectories observed in classical mechanics. Instead, electrons are confined to discrete, quantized energy states known as stationary states or energy levels. A transition occurs when a system shifts from an initial state $|i\rangle$ with energy $E_i$ to a final state $|f\rangle$ with energy $E_f$. This process is intrinsically linked to the exchange of energy with the electromagnetic field: the system must either absorb a photon to move to a higher energy level or emit a photon to drop to a lower one. The energy of the photon is strictly governed by the relation $\Delta E = E_f - E_i = h\nu$, where $h$ is Planck's constant and $\nu$ is the frequency.

However, the mere existence of two energy levels does not guarantee a transition. The likelihood of such an event is not arbitrary; it is dictated by the underlying symmetry of the wavefunctions and the strength of the coupling between the atom and the external field, quantified by the transition matrix element.

Time-Dependent Perturbation Theory

To mathematically describe how an atom responds to external radiation, we employ time-dependent perturbation theory. We begin by considering an atom in an unperturbed state described by the Hamiltonian $H_0$, where the eigenstates $|\psi_n^{(0)}\rangle$ satisfy the time-independent Schrödinger equation:
$$H_0 |\psi_n^{(0)}\rangle = E_n^{(0)} |\psi_n^{(0)}\rangle$$

When the atom is exposed to a weak, time-varying electromagnetic field, such as an oscillating electric field $\mathbf{E}(t) = \mathbf{E}_0 \cos(\omega t)$, the total Hamiltonian of the system is modified:
$$H = H_0 + H'(t)$$
Here, $H'(t)$ represents the perturbation Hamiltonian. Under the electric dipole approximation—which assumes the wavelength of the radiation is much larger than the atomic dimensions—the interaction is dominated by the coupling between the field and the atom's electric dipole moment operator $\hat{\boldsymbol{\mu}}$. The perturbation can be expressed as:
$$H'(t) = -\hat{\boldsymbol{\mu}} \cdot \mathbf{E}(t) = -\hat{\boldsymbol{\mu}} \cdot \mathbf{E}_0 \cos(\omega t)$$

In the first-order approximation of perturbation theory, the probability amplitude of finding the system in a final state $|f\rangle$ after a time $t$ is directly related to the matrix element $\langle f | H'(t) | i \rangle$. This term encapsulates how effectively the external field "connects" the initial and final quantum states.

Fermi's Golden Rule and Transition Rates

When dealing with a continuum of final states or a radiation field with a broad spectrum, we are often interested in the transition rate $W_{i \to f}$—the probability of transition occurring per unit time. This is elegantly provided by Fermi's Golden Rule.

For a harmonic perturbation, the transition rate is proportional to the square of the transition matrix element and the density of states $\rho(\omega)$ at the transition frequency. For an absorption process, the rate is given by:
$$W_{i \to f} = \frac{2\pi}{\hbar} |\langle f | \hat{\boldsymbol{\mu}} \cdot \mathbf{E}_0 | i \rangle|^2 \rho(\omega)$$

In the context of atomic physics and spectroscopy, this relationship is often expressed through the Einstein B coefficient, which characterizes the strength of the interaction between the radiation field and the atomic transition:
$$B_{if} \propto \frac{|\langle f | \hat{\boldsymbol{\mu}} | i \rangle|^2}{\hbar^2 \omega}$$

A critical takeaway from this derivation is that if the matrix element $\langle f | \hat{\boldsymbol{\mu}} | i \rangle$ vanishes due to symmetry, the transition rate becomes zero. Such transitions are termed forbidden transitions, meaning they cannot occur via the electric dipole mechanism, even if the energy conservation requirement is met.

Selection Rules and Symmetry

The vanishing of certain matrix elements leads to the establishment of selection rules. These rules act as "filters" that determine which transitions are "allowed" and which are "forbidden" based on the quantum numbers of the states involved. For electric dipole transitions in hydrogen-like atoms, these rules arise from the conservation of angular momentum and parity.

The primary selection rules for electric dipole transitions are:

• Change in Orbital Angular Momentum: The orbital angular momentum quantum number $l$ must change by exactly one unit, i.e., $\Delta l = \pm 1$. This is a consequence of the photon carrying one unit of intrinsic angular momentum.
• Change in Magnetic Quantum Number: The magnetic quantum number $m$ must satisfy $\Delta m = 0, \pm 1$, reflecting the orientation of the transition relative to the polarization of the light.
• Parity Change: The initial and final states must have opposite parity (one must be even, the other odd).

Case Study: The $2s \to 1s$ Transition
Consider an electron in a hydrogen atom transitioning from the $2s$ state ($n=2, l=0$) to the $1s$ ground state ($n=1, l=0$). While the energy difference $\Delta E$ is non-zero and could theoretically emit a photon, we observe that $\Delta l = 0 - 0 = 0$. Because this violates the $\Delta l = \pm 1$ rule, the $2s \to 1s$ electric dipole transition is strictly forbidden. In practice, such a transition can only occur through much slower, higher-order processes, such as two-photon emission or through collisions with other particles.

Conclusion and Scientific Significance

The derivation of transition probabilities reveals that the interaction between light and matter is not a chaotic process but one governed by rigorous mathematical and symmetry-based constraints. The transition rate is a delicate balance between the coupling strength (the matrix element) and the availability of final states (the density of states).

This theoretical foundation is indispensable to modern science. It provides the physical basis for interpreting atomic spectra, which allows us to identify elements in distant stars. Furthermore, it is the cornerstone of technologies such as lasers, where stimulated emission is precisely controlled, and atomic clocks, which rely on the extreme stability of specific "allowed" transitions. Understanding these quantum rules allows researchers to manipulate light-matter interactions with unprecedented precision, driving innovation in fields ranging from quantum computing to high-resolution fluorescence microscopy.