Transition Probabilities and Fermi's Golden Rule

One of the prominent failures of the Bohr model for atomic spectra was that it couldn't predict that one spectral line would be brighter than another. From the quantum theory came an explanation in terms of wavefunctions, and for situations where the transition probability is constant in time, it is usually expressed in a relationship called Fermi's golden rule.

波尔模型在解释原子光谱方面的一个显著失败是它无法预测某一条谱线会比另一条更明亮。从量子理论中产生了基于波函数的解释，而对于过渡概率在时间上恒定的情况，通常用一种称为费米黄金规则的关系来表达。

In general conceptual terms, a transition rate depends upon the strength of the coupling between the initial and final state of a system and upon the number of ways the transition can happen (i.e., the density of the final states). In many physical situations the transition probability is of the form

从概念上讲，过渡率取决于系统初始态与终态之间的耦合强度，以及过渡可以发生的方式数目（即终态的密度）。在许多物理情况下，过渡概率的形式为

The transition probability λ is also called the decay probability or decay constant and is related to the mean lifetime τ of the state by λ = 1/τ. The general form of Fermi's golden rule can apply to atomic transitions, nuclear decay, scattering ... a large variety of physical transitions.

过渡概率λ也称为衰变概率或衰变常数，并与态的平均寿命τ有关，由λ = 1/τ给出。费米黄金规则的一般形式可以应用于原子跃迁、核衰变、散射……等多种物理跃迁。

A transition will proceed more rapidly if the coupling between the initial and final states is stronger. This coupling term is traditionally called the "matrix element" for the transition: this term comes from an alternative formulation of quantum mechanics in terms of matrices rather than the differential equations of the Schrodinger approach. The matrix element can be placed in the form of an integral where the interaction which causes the transition is expressed as a potential V which operates on the initial state wavefunction. The transition probability is proportional to the square of the integral of this interaction over all of the space appropriate to the problem.

一个跃迁过程如果初始态与终态之间的耦合更强，就会进行得更快。这种耦合项传统上被称为“跃迁的矩阵元”：这个术语来源于对量子力学的另一种矩阵表述，而不是薛定谔方程的微分形式。矩阵元可以表示为一个积分形式，其中引起跃迁的相互作用被表达为一个势能V，该势能作用于初始态的波函数。跃迁概率与这个相互作用在整个问题适用空间上的积分的平方成正比。

This kind of integral approach using the wavefunctions is of the same general form as that used to find the "expectation value" or expected average value of any physical variable in quantum mechanics. But in the case of an expectation value for a property like the system energy, the integral has the wavefunction representing the eigenstate of the system in both places in the integral.

这种利用波函数的积分方法在形式上与在量子力学中求得任何物理量的“期望值”或期望平均值的方法相同。但当求一个像系统能量这样的量的期望值时，积分中的波函数在积分的两个地方都表示系统的本征态。

The transition probability is also proportional to the density of final states r_f. It is reasonably common for the final state to be composed of several states with the same energy - such states are said to be "degenerate" states. This degeneracy is sometimes expressed as a "statistical weight" which will appear as a factor in the transition probability. In many cases there will be a continuum of final states, so that this density of final states is expressed as a function of energy.

跃迁概率也与终态的密度 r_f 成正比。通常情况下，终态可能由多个具有相同能量的状态组成——这些状态被称为“简并态”。这种简并性有时会表示为“统计权重”，它会在跃迁概率中作为因子出现。在许多情况下，终态将包含连续的多个状态，因此终态的密度会表示为能量的函数。