Quantum Numbers and Atomic Energy Levels

量子数和原子能级

Bohr model

Schrodinger equation

principal quantum number

For hydrogen and other nuclei stripped to one electron, the energy depends only upon the principal quantum number n.

This fits the hydrogen spectrum unless you take a high resolution look at fine structure where the electron spin and orbital quantum numbers are involved. At even higher resolutions, there is a tiny dependence upon the orbital quantum number in the Lamb shift.

对于氢和其它被剥离至一个电子的核，能量仅取决于主量子数n。这符合氢谱线，除非你以高分辨率观察精细结构，其中涉及电子自旋和轨道量子数。在更高分辨率下，还有微小的依赖于轨道量子数的效应，即朗德位移。

For helium you have a pair of electrons, and the helium energy levels associated with one electron in an n=2 excited state give a sizable dependence on the orbital quantum number l. This can be attributed to the fact that the 2s electron penetrates more inside the charge of the 1s electron. The 2s is therefore less shielded from the attractive force of the nucleus than the 2p and is lower in energy. This dependence upon the orbital quantum number of a single excited state electron is accounted for by the penetration of the wavefunction and can be clearly seen in the energy levels of the alkali metals such as lithium and sodium.

对于氦原子，你有两电子。与n=2激发态中一个电子相关的氦能级对轨道量子数l有显著的依赖性。这可以归因于2s电子更深入地穿透了1s电子的电荷分布。因此，2s电子比2p电子更少受到核吸引力的屏蔽，其能量更低。单个激发态电子的轨道量子数依赖性可以通过波函数的穿透来解释，这种依赖性在碱金属（如锂和钠）的能级中可以明显观察到。

Another effect on the helium energy levels can be associated with the spin quantum number and arises from the symmetry requirements of the Pauli exclusion principle. Note that for given values of principal and orbital quantum numbers, the states with parallel spins are significantly lower than those with anti-parallel spins. This is sometimes referred to as the spin-spin effect and summarized in Hund's Rule #1.

另一个影响氦能级的因素可以与自旋量子数相关联，并源于泡利不相容原理的对称性要求。需要注意的是，对于给定的主量子数和轨道量子数，平行自旋的状态显著低于反平行自旋的状态。这种现象有时被称为自旋-自旋效应，并被总结为洪德规则#1。

For light atoms with multiple electrons outside a closed shell, we can combine the orbital angular moments to give a resultant angular momentum L. Considering the vector model, we expect that we can get different values of L for a given set of individual orbital angular momenta. It turns out that the largest possible L value has the lowest energy. This is sometimes referred to as the "orbit-orbit effect" (Hund's Rule #2)

对于具有多个外层电子的轻原子，我们可以将轨道角动量结合起来，得到总角动量L。考虑到矢量模型，我们期望对于给定的个体轨道角动量，可以得到不同的L值。实际上，最大的可能L值对应最低的能量。这种现象有时被称为‘轨道-轨道效应’（洪德规则#2）

With multiple electrons there is an additional source of splitting of the electron energy levels which is characterized in terms of another quantum number, the total anglular momentum quantum number J. The source of the splitting is called the spin-orbit effect. For light atoms, the spins and orbital angular momenta of individual electrons are found to interact with each other strongly enough that you can combine them to form a resultant spin S and resultant orbital angular momentum L (this is called Russell-Saunders or LS coupling). The S and L are combined to produce a total angular momentum quantum number J, and it is found that higher J values lie lower in energy (Hund's Rule #3)

在多电子系统中，电子能级的分裂还有一种额外的来源，这种来源由另一个量子数——总角动量量子数J来表征。这种分裂的来源称为自旋-轨道效应。对于轻原子，单个电子的自旋和轨道角动量被发现强 enough 相互作用，可以结合起来形成总自旋S和总轨道角动量L（这被称为Russell-Saunders或LS耦合）。S和L结合产生总角动量量子数J，发现较高的J值在能量上更低（Hund规则#3）

When an external interaction such as a magnetic field is applied, then further splitting of the energy levels occurs, and that splitting is characterized in terms of the magnetic quantum number associated with the z-component of angular momentum. This splitting is called the Zeeman effect. Effects on energy levels from applied electric fields are called Stark effects.

当施加如磁场这样的外部相互作用时，能量水平进一步分裂，这种分裂由与角动量z分量相关的磁量子数来表征。这种分裂被称为塞曼效应。由施加的电场对能量水平产生的影响称为斯托克效应。

从玻尔模型或氢原子的薛定谔方程解中可知，电子能级仅取决于主量子数。