Perturbaciones débiles y subespacios degenerados
GRE Subject Test · GRE Physics Tema 37 6:31 Narración en inglés · Subtítulos en inglés + 中文 quemados en pantalla
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Transcripción
A zero first-order energy shift does not mean nothing changes: the state can still mix, and a higher-order shift can remain.
一阶能量修正为零并不意味着什么都没有改变：态仍然可以混合，高阶修正也可以存在。
This topic computes first-order shifts as expectation values, uses symmetry and level gaps to judge weak coupling, and diagonalises the perturbation inside a degenerate level.
本主题把一阶修正计算为期望值，用对称性和能级间隔判断耦合是否足够弱，并在简并能级内部对角化微扰。
Let H equals H sub zero plus lambda W, where lambda is a small dimensionless parameter and the eigenstates of H sub zero are known and normalised.
设 H 等于 H 下标零加 lambda W，其中 lambda 是一个小的无量纲参数，且 H 下标零 的本征态已知并已归一化。
For a nondegenerate level n, the first-order energy correction is lambda times the expectation value of W in state n.
对于非简并能级 n，一阶能量修正为 lambda 乘以状态 n 中 W 的期望值。
In position space this is lambda times the integral of psi sub n star of x times W of x times psi sub n of x with respect to x for a multiplicative potential, with integration over the allowed domain.
在位置空间中，对于乘性势场，这等于 lambda 乘以 psi 下标 n 共轭乘以 W 乘以 psi 下标 n 关于 x 的积分，积分区域为允许域。
A potential value at one point is not an expectation value; the state’s probability density weights the whole domain.
单点的势值并非期望值；态的概率密度对全区域进行加权。
For an infinite well from zero to L, psi sub n equals the square root of two over L times sine of n pi x over L.
对于从 0 到 L 的无限深势阱，psi 下标 n 等于根号下 2 除以 L 乘以 sin(n pi x / L)。
Reflection about its centre makes the expectation value of x equal to L over 2, so a weak added potential gamma x has shift gamma L over 2 for every nondegenerate well level at first order.
关于中心的反射使得 x 的期望值为 L/2，因此在一阶近似下，微扰势 gamma x 使每个非简并能级移动 gamma L/2。
Gamma has units of energy per length.
Gamma 的单位是能量每长度。
For a perturbation odd about the centre, such as gamma times x minus L over 2, the probability density of an unperturbed well eigenstate is even, so its diagonal expectation vanishes.
对于关于中心为奇的扰动，例如 γ 乘以 x 减 L 的一半，未扰动势阱本征态的概率密度是偶函数，因此其对角期望值为零。
A centred quadratic perturbation, eta times x minus L over 2, squared, instead gives a shift of eta L squared times one twelfth minus 1 over 2 pi squared n squared, which is positive for positive eta and depends on n.
而居中的二次扰动，η 乘以 x 减 L 的一半的平方，给出的能量移动为 η L 平方乘以十二分之一减去 2π 平方 n 平方分之一，当 η 为正时该值为正，并依赖于 n。
Parity makes the first-order integral zero only for the specified state and operator symmetry.
宇称只对指定的态和算符对称性使一阶积分为零。
Off-diagonal matrix elements can still change the wavefunction.
非对角矩阵元仍可能改变波函数。
Do not turn a symmetry cancellation into a claim that every energy correction vanishes or that the original state remains exact.
不要把对称抵消说成所有能量修正都为零，或原来的态仍然精确。
For a nondegenerate state, the leading admixture of another unperturbed state m is proportional to lambda W sub mn divided by E sub n superscript zero minus E sub m superscript zero.
对于非简并态，另一未微扰态m的主要混合项正比于lambda W下标mn除以E下标n上标零减E下标m上标零。
The useful smallness condition therefore compares coupling matrix elements with the relevant level separations, not just with an arbitrary absolute energy zero.
因此，有用的小量条件是将耦合矩阵元与相关的能级间隔进行比较，而不仅仅是与任意绝对能量零点比较。
To second order, the energy correction contains the sum over m not equal to n of the absolute value of lambda W sub mn squared, all over E sub n superscript zero minus E sub m superscript zero.
到二阶时，能量修正包含对m不等于n的求和，即lambda W下标mn的绝对值平方除以E下标n上标零减E下标m上标零。
For the lowest nondegenerate state, all these denominators are negative, so the second-order correction is nonpositive in this model.
对于最低的非简并态，所有这些分母均为负数，因此在该模型中二阶修正为非正值。
Near a degeneracy, a small denominator defeats the nondegenerate expansion; use a coupled subspace instead of dividing by zero.
在简并附近，小的分母会使非简并展开失效；应使用耦合子空间代替除以零。
If a level of H₀ is exactly degenerate, choose an orthonormal basis within that subspace and form the Hermitian matrix of the perturbation there.
如果H零的某个能级恰好简并，请在该子空间内选取一组标准正交基，并构造该处微扰的厄米矩阵。
Its eigenvalues are the first-order energy shifts, and its eigenvectors specify the combinations that diagonalise the leading splitting.
其特征值即为一级能量移动，特征向量则确定了使主分裂对角化的线性组合。
For a two-state subspace with perturbation ε , ], the normalised symmetric and antisymmetric combinations have shifts 3ε and ε.
对于具有微扰epsilon乘以矩阵（第一行2、1，第二行1、2）的两态子空间，归一化对称与反对称组合的能量移动分别为3epsilon和epsilon。
Reading only the two diagonal entries would incorrectly predict two shifts of 2ε.
仅读取两个对角元会错误地预测出两个2epsilon的移动。
A common scalar multiple of the identity shifts both states equally and does not split their degeneracy; off-subspace couplings can matter at higher order.
恒等矩阵的公共标量倍数会使两个态同等移动，而不会分裂其简并；非子空间耦合在更高阶时可能产生影响。
In a degenerate level, diagonalise before reading shifts.
在简并能级中，要先对角化再读出修正。
With the perturbation epsilon times the matrix 2, 1; 1, 2, the determinant condition is 2 epsilon minus w, squared, minus epsilon squared, equals zero.
微扰为 ε 乘以矩阵 (2, 1; 1, 2) 时，行列式条件为 (2ε − w)² − ε² = 0。
The shifts are epsilon and 3 epsilon, for the antisymmetric and symmetric combinations.
修正为 ε 和 3ε，分别对应反对称和对称组合。
Reading only the diagonal would give 2 epsilon twice and miss the splitting.
只读对角元会得到两个 2ε，从而漏掉能级分裂。
A two-state Hamiltonian is the matrix with zero and epsilon in the first row, and epsilon and Delta in the second row.
一个两态哈密顿量是矩阵，第一行为零和epsilon，第二行为epsilon和Delta。
With Delta positive and the absolute value of epsilon much less than Delta.
其中Delta为正数，且epsilon的绝对值远小于Delta。
Find the exact eigenvalues and leading ground-energy shift.
求精确本征值和基态能量的一阶移动。
Explain why a zero diagonal perturbation is insufficient to predict zero shift.
解释为何对角微扰为零不足以预测能量移动为零。
The characteristic equation is E squared minus delta E minus epsilon squared equals zero.
特征方程为 E 平方减去 delta E 减去 epsilon 平方等于零。
The roots are E plus and E minus, equal to delta plus or minus the square root of delta squared plus 4 epsilon squared, all over 2.
根为 E 正和 E 负，等于 delta 加减根号下 delta 平方加 4 epsilon 平方，整体除以 2。
Expanding the lower root gives E sub negative approximately equal to negative epsilon squared divided by delta.
展开较低根可得 E 下标负约等于负的 epsilon 平方除以 delta。
The first-order shift is zero, but off-diagonal mixing produces a negative second-order shift.
一阶移动为零，但非对角混合产生负的二阶移动。
Award points for the equation, the roots, the expansion, the sign, and the mention of mixing or validity.
对公式、根、展开式、符号以及提及的混合或有效性给予评分。
At delta equals zero, use degenerate theory instead.
在 delta 等于零时，改用简并理论。
Use normalised state weights, keep perturbation units, and diagonalise a degenerate block.
使用归一化态权重。 保留微扰单位。
First-order cancellation and exact invariance are different claims.
并对简并块进行对角化。 一阶抵消与精确不变性是不同的论断。
first-order energy shift: Leading weak-perturbation correction given by the unperturbed state’s expectation value.
一阶能量修正：由未微扰态的期望值给出的主要弱微扰修正。
degenerate perturbation theory: Method that first diagonalises the perturbation within an unperturbed degenerate subspace.
简并微扰理论：首先在未微扰简并子空间内对微扰进行对角化的方法。
Retrieve the meanings without reading the definitions, then change a parameter in the original transfer and test the limiting case.
不阅读定义，凭记忆回顾相关含义；随后在原传递函数中修改一个参数，并检验极限情况。