التشتت، التدهور وعمليات باولي
GRE Subject Test · GRE Physics الموضوع 24 6:31 سرد باللغة الإنجليزية · ترجمة مدمجة بالإنجليزية + الصينية
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A quantum wave can reflect from a downward potential step, even though a classical particle has no turning point there.
量子波即使遇到向下的势阶也会发生反射，尽管经典粒子在那里没有转折点。
This topic matches travelling waves and probability fluxes across finite potential steps, counts the degenerate states of an isotropic oscillator, and evaluates Pauli products, telling anticommutation apart from equality.
本主题在有限势阶两侧匹配行波和概率流，计算各向同性振子的简并态数目，并计算泡利矩阵乘积，区分反对易与相等。
For constant potential V and energy E greater than V, the spatial solutions are traveling factors e to the plus i k x and e to the minus i k x, with k equal to the square root of 2 m times E minus V, all over h-bar.
对于恒定势场V和能量E大于V的情况，空间解为行波因子e的正ikx次方和e的负ikx次方，其中k等于根号下2m乘以E减V，再除以h-bar。
At finite steps with the same particle mass, the wavefunction and its first derivative are continuous.
在粒子质量相同的有限阶跃处，波函数及其一阶导数连续。
In a left-incident scattering problem with no incoming beam from the right, the far-right solution contains only the right-traveling factor A e to the i k x.
在左入射散射问题中，若右侧无入射束，则最右侧解仅包含右行波因子A e的ikx次方。
If a finite well returns to the original external potential, the transmitted external wave number equals the incident one, even though the interior wave number differs.
若有限深势阱回归到原始外部势场，则透射外部波数等于入射波数，尽管内部波数不同。
Exponentially decaying solutions describe E less than V regions, not every potential well.
指数衰减解描述的是E小于V的区域，而非所有势阱。
For a step from V 1 to V 2 with both regions classically allowed, write the incident-plus-reflected amplitude in region 1 and the transmitted amplitude in region 2.
对于从 V1 到 V2 的势阶，且两个区域都是经典允许区，在区域 1 写出入射加反射的振幅，在区域 2 写出透射振幅。
Continuity gives r equals k 1 minus k 2 over k 1 plus k 2, and t equals 2 k 1 over k 1 plus k 2.
连续性给出 r 等于 k1 减 k2 除以 k1 加 k2，t 等于 2 k1 除以 k1 加 k2。
The reflection probability is R equals the magnitude of r squared; transmission is T equals k 2 over k 1 times the magnitude of t squared, because probability current depends on wave number.
反射概率 R 等于 r 的模的平方；透射概率 T 等于 k2 除以 k1 再乘以 t 的模的平方，因为概率流依赖于波数。
Thus R plus T equals 1 for a lossless step.
因此对于无损耗势阶，R 加 T 等于 1。
Do not add the raw squared transmitted amplitude to R without its current factor.
不要把未乘流因子的透射振幅平方直接与 R 相加。
With k 2 equal to 3 k 1, R is one quarter and T is three quarters despite a downward step.
当 k2 等于 3 k1 时，尽管是向下的势阶，R 为四分之一，T 为四分之三。
A three-dimensional isotropic oscillator separates into x, y, z modes with nonnegative integers n sub x, n sub y, n sub z.
三维各向同性谐振子分离为x、y、z模式，具有非负整数n_x、n_y、n_z。
Its energy is N plus three halves, times h-bar omega, where N equals n sub x plus n sub y plus n sub z.
其能量为(N+3/2)ℏω，其中N等于n_x加n_y加n_z。
For a spin-zero distinguishable single particle, the spatial degeneracy is the number of such triples: N plus 1, times N plus 2, all over 2.
对于自旋为零的可区分单粒子，空间简并度是此类三元组的数量：(N+1)(N+2)/2。
This counts different assignments, not just different unordered partitions.
这计算的是不同的分配方式，而不仅仅是不同的无序划分。
For N equals 2, the six states are permutations of 2, 0, 0 and 1, 1, 0.
当N等于2时，六个状态是(2,0,0)和(1,1,0)的排列。
For N equals 3, the degeneracy is 10.
当N等于3时，简并度为10。
Additional spin or identical-particle constraints would change the counting problem.
额外的自旋或全同粒子约束会改变计数问题。
Pauli matrices are sigma sub x equals the matrix with first row zero comma one, second row one comma zero.
泡利矩阵为σ_x等于第一行零、一，第二行一、零的矩阵。
Sigma sub y equals the matrix with first row zero comma negative i, second row i comma zero.
σ_y等于第一行零、负i，第二行i、零的矩阵。
Sigma sub z equals the matrix with first row one comma zero, second row zero comma negative one.
σ_z等于第一行一、零，第二行零、负一的矩阵。
Each squares to identity.
每个矩阵平方后均等于单位矩阵。
Direct multiplication gives sigma sub x times sigma sub y equals i sigma sub z and sigma sub y times sigma sub x equals negative i sigma sub z, so distinct Pauli matrices anticommute.
直接相乘可得σ_x乘以σ_y等于iσ_z，而σ_y乘以σ_x等于负iσ_z，因此不同的泡利矩阵互为反对易。
Cyclic products from x to y to z have positive i; reversing order changes the sign.
按x到y再到z的循环乘积结果为正i；若反转顺序则符号改变。
The general identity is sigma sub i times sigma sub j equals delta sub ij I plus i times the sum over k of epsilon sub ijk sigma sub k.
一般恒等式为σ_i乘以σ_j等于δ_ij乘以I加上i对k求和ε_ijk乘以σ_k。
Matrix order matters: sigma sub x times sigma sub z equals negative i sigma sub y, not i sigma sub y or simply sigma sub y.
矩阵的乘法顺序至关重要：σ_x乘以σ_z等于负iσ_y，而非iσ_y或单纯的σ_y。
These dimensionless matrices become spin operators S sub i equals h-bar sigma sub i divided by 2, adding physical units and factors.
这些无量纲矩阵演变为自旋算符S_i等于h-bar乘以σ_i除以2，从而引入了物理量纲与系数。
At a finite step, match the wavefunction and its derivative.
在有限势阶处，匹配波函数及其导数。
The amplitudes are r equals k 1 minus k 2 over k 1 plus k 2, and t equals 2 k 1 over k 1 plus k 2.
振幅为 r 等于 k1 减 k2 除以 k1 加 k2，t 等于 2 k1 除以 k1 加 k2。
For k 2 equal to 3 k 1, r is minus one half and t is one half.
当 k2 等于 3 k1 时，r 等于负二分之一，t 等于二分之一。
R is one quarter, and T is k 2 over k 1 times t squared: three quarters.
R 等于四分之一，T 等于 k2/k1 乘 t 的平方：四分之三。
They add to one: even a downward step reflects.
两者之和为一：即使向下的势阶也会反射。
A spin-half particle has state open parenthesis square root of 3, i close parenthesis transpose divided by 2 in the z-spin basis.
一个自旋二分之一粒子在z自旋基下的态为(√3, i)转置除以2。
Check normalisation and find probabilities for Sz and Sx outcomes. Five marks.
检查归一化并求出Sz和Sx测量结果的概率。
Norm is three plus one, all over four, equals one.
归一化系数为三加一除以四，等于一。
Sz gives plus h-bar over two with probability three quarters and minus h-bar over two with probability one quarter.
Sz测得正h-bar除以二，概率为四分之三；测得负h-bar除以二，概率为四分之一。
The x eigenstates are one plus or minus one, transpose, all over the square root of two.
x本征态为一加或减一，转置，再除以根号二。
Their amplitudes are square root of three plus or minus i, all over two times the square root of two, each with squared modulus one half.
其振幅为根号三加减i，再除以二倍根号二，每个的模平方均为二分之一。
Award norm, z results and probabilities, x eigenbasis, projections and x probabilities.
请给出归一化系数、z测量结果与概率、x本征基、投影及x测量概率。
Complex conjugation is required in the inner product; the imaginary relative phase matters.
内积中必须进行复共轭运算；相对虚相位至关重要。
Transmission probability needs the wave-number current ratio.
透射概率需要波数电流比。
Oscillator degeneracy counts ordered mode triples.
振子简并度计算有序模式三元组。
Do not reverse Pauli matrix order without changing its sign.
在不改变符号的情况下，不要反转泡利矩阵的顺序。
reflection coefficient: Reflected probability-current fraction relative to incident current.
反射系数：相对于入射流的反射概率流分数。
degeneracy: Number of independent states sharing the specified energy.
简并度：共享指定能量的独立状态数量。
Retrieve the meanings without reading the definitions, then change a parameter in the original transfer and test the limiting case.
在不阅读定义的情况下回忆其含义，然后改变原始传递中的一个参数并检验极限情况。