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GRE Subject Test · GRE Physics Chủ đề 39 7:41 Giọng đọc tiếng Anh · phụ đề tiếng Anh + 中文 được ghi trực tiếp
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A bound wavefunction reaches outside a finite well, even though the particle's energy is below the potential there.
束缚态波函数会延伸到有限深势阱之外，尽管粒子的能量低于那里的势能。
This topic matches decaying tails to find the allowed energies, solves the dimensionless root, and reads hydrogen's quantum numbers and radial probability.
本主题通过匹配衰减的尾部求出允许的能量，求解无量纲方程的根，并解读氢原子的量子数和径向概率。
Declare a symmetric well V equals negative V sub 0 for the absolute value of x less than a, and V equals 0 outside, with V sub 0 greater than 0 and the same mass throughout.
定义一个对称势阱，当 |x| < a 时 V = −V₀，外部 V = 0，其中 V₀ > 0，且质量处处相同。
A bound energy lies between negative V sub 0 and zero.
束缚态能量介于 −V₀ 与零之间。
Inside, k equals the square root of 2 m E plus V sub 0, all over h-bar, gives oscillatory solutions. Outside, kappa equals the square root of negative 2 m E, all over h-bar, gives decaying tails.
在势阱内部，k = √ /ℏ 给出振荡解；在外部，κ = √(−2mE)/ℏ 给出衰减尾波。
Reject growing exponentials to obtain a normalisable state.
舍去发散的指数项以获得可归一化态。
Reflection symmetry permits even interior cos of k x or odd sin of k x states.
反射对称性允许偶宇称的 cos(kx) 或奇宇称的 sin(kx) 态。
At finite boundaries without delta interactions, both psi and psi prime are continuous.
在无 δ 相互作用的有限边界处，ψ 及其导数 ψ′ 均连续。
Even matching gives k tan of k a equals kappa. Odd matching gives negative k cot of k a equals kappa.
偶宇称匹配条件为 k tan(ka) = κ；奇宇称匹配条件为 −k cot(ka) = κ。
A finite well does not require psi to vanish at its edges as an infinite wall does.
有限深势阱不要求波函数在边界处为零，这与无限深势阱不同。
Introduce z equals k a and rho equals a times the square root of 2 m V sub 0, all over h-bar.
引入 z = ka 和 ρ = a√(2mV₀)/ℏ。
Then kappa a equals the square root of rho squared minus z squared, and 0 is less than z which is less than rho.
则 κa = √(ρ²−z²)，且 0 < z < ρ。
Solve z tan z equals the square root of rho squared minus z squared for an even state, or negative z cot z equals the square root of rho squared minus z squared for an odd one, on intervals with the appropriate sign and away from tangent poles.
在具有适当符号且避开正切极点的区间上，求解 z tan z = √(ρ²−z²) 以得到偶宇称态，或 −z cot z = √(ρ²−z²) 以得到奇宇称态。
Energy follows as E divided by V sub 0 equals z squared divided by rho squared minus 1.
能量由 E/V₀ = z²/ρ² − 1 给出。
For rho equals 1, the ground root lies in 0 is less than z which is less than 1, and the even equation is equivalent there to z equals cos z.
当 ρ = 1 时，基态根位于 0 < z < 1 区间内，此时偶宇称方程等价于 z = cos z。
Bisection or a converged root finder gives z approximately equal to 0.739085 and E divided by V sub 0 approximately equal to negative 0.453753.
使用二分法或已收敛的求根器可得 z ≈ 0.739085，E/V₀ ≈ −0.453753。
Since rho is less than pi over 2, no odd bound-state branch fits.
由于 ρ < π/2，不存在满足条件的奇宇称束缚态分支。
In one dimension an attractive square well has an even bound ground state however shallow it is; an E equals 0 threshold tail is not square integrable.
在一维情况下，无论势阱多浅，吸引方势阱总存在一个偶宇称束缚基态；而 E = 0 处的阈值尾部不是平方可积的。
For an even bound state, write psi equals A cosine k x inside, and psi equals A cosine k a times e to the minus kappa times the size of x minus a, outside.
对于偶宇称束缚态，在阱内写 ψ 等于 A cos kx，在阱外写 ψ 等于 A cos ka 乘以 e 的负 κ 乘以 x 的绝对值减 a 次方。
Continuity sets the relative tail amplitude; integrate the square of psi over both the interior and the tails to determine A.
连续性确定尾部振幅的相对大小；对阱内和尾部的 ψ 模平方一起积分来确定 A。
Probability outside the well is nonzero and depends on its depth, width and the selected state.
阱外概率不为零，取决于势阱的深度、宽度和所选的态。
Bound states have discrete energies and normalisable decaying asymptotes.
束缚态能量离散，渐近衰减且可归一化。
A scattering state with positive E has propagating external waves and is normalised or interpreted using a continuum or flux convention.
E 为正的散射态有向外传播的波，需要用连续谱或通量约定来归一化或解释。
A decaying tail is not evidence that the state violates energy conservation, and raw tail amplitude is not the outside probability until it has been integrated and normalised.
衰减的尾部并不说明该态违反能量守恒；原始尾部振幅只有在积分并归一化之后才是阱外概率。
For a one-electron Coulomb ion with nuclear charge Z e, use the nonrelativistic central-potential model and a heavy nucleus approximation unless reduced mass is specified.
对于核电荷为Ze的单电子库仑离子，除非指定了约化质量，否则请使用非相对论中心势模型和重核近似。
E sub n is approximately negative 13.6 Z squared over n squared electron volts.
E_n约等于负13.6乘以Z平方除以n平方电子伏特。
Allowed orbital numbers are n greater than or equal to 1, l equals 0 through n minus 1, and m sub l equals negative l through l.
允许的轨道量子数为n大于等于1，l等于0到n减1，m_l等于负l到l。
L squared has eigenvalue l times l plus 1, h-bar squared, and L sub z equals m sub l times h-bar.
L平方具有本征值l乘(l加1)乘以h-bar平方，L_z等于m_l乘以h-bar。
The ground 1s orbital has l equals 0, despite the historical Bohr circular-orbit rule.
基态1s轨道的l等于0，尽管有历史上的玻尔圆轨道规则。
Spatial degeneracy at fixed n is the sum over l of 2l plus 1, which equals n squared before spin and fine-structure corrections.
固定n时的空间简并度是l从0到n减1对(2l加1)求和，即n平方，这是在考虑自旋和精细结构修正之前。
If psi equals R sub n sub l of r times Y sub l sub m, with angular part normalized, radial probability in dr is r squared times the absolute value of R squared times dr, not just the absolute value of R squared times dr.
若波函数psi等于R_nl(r)乘以Y_lm，且角向部分已归一化，则径向概率元为r平方乘以|R|平方乘以dr，而不仅仅是|R|平方乘以dr。
For hydrogen 1s, the radial density is 4 r squared e to the minus 2r over a naught cubed: it peaks at r equals a naught and has mean 3a naught over 2, while the three-dimensional point density is largest at the origin.
对于氢原子1s态，径向密度为4r平方e^(-2r/a_0)除以a_0立方：它在r等于a_0处达到峰值，均值为3a_0除以2，而三维点密度在原点处最大。
Distinguish these measures before locating a most probable radius.
在确定最概然半径之前，请区分这些概率测度。
A bound energy lies between minus V naught and zero.
束缚能量位于负 V0 和零之间。
Inside the well, k is root 2 m times E plus V naught, over h bar; outside, kappa is root minus 2 m E, over h bar.
阱内 k 等于根号下 2m 乘 E 加 V0，再除以 ħ；阱外 κ 等于根号下负 2mE，再除以 ħ。
Even parity gives a cosine inside and decaying tails outside.
偶宇称时阱内为余弦，阱外为衰减的尾部。
Matching the value and the slope at a gives k tan k a equals kappa.
在 a 处匹配函数值和斜率，得到 k tan ka 等于 κ。
Odd states use minus k cot k a instead.
奇态则改用负 k cot ka。
For an even bound state, the interior wavefunction is A cos of k x, and the exterior tails are A cos of k a times e to the minus kappa times the absolute value of x minus a.
对于偶宇称束缚态，内部波函数为 A cos(kx)，外部尾部为 A cos(ka) 乘以 e 的负 kappa 乘绝对值 x 减 a 次方。
Derive its normalization and outside probability.
推导其归一化及外部概率。
Explain what fails at zero binding.
解释在零结合能时发生了什么失效。
Interior norm is A squared, times a plus sine of 2 k a, all over 2 k.
内部范数为 A 平方乘以 a 加 sin(2ka) 除以 2k。
Both tails contribute A squared, times cosine squared of k a, all over kappa.
两个尾部共同贡献 A 平方乘以 cos²(ka) 除以 κ。
Thus A to the minus 2 equals a plus sine of 2 k a, all over 2 k, plus cosine squared of k a, all over kappa.
因此 A 的负二次方等于 a 加 sin(2ka) 除以 2k，再加 cos²(ka) 除以 κ。
And P sub out equals cosine squared of k a, all over kappa, divided by a plus sine of 2 k a, all over 2 k, plus cosine squared of k a, all over kappa.
而 P_out 等于 cos²(ka) 除以 κ，除以 a 加 sin(2ka) 除以 2k，再加 cos²(ka) 除以 κ。
Bound states require kappa positive; a nondecaying threshold tail is not square integrable.
束缚态要求 κ 为正；非衰减的阈值尾部不是平方可积的。
Award interior integral, both tails, A, probability and threshold.
给分点包括：内部积分、两个尾部、A、概率和阈值。
The matching equation still applies.
匹配方程仍然适用。
Use the declared potential zero, match ψ′ as well as ψ, and include the radial shell measure.
使用声明的电势零点，匹配ψ′与ψ，并包含径向壳层测度。
A bound tail and a continuum travelling wave have different energy regimes.
束缚尾波与连续行波处于不同的能量区域。
bound state: Normalisable stationary state with confined probability and discrete energy in the specified model.
束缚态：在指定模型中，具有归一化、概率局域化和离散能级的定态。
radial probability density: Probability per radial distance, including the spherical-shell measure after angular integration.
径向概率密度：角向积分后包含球壳测度的径向距离单位概率。
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；随后在原传递过程中修改一个参数，并检验极限情况。