Scattering, degeneracy and Pauli operators
| English | 中文 | Pinyin |
|---|---|---|
| degeneracy | 简并度 | jiǎn bìng dù |
| reflection coefficient | 反射系数 | fǎn shè xì shù |
A decision before an answer
- A quantum wave can reflect from a downward potential step even though a classical particle has no turning point there.
- Your goal: Match travelling waves and fluxes across finite potential changes.
Read the relationship
- For constant potential V and energy E>V, the spatial solutions are travelling factors exp(±ikx), with k=sqrt[2m(E−V)]/ℏ. At finite steps with the same particle mass, wavefunction and its first derivative are continuous. In a left-incident scattering problem with no incoming beam from the right, the far-right solution contains only the right-travelling factor Ae^(ikx). 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. Exponentially decaying solutions describe E<V regions, not every potential well.
- Count degenerate isotropic-oscillator states.
For a lossless potential step with k2=2k1, reflection probability is:
r=(1−2)/(1+2)=−1/3; R=1/9.
Use the defining rule
- For a step from V1 to V2 with both regions classically allowed, write incident-plus-reflected amplitude in region 1 and transmitted amplitude in region 2. Continuity gives r=(k1−k2)/(k1+k2) and t=2k1/(k1+k2). Reflection probability is R=|r|²; transmission is T=(k2/k1)|t|² because probability current depends on wave number. Thus R+T=1 for a lossless step. Do not add raw squared transmitted amplitude to R without its current factor. With k2=3k1, R=1/4 and T=3/4 despite a downward step.
- Evaluate Pauli products and distinguish anticommutation from equality.
A spin-zero three-dimensional isotropic oscillator at E=(5/2)ℏω has degeneracy:
N=1 gives triples (1,0,0),(0,1,0),(0,0,1).
Check the conditions
- A three-dimensional isotropic oscillator separates into x,y,z modes with nonnegative integers n_x,n_y,n_z. Its energy is (N+3/2)ℏω, where N=n_x+n_y+n_z. For a spin-zero distinguishable single particle, the spatial degeneracy is the number of such triples: (N+1)(N+2)/2. This counts different assignments, not just different unordered partitions. For N=2, the six states are permutations of (2,0,0) and (1,1,0). For N=3, the degeneracy is 10. Additional spin or identical-particle constraints would change the counting problem.
- Evaluate Pauli products and distinguish anticommutation from equality.
At a lossless step with k2=3k1, r=−1/2 and t=1/2. R=1/4, T=3·1/4=3/4 and their sum is one. An isotropic spin-zero oscillator at E=(9/2)ℏω has N=3 and degeneracy 10. Multiplying σ_x by σ_y gives diag(i,−i)=iσ_z.
For a spin-zero isotropic oscillator with N=4, spatial degeneracy is ____.
(N+1)(N+2)/2=5·6/2=15.
Apply the task format
- Pauli matrices are σ_x=[[0,1],[1,0]], σ_y=[[0,−i],[i,0]], σ_z=[[1,0],[0,−1]]. Each squares to identity. Direct multiplication gives σ_xσ_y=iσ_z and σ_yσ_x=−iσ_z, so distinct Pauli matrices anticommute. Cyclic products x→y→z have positive i; reversing order changes the sign. The general identity is σ_iσ_j=δ_ijI+iΣε_ijkσ_k. Matrix order matters: σ_xσ_z=−iσ_y, not iσ_y or simply σ_y. These dimensionless matrices become spin operators S_i=ℏσ_i/2, adding physical units and factors.
- Evaluate Pauli products and distinguish anticommutation from equality.
Transmission probability needs the wave-number current ratio. Oscillator degeneracy counts ordered mode triples. Do not reverse Pauli matrix order without changing its sign.
Which answer fits this case?
Match travelling waves and fluxes across finite potential changes
Distinct Pauli matrices commute with each other.
Their products change sign on reversing order; they anticommute.
Keep the distinctions
- reflection coefficient 反射系数 — Reflected probability-current fraction relative to incident current.
- degeneracy 简并度 — Number of independent states sharing the specified energy.
- Match travelling waves and fluxes across finite potential changes.
- Count degenerate isotropic-oscillator states.
- Evaluate Pauli products and distinguish anticommutation from equality.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.