散乱、縮重およびパウリ演算子
導入済み| English | 意味 |
|---|---|
| degeneracy/dɪˈdʒenərəsi/ | 縮重度 |
| reflection coefficient/rɪˈflekʃn ˌkəʊɪˈfɪʃənt/ | 反射係数 |
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:
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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:
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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 ____.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
(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.
このケースに当てはまる答えはどれか？
有限なポテンシャル変化における伝搬波と電流の一致
Distinct Pauli matrices commute with each other.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
難易度 3/5 · 200 コイン + 今日初回完了時に小ボーナス付与
The order follows the stated process; check each stage before the next.