به انگلیسی

Ladders built from commutators

Science · Quantum & Modern Physics · ages 22-24

Name ______________________ Date ____________

When a ladder operator gives a nonzero new state, by how much does it change m?
- Exactly j
- Exactly one half always
- Exactly one
For Hermitian angular-momentum operators, which relations start the algebraic ladder derivation?
- A separated orbital differential equation
- The angular-momentum commutation relations
- An observed atomic emission-line spectrum
The ladder algebra permits half-integer j, which can describe intrinsic spin but not ordinary orbital spherical harmonics.
Circle one: True False
For j equal to 3/2, which m values appear?
- 3/2, 1, 1/2, 0, -1/2
- 2, 1, 0, -1, -2
- 3/2, 1/2, -1/2, -3/2
A system has j equal to 2. Type how many m states its ladder holds.
Answer: ______________
Why would integer-only values be a problem for physics?
- Spin-half would be excluded
- Spherical harmonics would disappear
- Commutators would stop closing
For Hermitian angular-momentum operators in a finite normalized ladder, how do you obtain the allowed m values algebraically?
- Solve an orbital differential equation for every possible spin
- Use ladder norms to find endpoints, then take unit steps
- Infer every spin state from an observed atomic line spectrum
A student says ordinary orbital spherical harmonics disprove half-integer intrinsic spin. Which distinction resolves the claim?
- They cover orbital motion, not spin
- Harmonics forbid all integers too
- Ladders cannot produce any harmonics

Answer key

For grown-ups. Fold this page away before handing over the rest.

Exactly one · A nonzero action of J+ raises m by one; J- lowers it by one.
The angular-momentum commutation relations · The derivation uses the component commutators and Hermiticity, rather than solving an orbital differential equation.
True · A finite ladder needs integer 2j, not necessarily integer j. Orbital angular momentum has integer l.
3/2, 1/2, -1/2, -3/2 · The m values descend from 3/2 to minus 3/2 in unit steps.
5 · The count is 2j plus 1, which is 5.
Spin-half would be excluded · Electron spin requires a half-integer ladder. Intrinsic spin in general may be integer or half-integer.
Use ladder norms to find endpoints, then take unit steps · Nonnegative ladder norms and zero endpoint norms give m=j and m=-j. Unit steps enumerate the states between them.
They cover orbital motion, not spin · Orbital spherical harmonics realize integer angular-momentum ladders. They are not intrinsic spin wave functions.