HTI5205 Chap.6 Quantum Numbers and Electron Transitions
Quantum Numbers and Electron Transitions
Quantum numbers form nested constraints: n limits the allowed subshell values, the chosen subshell limits orbital orientation, and spin has two permitted projections. Invalid states should be rejected before electron filling begins. Electron transitions add selection rules to the energy difference, so a candidate line needs both an available energy gap and an allowed change.
The chapter applies the stated orbital and total-angular-momentum rules to characteristic X-ray transitions and uses the photon energy difference to connect a permitted transition with wavelength.
Validate a state from left to right
The principal quantum number is a positive integer. Once n is chosen, the orbital number runs from zero to n minus one.
The magnetic number then runs through integer values from minus the orbital number to plus it, and spin projection takes either sign of one half.
Because each range depends on an earlier choice, checking the tuple from left to right exposes an impossible state before any energy or filling calculation begins.
Separate occupancy from transition permission
Electron configuration asks which allowed states are occupied. A transition question also needs a vacancy in the destination and a permitted change between labelled states.
An energy difference alone does not guarantee that a spectral line occurs, and an allowed transition does not determine its energy without the levels. Keep state existence, occupancy, vacancy, permission and energy as separate filters.
Build a candidate-state table
Shell labels such as K alpha and K beta identify principal-shell movement but not every fine-structure state.
List each possible initial label, the final label and the changes in orbital and total angular momentum. Apply the stated selection rules to every row and preserve the forbidden rows in the working.
Counting only after the table is complete prevents one shell from being mistaken for one transition.
Calculate the photon last
For a surviving candidate, photon energy equals the difference between starting and ending levels, and wavelength follows from the energy-wavelength relation. This order keeps a plausible numerical answer from legitimising an excluded transition.
In a written explanation, name the violated rule for an absent line or show the changes that permit a present line, then attach the calculated energy.
What this chapter covers
- 01
Principal quantum number
- 02
Orbital and magnetic numbers
- 03
Spin projection
- 04
Shell and subshell notation
- 05
Transition selection rules
- 06
Characteristic line counting
Worked example · free
Reject an impossible state
- 2Find the permitted l range.
- 2Compare the proposed value.
Key terms
- Principal Quantum Number
- The positive integer that labels the main electron shell.
- Orbital Quantum Number
- The subshell label constrained to values below the principal number.
- Magnetic Quantum Number
- The integer label for orbital orientation within a subshell.
- Spin Projection
- The two-valued projection of electron spin.
- Selection Rule
- A condition determining whether a transition is permitted under the model.
Quantum Numbers and Electron Transitions FAQ
Why check the state before its energy?
A tuple outside the allowed quantum-number ranges represents no state in the model. Calculating its energy wastes time and can produce a plausible-looking number for an impossible configuration.
What creates a characteristic X-ray line?
An inner-shell vacancy is filled by an electron from a higher state, and the energy difference leaves as a photon. The line energy is tied to the target atom's allowed states and transition rules.
Why can two transitions have similar shell labels?
Fine structure resolves states with additional angular-momentum labels. Candidates sharing principal shells can differ in orbital or total angular momentum, and selection rules may permit more than one line.
What is the safest order for line counting?
List candidate states, validate quantum labels, apply each selection rule and keep forbidden rows visible. Count survivors only after the table is complete, then calculate energy and wavelength.
Exam move
Build a permission tree beginning with n, then list l, ml and spin ranges beneath it. For transitions, write initial and final quantum labels in two columns and subtract each relevant number. Only after the rules pass should you calculate the energy difference and wavelength. Generate a mixture of valid and invalid tuples and reject each invalid one by naming the first broken range.
For line counting, keep forbidden candidate rows visible and annotate the exact rule that removes them. Separate three questions on the page: does the state exist, is it occupied with a destination vacancy, and is the change permitted? This prevents an energy difference from being treated as sufficient evidence for a line.
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