An infinite well runs from x=0 to x=8 nm. Add V(x)=F x with F=0.03 eV/nm. What first-order energy shift does a reference energy eigenstate receive?

• 0.24 eV
• 0.03 eV
• 0.12 eV

H0 describes a solved infinite well. The potential V(x)=F x is added inside it. Which expression describes the perturbed Hamiltonian?

• H=H0+V, adding the stated slope
• H=V alone, discarding the reference well
• H=H0-V, reversing the stated addition

If <n|V|n>=0, the perturbation cannot mix |n> with any other reference state.

Circle one:   True   False

V=C I adds a constant energy without changing the boundary conditions. What happens to every reference energy eigenstate?

• Its energy stays fixed while its eigenstate changes.
• Its energy shifts by C and its eigenstate is unchanged.
• Its energy shifts by 2C and its eigenstate is unchanged.

An isolated, nondegenerate reference level has E_n^0=7.2 eV. In its normalized eigenstate |n>, <n|V|n>=-0.3 eV. The perturbation expansion is valid. What is its energy through first order?

• 7.5 eV
• 6.9 eV
• -0.3 eV

Two nondegenerate reference levels have |<m|V|n>|=0.4 eV and |E_n^0-E_m^0|=0.8 eV. What does their coupling-to-gap ratio tell you?

• The ratio is 0.5; weak mixing is not justified.
• The ratio is 0.05; these values support weak mixing.
• The ratio is 2; the coupling is smaller than the gap.

Two reference states have exactly equal energy. Coupling to outside levels is weak. How do you find their first-order energy shifts?

• Divide their coupling by their zero reference energy gap.
• Use diagonal averages in any basis and omit their coupling.
• Diagonalize V within their shared-energy subspace.

For a nondegenerate level, a student calls <phi|V|phi> the first-order energy shift, using a guessed corrected state phi. Which calculation implements the first-order rule?

• Average H0+V in phi and call that full energy the shift.
• Evaluate <n|V|n> in the known reference state.
• Average V uniformly over positions without a reference density.