KVPY2019PhysicsDual Nature of Matter
Four electrons, each of mass m e are in a one dimensional box of size L . Assume that, the electrons are non-interacting, obey the Pauli exclusion principle and are described by standing de Broglie waves confined within the box. Define α = h 2 / 8 m e L 2 and U 0 to be the ground state energy. Then,
Options
- Athe energy of the highest occupied state is 16 α
- BU 0 = 30 α
- Cthe total energy of the first excited state is U 0 + 9 α
- Dthe total energy of the second excited state is U 0 + 8 α
Correct answer
D. the total energy of the second excited state is U 0 + 8 α
Step-by-step solution
For an electron described by standing waves confined within a box of length L , the wave can occupy only states for which L = n λ 2 , where n = 1 , 2 , 3 … Now, by de Broglie wavelength of this electron, λ = h p = h 2 m E From above two equations, we have E n = h 2 9 m L 2 n 2 (where, n = 1 , 2 , 3 … ) Now, ground state energy is obtained by substituting n = 1 . ∴   U 0 = E 1 = h 2 8 m L 2 Energy of first excited state, E 2 = h 2 8 m L 2 × 2 2 = 4 U 0 = U 0 + 3 α Also, ene