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Application of Quantum Mechanics: Translational Motion of a Particle in a Box For L of same order of magnitude as Particle trapped in 1-D box Boundary condition: Particle wave must fit into the box exactly h λ= : p 2 λ = 2 L, L, L,... 3 2 or λ = L n n = 1, 2, 3,... Nils Walter: Chem 260 The Permitted Wavefunctions of a Particle in a Box Permitted wavefunctions are: nπx Ψn = N sin ; n = 1, 2,... L N= 2 L normalization constant (for scaling) n=6 L ³ According to Born’s interpretation: Ψ dx = 1 n=5 nπx 2 1 N sin dx = 1 = N L ³0 L 2 L 2 n=4 n=3 n=2 n=1 0 2 L L 1 sin 2ax b/c of ³ sin ( ax ) dx = x − 2 4a 0 0 2 Nils Walter: Chem 260 2 The Permitted (Quantized!) Energies of a Particle in a Box n=7 n=6 n=5 n=4 n=3 n=2 h nh ; n = 1, 2,... According to de Broglie: p = = λ 2L p2 and Etotal = Ekin = n is a quantum 2m number specifying the n2h2 En = ; n = 1, 2,... energy of the particle 2 8mL The same result is obtained when applying the Schrödinger equation to the wavefunction: 2 d 2Ψ nπx − + VΨ = EΨ; Ψn = N sin 2 2m dx L n=1 Only the n=1 state (lowest energy) Nils Walter: Chem 260 has no nodes; state n has n-1 nodes E The Behavior of a Particle in a Box n=7; 49E1 n=6; 36E1 ∆E = En +1 − En n2h2 En = ; n = 1, 2,... 2 2 h 8mL = (2n + 1) 8mL2 L small n=5; 25E1 n=4; 16E1 n=3; 9E1 n=2; 4E1 n=1; E1 Examples: • electron in NH3(l) • conjugated polyenes L large Zero-point energy (in accord with Nils Walter: Chem 260 Heisenberg’s uncertainty principle!) What about a Particle Travelling on a Ring? Linear momentum p → angular momentum J J = pr = mvr Boundary condition: m↑ ↑, v↑ ↑J↑ 2 2 1 p J = Etotal = Ekin = mv 2 = 2 2m 2mr 2 hr moment of inertia I J = pr = λ de Broglie 2πr λ= ; n = 0, 1,... unacceptable n acceptable ( nh / 2π ) = 2 En 2I Nils Walter: Chem 260 n 2 2 = 2I n = 0, ±1, ±2,...