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Unformatted text preview: 1/26/11 1 PHYS 360 Quantum Mechanics Wed Feb 2, 2011 Lecture 10: How do the boundary condiCon alone give quanCzed energy levels? The Infnite Square Well V x ( ) = 0, if 0 ≤ x ≤ a, ∞ , otherwise ⎧ ⎨ ⎪ ⎩ ⎪ − 2 2 m d 2 ψ n dx 2 + V ψ n = E n ψ n We will solve this equaCon frst outside the well, then inside, then match up the soluCons. − 2 2 m d 2 ψ n dx 2 = E n ψ n d 2 ψ n dx 2 = − k n 2 ψ n , where k n ≡ 2 mE n ψ n x ( ) = A sin k n x + B cos k n x Inside, V(x)=0, so…. Outside the well V(x)=∞, so the wave ¡uncCon must be zero. ψ n ( x ) = 0 outside the well Need to match up these ¡uncCons at the boundaries…. 1. Any physically valid wave ¡uncCon must be conCnuous at a boundary. 2. I¡ the potenCal V(x) is physically valid, then the slopes must be conCnuous at a boundary. HOWEVER, a potenCal jumping abruptly ¡rom zero to infnity is NOT physically valid (but a nice model anyway)....
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This note was uploaded on 04/23/2011 for the course PHYS 360 taught by Professor Durbin,stephen during the Spring '11 term at Purdue.
- Spring '11