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Quantum Physics Schrödinger Quantum Physics Particle in 1D box - Infinite Potential Classical p Classical: x0 p mv 1 2 1 p2 2 E mv (mv) 2 2m 2m p xL x p 2mE x p 2mE Bohr Half Quantum Physics: pdx nh Bohr n 2 2 h 2 2 L 2mE n2h E 2mL2 Quantum Physics Particle in 1D box - Infinite Potential Scrödinger Equation h2 d 2 ) x ( V ( x) E ( x) 2 dx m 2 d2 2mEn 2 2 kn k n n ( x ) 0 2 h2 dx n ( x) Aeik x Be ik x C cos k n x D sin k n x n ( 0) n ( L ) 0 n n C 0 sin k n L 0 k n n L n ( x) nx 2 sin L L n 2 2 h 2 En 2mL2 Quantum Physics Particle in 1D box - Infinite Potential Scrödinger Equation h2 d 2 V ( x ) ( x) E ( x) 2 2 m dx d2 2mEn 2 2 k ( x ) 0 k n n n 2 h2 dx n ( x) Aeik x Be ik x C cos k n x D sin k n x n (0) n ( L) 0 n n C 0 sin k n L 0 k n n L 2 nx sin L L n ( x) n ( x) n 2 2 h 2 En 2mL2 2 nx sin L L ( x, t ) e i En t h n ( x) e i En t h 2 nx sin L L U0 Quantum Physics Particle in 1D box - Finite Potential Scrödinger Equation h2 d 2 V ( x ) ( x) E ( x) 2 2 m dx Inside: V(x) = 0 d2 2mEn 2 2 k kn n n ( x ) 0 2 h2 dx n ( x) Aeikn x Be ikn x C cos k n x D sin k n x h2 d 2 V ( x ) ( x) E ( x) 2 2 m dx d2 2m(U 0 E ) 2 ( x ) 0 2 h2 dx ( x) Re x Se x Outside: V(x) = U0 x 0 : S 0x L : R 0 The solutions and it’s derivative must match at the boundary Quantum Physics Tunneling - Def h2 d 2 V ( x ) ( x) E ( x) 2 2 m dx Outside: V(x) = 0 d2 2mEn 2 2 k kn n n ( x ) 0 2 h2 dx n ( x) Aeikn x Be ikn x C cos k n x D sin k n x h2 d 2 V ( x ) ( x) E ( x) 2 2 m dx d2 2m(U 0 E ) 2 ( x ) 0 2 h2 dx Inside: V(x) = U0 ( x) Re x Se x Tunneling probability T that the particle gets through the barrier is proportional to the square of the ratio of the amplitudes of the sinusoidal wave function on the two sides of the barrier. T Ge2L G 16 E E 1 U0 U0 2mU 0 E h Quantum Physics Tunneling - Example - Electron A 2.0 eV electron encounters a barrier 5.0 eV heigh. What is the probability that it will tunnel through the barrier if the barrier width is a) 1.00 nm b) 0.50 nm ? T Ge 2L E E 2 1 e 16 U0 U0 2 m U 0 E h L U 0 5.0eV 5.0 1.602 10 19 J E 2.0eV 2.0 1.602 10 19 J m 9.1110 31 kg h 6.626 10 34 Js h h 2 a) G 16 b) L 1.00m 1.00 10 9 m T 7.1 10 T Ge2L 8 L 0.50nm 0.50 10 9 m T 5.2 10 4 E E 1 U0 U0 2mU 0 E h