05_02_18 - finding it on the right The particle can come...

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[email protected] 44 String attached with infinite force: only string vibrates with sin( w t) . String attached at a realistic fixture: fixture also vibrates a little bit. Infinite potential well: Wave function is 0 outside well. Finite potential well : There is a wave function with small amplitude outside the well . Particles in nonrigid boxes ) ( ) ( ) ( ) ( 2 2 x E x x V x x x m Φ = Φ + Φ - x ) ( x V 0 L 0 0 E 0 Φ No classical particle since E- V = ½mv 2 > 0 ) ( ) ( 2 ) ( 2 x x V E m Φ - = Φ - E > V: Curvature towards x-axis E < V: Curvature away from x-axis with to have 0 ) ( lim = Φ ±∞ x x - = Φ 1 | ) ( | 2 dx x 05/18/2004
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[email protected] 45 Tunneling x ) ( x V 0 L 0 0 E 0 Φ No classical particle since E- V = ½mv 2 > 0 0 ) ( lim 1 | ) ( | ) ( lim ) ( lim ) ( ) ( 2 0 0 ) ( 2 2 = Φ = Φ Φ = Φ Φ - = Φ ±∞ - - + - x dx x x x x x x x x V E m Properties of the wave function: Stationary Schrödinger equation is satisfied: The first derivatives are continuous: The wave function is normalized: and the wave function is therefore bounded How can there be a probability of finding the particle on the left as well as a probability of
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Unformatted text preview: finding it on the right ? The particle can come from the left to the right by tunneling . 02/18/2005 [email protected] 46 E E < E E E E = 2 E E < 2 E E 2 E E = Fulfilling the boundary conditions The normalization requirement leads to boundary conditions at infinity and they lead to energy quantization. 02/18/2005 ) ( ) ( ’ ’ 2 ) ( 2 x x V E m Φ-= Φ-) ( lim = Φ ±∞ → x x ) ( x V ) ( x V ) ( x Φ ) ( 2 x Φ [email protected] 47 2 E E = E E = Energy of a stationary wave function 02/18/2005 The “curvature” of the wave function is less for the finite potential well and therefore the energy levels E n of stationary states are lower. ) ( ) ( / ) ( ’ ’ V E x x--∝ Φ Φ ) ( x V ) ( x V ) ( x Φ ) ( 2 x Φ...
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