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05_02_23 - well has states with energies below V V k k 3 V...

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[email protected] 53 02/23/2005 + + = + = + = ± = = = + + = - + = = - - - - ) tan( 0 | ) sin( | ) cos( ) sin( ) ( ) cos( ) sin( : 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 2 2 2 2 ϕ ϕ ϕ α ϕ α ϕ α ϕ α α α α kL kL k k L k k L k k k k V E L B A L L L L V V V V L L k k k e B k k e A k kB Ae k kB kAe z # Stationary states in the square well 0 2 2 0 2 2 2 2 , ) ( V k E k E V m V m m = = - = α ) sin( )] [cos( sign 2 2 ϕ ϕ + + - = L k k k L k k k k k V V V V V x 0 V 2 L - 2 L 0 E Φ
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[email protected] 54 02/23/2005 2 2 2 or 0 with ) tan( 0 and | ) sin( | π ϕ ϕ ϕ ϕ = = + + = kL kL k k V Graphical determination of possible wave numbers V k k 2 kL 2 kL V k k 2 kL | ) sin( | 2 kL k k V = ) tan( 0 2 kL
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[email protected] 55 02/23/2005 Square well versus Infinite well Number of solutions = There is at least one bound state, and it is symmetric. 1 ] int[ 1 ] int[ 0 2 2 + = + V m L L k V π π Stationary states for the infinite well: 0 ] , mod[ , 2 2 = = π π kL L n k V k k 2 kL 2 π 2 2 π 2 3 π 2 4 π 2 5 π 2 6 π 0 0 V E = The finite potential well with height
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Unformatted text preview: well has states with energies below V . V k k 3 V k k 2 V k k 1 V k k Stationary states for the finite well: ]) , cos(mod[ 2 2 kL k k V = 2 kL 2 2 2 2 3 2 4 2 5 2 6 1 [email protected] 56 02/23/2005 Graphic determination of states for the square well 2 π L Φ 1 Φ 2 Φ 3 Φ 2 2 2 k E m = L 2 L 3 Infinite well: L n k = Finite well: ]) , cos(mod[ 2 2 kL V k k = L 4 ) cos( ϑ V k...
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