Lecture 4 - Finite Quantum Well Systems with two types of...

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Finite Quantum Well Systems with two types of spectrum
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Finite potential well: Two energy regions x 0 V -a/2 a/2 V E < 0 E > 0 For E < 0 classical motion is bounded, in quantum mechanics states are normalizable and energy spectrum is discrete For E > 0 classical motion is unbounded, in quantum mechanics states are not normalizable, and energy spectrum is continues.
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Finite potential well: three space regions 0 V -a/2 a/2 V I II III In the case of a piece-wise constant potential one needs to consider SE separately in each interval of continuity . /2 Ix a II a x a III x a < − −< < >
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Discrete spectrum. I /2 xa < − 22 2 2 11 1 1 2 ,0 ( 0 ) 2 ( 0 to ensure normalizability) xx x dd m E EE mdx dx Ae Be A κκ κ ψψ ψκ ψ −= = = > < =+ = = = = 0 V -a/2 a/2 V I II III E
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Discrete spectrum. III /2 xa > 22 2 2 33 3 3 2 ,0 ( 0 ) 2 ( 0 to ensure normalizability) xx x dd m E EE mdx dx Ae Be B κκ κ ψψ ψκ ψ −− −= = = > < =+ = = = = 0 V -a/2 a/2 V I II III E
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Discrete spectrum. II 0 V -a/2 a/2 V I II III () 22 2 2 0 2 0 2 , 2 2 0( 0) cos sin dd VE k mdx dx mE V
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Lecture 4 - Finite Quantum Well Systems with two types of...

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