L16 - Lecture 16: The Schrodinger Equation in 1D 01 April,...

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PHYS126, L16: page 1 of 9 Lecture 16: The Schrodinger Equation in 1D 01 April, 2011 1. The Schrodinger equation in 1D: ) , ( ) ( 2 ˆ ) , ( 2 t x x U m p t x t i x Ψ + = Ψ h (16-1) Recall: x i p x = h ˆ , we have: ) , ( ) ( 2 ) , ( 2 2 2 t x x U x m t x t i Ψ + = Ψ h h (16-2) with initial condition ) ( ) 0 , ( 0 x t x Ψ = = Ψ Requirement of continuity: The wavefunction ) , ( t x Ψ must be a continuous and single valued function in time and space. ) , ( t x x Ψ must be also continuous and single valued within the defined time and space. It can be shown that, a specific solution to equation (16-2) can take the form: h / ) ( ) , ( iEt E e x t x = Ψ ψ (16-3) Then equation (16-2) becomes h h h h / 2 2 2 / ) ( ) ( 2 ) ( iEt E iEt E e x x U x m e x t i + = h h h / 2 2 2 / ) ( ) ( 2 ) ( iEt E iEt E e x x U x m e x E + = ) ( ) ( 2 ) ( 2 2 2 x x U dx d m x E E E + = h (16-4) Equation (16-4) is called time-independent Schrodinger equation. E is the energy eigen value. We can show that, if both h / 1 1 1 ) ( ) , ( t iE E e x t x = Ψ and h / 2 2 2 ) ( ) , ( t iE E e x t x = Ψ are solutions to the time-dependant Schrodinger equation (16-2), ) , ( ) , ( 2 1 t x t x Ψ + Ψ is also a solution to the equation (16-2). Proof: ) , ( ) ( 2 ) , ( 1 2 2 2 1 t x x U x m t x t i Ψ + = Ψ h h Classical Mechanics: x dt d m ma F 2 2 = = with initial condition: 0 ) 0 ( x t x = = 0 ) 0 ( v t v = =
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PHYS126, L16: page 2 of 9 ) , ( ) ( 2 ) , ( 2 2 2 2 2 t x x U x m t x t i Ψ + = Ψ h h Adding the above two equations together, we have: [] ) , ( ) , ( ) ( 2 ) , ( ) , ( 2 1 2 2 2 2 1 t x t x x U x m t x t x t i Ψ + Ψ + = Ψ + Ψ h h Î ) , ( ) , ( 2 1 t x t x Ψ + Ψ is also a solution. In general, the solution to the Schrodinger equation can be expressed as: = Ψ E iEt E E e x c t x h / ) ( ) , ( ψ (16-5) There are two cases depending on whether the energy E is quantized or continuous [Powerpoint slides] 1.1 Discrete energy spectrum ----- system with boundary When a particle is confined within a finite volume, i.e., 0 ) ( ±∞ x , or there is a boundary, the energy spectrum is quantized due to the boundary condition. (1) Solve time-independent Schrodinger equation and find all possible solutions of stationary states ) ( x E : ) ( ) ( 2 ) ( 2 2 2 x x U x m x E E E + = h ) ( x E satisfy: Boundary condiction, Continuity, and normalization-orthogonal condition.
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L16 - Lecture 16: The Schrodinger Equation in 1D 01 April,...

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