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Lecture7 - 5.61 Fall 2007 Lecture#7 page 1 SOLUTIONS TO THE SCHRÖDINGER EQUATION Free particle and the particle in a box Schrödinger equation is

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Unformatted text preview: 5.61 Fall 2007 Lecture #7 page 1 SOLUTIONS TO THE SCHRÖDINGER EQUATION Free particle and the particle in a box Schrödinger equation is a 2 nd-order diff. eq. x − + V x x x ! 2 ∂ 2 ψ ( ) ( ) ψ ( ) = E ψ ( ) 2 m ∂ x 2 We can find two independent solutions φ 1 x x ( ) and φ 2 ( ) The general solution is a linear combination ψ x A φ 1 ( ) + B φ 2 ( ) x ( ) = x A and B are then determined by boundary conditions on ψ x ψ ′ x ( ) and ( ) . Additionally, for physically reasonable solutions we require that ψ x ( ) and ψ ′ x be continuous function. ( ) (I) Free particle V ( x ) = 0 ! 2 ∂ 2 ψ x ( ) − = E ψ x ( ) 2 m ∂ x 2 2 mE ! 2 k 2 or E = Define k 2 = ! 2 2 m 2 V x E = p ⇒ p 2 = ! 2 k 2 ⇒ ( ) = 0, 2 m h 2 π de Broglie p = ⇒ k = λ λ p = ! k 5.61 Fall 2007 Lecture #7 page 2 ∂ 2 ψ x ( ) = − k 2 ψ x The wave eq. becomes ( ) ∂ x 2 with solutions ψ ( ) = ( ) + B sin kx x A cos kx ( ) Free particle ⇒ no boundary conditions !...
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This note was uploaded on 10/01/2011 for the course MATH 1090 taught by Professor Greenwood during the Fall '08 term at MIT.

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Lecture7 - 5.61 Fall 2007 Lecture#7 page 1 SOLUTIONS TO THE SCHRÖDINGER EQUATION Free particle and the particle in a box Schrödinger equation is

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