Chapter2_17 - i.e. orthonormal like the discrete solutions...

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PHY4604 R. D. Field Department of Physics Chapter2_17.doc University of Florida The Free Particle A stationary state free particle ( i.e. V(x) = 0) with energy E must satisfy ) ( ) ( 2 2 2 2 x E dx x d m ψ = h and hence ikx Ae x ± = ) ( where A is a constant and E m k = ) 2 /( 2 2 h . But ψ (x) is also an eigenstate of (p x ) op as follows: > ± >= | | ) ( k p op x h . Thus, ) ) ( ( ) , ( t k kx i k Ae t x ω = Ψ corresponds free particle with definite momentum k p x h = (all values of k allowed) and definite energy ) 2 /( ) ( 2 2 m k k E h h = = (k > 0 corresponds to the wave moving to the right and k < 0 corresponds to the wave moving to the left). However, these solutions are not normalizable and hence not allowed. A free particle cannot exist in a stationary state ; or put another way, there is no such thing as a free particle with definite energy . The overlap of one solution with another is given by ) ' ( ) ' ( | | 2 | | | 2 ) ' ( )) ' ( ) ( ( 2 ' k k k k A dx e e A x k k i t k k i k k = = >= Ψ Ψ < +∞ δ π , with 2 / 1 = A . If k = k' we get infinite and if k k' we get zero (
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Unformatted text preview: i.e. orthonormal like the discrete solutions &lt; n | m &gt;= nm ). Most General Solution: The most general solution is a superposition of the free particle eigenfunctions as follows + + = = dk e k f dk t x k f t x t k kx i k ) ) ( ( ) ( 2 1 ) , ( ) ( ) , ( , where ) 2 /( ) ( 2 m k k h = . The dependence of on k is called the dispersion . The coefficient function f(k) (analogous to the overlap constants c n =&lt; n | &gt;) is given by + = dx e x k f ikx ) , ( 2 1 ) ( . Fourier Transforms: f(k) is the Fourier transform of F(x) and F(x) is the inverse Fourier transform of f(k) as follows: + = dk e k f x F ikx ) ( 2 1 ) ( + = dx e x F k f ikx ) ( 2 1 ) (...
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.

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