Calculation_of_x_for_Stationary

# Calculation_of_x_for_Stationary - Calculation of x for...

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Unformatted text preview: Calculation of x for Stationary and Nonstationary States of the 1D Harmonic Oscillator A. The Stationary State ( x,t ) The wavefunction for the ground ( v = 0 ) stationary state is i ( x, t ) = exp E v=0 t v=0 ( x ) . - h Note ( x, t = 0 ) = v=0 ( x ) = real function and thus x Next h i E x t = * ( x, t ) x ( x, t ) dx = E v=0 t v=0 ( x ) x v=0 ( x ) exp i v=0 t exp + - . - - h h t =0 h - = x = x 2=0 ( x ) dx . v i i + exp - = But exp E v=0 t E v=0 t 1 . h h So x t= Finally note x t = 0 since x t =0 h - 2 x 0 ( x ) dx = x t =0 . = 0. B. The Nonstationary State ( x,t ) We define the nonstationary state ( x, t ) by the wavefunction ( x, t ) = 1 i i - + - 0 ( x ) exp h E v=0 t 1 ( x ) exp h E v=1t . 2 Note P ( x, t ) for this state is P ( x, t ) = * ( x, t ) ( x, t ) = 1 i i exp E v=0 t v=0 ( x ) + exp E v=1t v=1 ( x ) h h 2 1 i i exp - - h E v=0 t v=0 ( x ) + exp h E v=1t v=1 ( x ) 2 or P ( x, t ) = 1 2 i i exp + - { v=0 ( x ) + 2v=1 ( x ) + ( E v=1 - E v=0 ) t exp h( E v=1 - E v=0 ) t 2 h v=0 ( x ) v=1 ( x ) } . 1 v h To simplify note E v=1 - E v=0 = h since E v = + . 2 Also since exp ( ix ) + exp ( -ix ) = 2cos x , P ( x, t ) becomes . Check normalization 1 1 P ( x, t ) dx = 2 ( x ) dx + 2 ( x ) dx + cos t ( x ) ( x ) dx h -TM - 2 v =0 -* 2 v =1 - 0 1 = 1 1 ( 1) + ( 1) + cos t ( 0 ) = 1 . 2 2 h - Therefore P ( x, t ) dx = 1 is normalized. t Finally, compute x h - from 1 1 2 2 x v=0 ( x ) + 2 x v=1 ( x ) dx + cos t x v=0 ( x ) v=1 ( x ) dx , -* - 2 - h - x t = ( x, t ) dx = xP or x t = cos t x v=0 ( x ) v=1 ( x ) dx . Summary Stationary i ( x, t ) = exp E v=0 t v=0 ( x ) - h P ( x, t ) = 2=0 ( x ) = P ( x, t = 0 ) v x = h - Nonstationary 1 i 1 ( x, t ) = - + - 0 ( x ) exp h E v=0 t 1 ( x ) exp h E v=1t 2 1 P ( x, t ) = 2=0 ( x ) + 2=1 ( x ) + 2cos t 0 ( x ) 1 ( x ) v v 2 x t = cos t h - x 2=0 ( x ) dx = x v v=0 =0 x v=0 ( x ) v=1 ( x ) dx . ...
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