HW1_solutions - Homework 1: Solutions 1. Consider how the...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
H o m e w o r k 1 : S o l u t i o n s 1 . C o n s i d e r h o w t h e d e B r o g l i e s s u g g e s t i o n m i g h t e x p l a i n s o m e p r o p e r t i e s o f t h e h y d r o g e n a t o m . a . S h o w t h a t t h e a s s u m p t i o n p = m υ = h λ a n d t h e q u a n t i z a t i o n c o n d i c t i o n t h a t t h e l e n g t h o f a c i r c u l a r o r b i t b e a n i n t e g e r m u l t i p l e o f t h e l e n g t h o f t h e e l e c t r o n w a v e l e n t g h ( t h a t i s : n λ = 2 π r , w h e r e r i s t h e r a d u i s o f t h e o r b i t a n d n a n i n t e g e r ) i m p l y t h a t o n l y d i s c r e t e o r b i t s a r e a l l o w e d . b . C a l c u l a t e t h e t o t a l e n e r g y ( k i n e t i c p l u s p o t e n t i a l ) o f t h e e l e c t r o n i n e a c h o r b i t c h a r a c t e r i z e d b y n . H i n t : I n p a r t a F n d t w o e q u a t i o n s d e s c r e i b i n g t h e b a l a n c e b e t w e e n t h e c e n t r i f u g a l a n d t h e C o u l o m b ( c e n t r i p e t a l ) f o r c e . S o l v e f o r t h e r a d i u s r a n d f o r t h e a n g u l a r v e l o c i t y ω . N o w , i n b i n s e r t t h e s e e x p r e s s i o n s i n t o t h e f o r m u l a e f o r t h e k i n e t i c a n d p o t e n t i a l e n e r g y . S o l u t i o n : S e e E C E 6 1 8 N o t e s , P a r t 1 . 2 . P r o b l e m . C o n s i d e r t h e t u n n e l i n g p r o b l e m w i t h t h e p o t e n t i a l b a r r i e r : V ( z ) = 0 f o r z 0 V > 0 f o r 0 < z < L 0 f o r z L . W r i t e t h e w a v e f u n c t i o n a s : ψ ( z ) = A e i k z + B e i k z f o r z 0 C e κ z + D e κ z f o r 0 < z < L F e i k z f o r z L , w i t h k = ( 2 m E ) 1 / 2 / ¯ h , κ = [ 2 m ( V E ) ] 1 / 2 / ¯ h . a . W r i t e t h e s y s t e m o f f o u r e q u a t i o n s e x p r e s s i n g t h e c o n t i n u i t y o f t h e w a v e f u n c t i o n a n d i t s d e r i v a t i v e a t z = 0 a n d a t z = L . b . ± i n d t h e t r a n s m i s s i o n c o e c i e n t , T = | F | 2 / | A | 2 . T h e r e s n o n e e d t o s o l v e t h e f u l l s y s t e m . B e c r e a t i v e . E C E 6 0 9 S p r i n g 2 0 1 0 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
H i n t : M u l t i p l y t h e e q u a t i o n e x p r e s s i n g c o n t i n u i t y o f ψ a t z = 0 b y i k a n d a d d a n d s u b s t r a c t i t f r o m t h e e q u a t i o n e x p r e s s i n g c o n t i n u i t y o f t h e d e r i v a t i v e s a t z = 0 . D o a s i m i l a r t h i n g w i t h t h e o t h e r t w o e q u a t i o n s ( b y m u l t i p l y i n g o n e b y κ ) . N o w i t s h o u l d b e r e l a t i v e l y e a s y t o s o l v e f o r F i n t e r m s o f A a l o n e . T h i s g i v e s y o u T . S
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 02/12/2010.

Page1 / 20

HW1_solutions - Homework 1: Solutions 1. Consider how the...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online