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Unformatted text preview: Lecture 15, p.1 Lecture 15: TimeDependent QM & Tunneling Review and Examples, Ammonia Maser 0 L U x U(x) E x  (x,t ) 2 U= U= x L  (x,t=0) 2 U= U= x L Lecture 13, p 2 Measurements of Energy What happens when we measure the energy of a particle whose wave function is a superposition of more than one energy state? If the wave function is in an energy eigenstate (E 1 , say), then we know with certainty that we will obtain E 1 (unless the apparatus is broken) . If the wave function is a superposition ( = a 1 +b 2 ) of energies E 1 and E 2 , then we arent certain what the result will be. However: We know with certainty that we will only obtain E 1 or E 2 !! To be specific, we will never obtain (E 1 +E 2 )/2 , or any other value. What about a and b? a 2 and b 2 are the probabilities of obtaining E 1 and E 2 , respectively. Thats why we normalize the wave function to make a 2 + b 2 =1 . Lecture 15, p.3 Example An electron in an infinite square well of width L = 0.5 nm is (at t=0) described by the following wave function: Determine the time it takes for the particle to move to the right side of the well. 2 2 ( , 0) sin sin x t A x x L L L = = +  (x,t) 2 U= U= x L  (x,0) 2 U= U= x L Lecture 15, p.4 Solution Determine the time it takes for the particle to move to the right side of the well.  (x,t) 2 U= U= x L  (x,0) 2 U= U= x L ( 29 ( 29 15 16 2 1 4.136 10 eV sec 4.6 10 sec 2 2 2 4.515 T h t E E eV = = = =  T = 1/f , where f = (E 2E 1 )/h 2 1 2 2 1 1.505 eV nm 1.505 eV 4 E 4 6.020 eV E L E = = = = Half a period. An electron in an infinite square well of width L = 0.5 nm is (at t=0) described by the following wave function: 2 2 ( , 0) sin sin x t A x x L L L = = + Lecture 15, p.5 ACT 1 An electron in an infinite square well of width L = 0.5 nm is (at t=0) described by the following wave function: 2 2 ( , 0) sin sin x t A x x L L L = = + 1 ) Suppose we measure the energy. What results might we obtain? a ) E 1 b ) E 2 c ) E 3 d ) Any result between E 1 and E 2 2 ) How do the probabilities of the various results depend on time? a ) They oscillate with f = (E 2E 1 )/h b ) They vary in an unpredictable manner. c ) They alternate between E 1 and E 2 . ( i.e. , its always either E 1 or E 2 ). d ) They dont vary with time.  (x,t) 2 U= U= x L  (x,0) 2 U= U= x L Lecture 15, p.6 Solution  (x,t) 2 U= U= x L  (x,0) 2 U= U= x L An electron in an infinite square well of width L = 0.5 nm is (at t=0) described by the following wave function: 2 2 ( , 0) sin sin x t A x x L L L = = + 1...
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This note was uploaded on 04/04/2011 for the course PHYSICS 214 taught by Professor Mestre during the Spring '11 term at University of Illinois at Urbana–Champaign.
 Spring '11
 MESTRE
 Magnetism, Energy

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