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# Lect15 - Lecture 15 Time-Dependent QM Tunneling Review and...

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Lecture 15, p.1 Lecture 15: Time-Dependent QM & Tunneling Review and Examples, Ammonia Maser 0 L U 0 x U(x) E x | ψ (x,t 0 )| 2 U= U= 0 x L | ψ (x,t=0)| 2 U= U= 0 x L

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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 ( ψ = a ψ 1 +b ψ 2 ) of energies E 1 and E 2 , then we aren’t 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. That’s 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 π π Ψ = = + | Ψ (x,t)| 2 U= U= 0 x L | Ψ (x,0)| 2 U= U= 0 x L

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Lecture 15, p.4 Solution Determine the time it takes for the particle to move to the right side of the well. | Ψ (x,t)| 2 U= U= 0 x L | Ψ (x,0)| 2 U= U= 0 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 2 -E 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 π π Ψ = = +
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 π π Ψ = = + 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 2 -E 1 )/h b ) They vary in an unpredictable manner. c ) They alternate between E 1 and E 2 . ( i.e. , it’s always either E 1 or E 2 ). d ) They don’t vary with time. | Ψ (x,t)| 2 U= U= 0 x L | Ψ (x,0)| 2 U= U= 0 x L

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Lecture 15, p.6 Solution | Ψ (x,t)| 2 U= U= 0 x L | Ψ (x,0)| 2 U= U= 0 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 π π Ψ = = + 1 ) Suppose we measure the energy.
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Lect15 - Lecture 15 Time-Dependent QM Tunneling Review and...

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