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Heisenburg’s Uncertainty Principle

Heisenburg’s Uncertainty Principle. Schrödenger’s Equation for a free electron -( h 2 /2m)( ¶ 2 y (x,t) /¶ x 2 ) = i h ( ¶ y (x,t) /¶ t). Y (x,t) = y (x)e- i E t/ h -( h 2 /2m)(d 2 y (x) / d x 2 ) = E y (x) k 2 =2mE/ h 2 (d 2 y (x) / d x 2 ) + k 2 y (x) = 0

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Heisenburg’s Uncertainty Principle

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  1. Heisenburg’s Uncertainty Principle Schrödenger’s Equation for a free electron -(h2/2m)(¶2 y(x,t)/¶x2) = ih(¶ y(x,t)/¶t)

  2. Y(x,t) = y(x)e-iEt/h • -(h2/2m)(d2 y(x)/dx2) = Ey(x) • k2=2mE/h2 • (d2 y(x)/dx2) + k2y(x) = 0 • Looks like classical harmonic oscillator equation • (d2y(t)/dt2) + w2y(t) = 0 • Y(x,t)= y(x)e-iwt = C ei(kx-wt) = C ei(px-Et)/h • Solution for wave propagating to in + x direction • The above solution is a plane wave

  3. We can find the constant C by using the probability equation discussed last lecture and forcing the solution to equal one: ¥ ¥ • êY(x,t)ê2dx = êCê2 ó ei(px-Et)/hdx = 1 -¥ -¥

  4. Begin with an infinite Fourier integral solution for the wave function: ¥ Y(x,t)= óA(p)ei(px-Et)/hdp -¥ 2phêA(p)ê2dp is the probability that the momentum will be found in a range dp at the value p

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