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Unformatted text preview: 1a One way to see this is to use the uncertainty principle. If the kinetic energy of the particle is zero, then Δ p = 0, and Δ x would have to be ∞ . But we know the particle is confined in a region of size L . So Δ p > 0, and KE ≥ h p 2 i / 2 m . 1b Bohrs correspondence principle says that the classical physics is recovered as the large n limit of the quantum mechanical problem. For very large n , the wave function has many wiggles (because large energy means that the second derivative is large) and the probability between one spatial point and another becomes indistinguishable. See the below figure. 1 2a P = Ax (1- x ) A is found by normalizing the wavefunction. 1 = Z 1 dxAx (1- x ) 1 = A x 2 2- x 3 3 1 A = 6 The expectation value of x is given by h x i = Z ∞-∞ dx Ψ( x,t ) * x Ψ( x,t ) h x i = 6 Z 1 dxx 2 (1- x ) h x i = 6 x 3 3- x 4 4 1 h x i = 1 2 The uncertainty in x is given by Δ x = p h x 2 i - h x i 2 ....
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- Spring '10
- Uncertainty Principle, quantum mechanical problem, dx x2, KE p2 /2m, Bohrs correspondence principle